Talk presentations
Below you can see a list of all uploaded presentations to the 8th Congress of Mathematics. Use the search field to narrow down the display.
Plenary speakers
Stable solutions to semilinear elliptic equations are smooth up to dimension 9
Prof. Xavier Cabré, ICREA and Universitat Politecnica de Catalunya
The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist stable energy solutions which are singular. In this talk I will describe a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems.
Minimal surfaces from a complex analytic viewpoint
Prof. Franc Forstnerič, University of Ljubljana
In this talk, I will describe some recent developments in the theory of minimal surfaces in Euclidean spaces which have been obtained by complex analytic methods. After a brief history and background of the subject, I will present a new Schwarz-Pick lemma for minimal surfaces, describe approximation and interpolation results for minimal surfaces, and discuss the current status of the Calabi-Yau problem on the existence of complete conformal minimal surfaces with Jordan boundaries.
Statistical Learning: Causal-oriented and Robust
Prof. Peter Bühlmann, ETH Zurich
Reliable, robust and interpretable machine learning is a big emerging theme in data science and artificial intelligence, complementing the development of pure black box prediction algorithms. Looking through the lens of statistical causality and exploiting a probabilistic invariance property opens up new paths and opportunities for enhanced robustness, with wide-ranging prospects for various applications.
Geometric Valuation Theory
Prof. Monika Ludwig, TU Wien
Valuations on compact convex sets in $\mathbb{R}^n$ play an active and prominent role in geometry. They were critical in Dehn's solution to Hilbert's Third Problem in 1901. They are defined as follows. A function $\operatorname{\rm Z}$ whose domain is a collection of sets $\mathcal{S}$ and whose co-domain is an Abelian semigroup is called a {\em valuation} if $$ \operatorname{\rm Z}(K) + \operatorname{\rm Z}(L) = \operatorname{\rm Z}(K\cup L) + \operatorname{\rm Z}(K\cap L),$$ whenever $K, L, K\cup L, K \cap L \in \mathcal{S}$. The first classification result for valuations on the space of compact convex sets, $\mathcal{K}^n$, in $\mathbb{R}^n$ (where $\mathcal{K}^n$ is equipped with the topology induced by the Hausdorff metric) was established by Blaschke. \begin{theorem}[Blaschke]\label{wb} A functional $\,\operatorname{\rm Z}:\mathcal{K}^n\to \mathbb{R}$ is a continuous, translation and SL$(n)$ invariant valuation if and only if there are $c_0, c_n\in\mathbb{R}$ such that $$\operatorname{\rm Z}(K) = c_0 V_0(K)+c_n V_n(K)$$ for every $K\in\mathcal{K}^n$. \end{theorem} Probably the most famous result in the geometric theory of valuations is the Hadwiger characterization theorem. \begin{theorem}[Hadwiger]\label{hugo} A functional $\,\operatorname{\rm Z}:\mathcal{K}^n\to \mathbb{R}$ is a continuous and rigid motion invariant valuation if and only if there are $c_0,\ldots, c_n\in\mathbb{R}$ such that $$\operatorname{\rm Z}(K) = c_0 V_0(K)+\dots+c_n V_n(K)$$ for every $K\in\mathcal{K}^n$. \end{theorem} Here $V_0(K),\ldots,V_n(K)$ are the intrinsic volumes of $K\in\mathcal{K}^n$. In particular, $V_0(K)$ is the Euler characteristic of $K$, while $2\, V_{n-1}(K)$ is the surface are of $K$ and $V_n(K)$ the $n$-dimensional volume of $K$. Hadwiger's theorem shows that the intrinsic volumes are the most basic functionals in Euclidean geometry. It finds powerful applications in Integral Geometry and Geometric Probability. The fundamental results of Blaschke and Hadwiger have been the starting point of the development of Geometric Valuation Theory. Classification results for valuations invariant (or covariant) with respect to important groups are central questions. The talk will give an overview of such results including recent extensions to valuations on function spaces. \begin{thebibliography}{10} \bibitem{BoeroeczkyLudwig} K.J. B\"or\"oczky, M. Ludwig, {\em Minkowski valuations on lattice polytopes}. J. Eur. Math. Soc. (JEMS) {\bf 21} (2019), 163–197. \bibitem{Colesanti-Ludwig-Mussnig-2} A. Colesanti, M. Ludwig, F. Mussnig, {\em Minkowski valuations on convex functions}, Calc. Var. Partial Differential Equations {\bf 56} (2017), 56:162. \bibitem{Colesanti-Ludwig-Mussnig-5} A.~Colesanti, M.~Ludwig, and F.~Mussnig, {\em The {H}adwiger theorem on convex functions.~{I}}, arXiv:2009.03702 (2020). \bibitem{Haberl:Parapatits_centro} C.~Haberl and L.~Parapatits, {\em The centro-affine {H}adwiger theorem}, J. Amer. Math. Soc. {\bf 27} (2014), 685--705. \bibitem{Hadwiger:V} H.~Hadwiger, {\em {V}or\-lesungen \"uber {I}nhalt, {O}ber\-fl\"ache und {I}so\-peri\-metrie}, Springer, Berlin, 1957. \bibitem{Klain:Rota} D.~A. Klain and G.-C. Rota, {\em Introduction to geometric probability}, Cambridge University Press, Cambridge, 1997. \bibitem{Ludwig:projection} M.~Ludwig, {\em Projection bodies and valuations}, Adv. Math. {\bf 172} (2002), 158--168. \bibitem{Ludwig:sobval} M.~Ludwig, {\em Valuations on {S}obolev spaces}, Amer. J. Math. {\bf 134} (2012), 827--842. \bibitem{Ludwig:Reitzner} M.~Ludwig and M.~Reitzner, {\em A characterization of affine surface area}, Adv. Math. {\bf 147} (1999), 138--172. \bibitem{Ludwig:Reitzner2} M.~Ludwig and M.~Reitzner, {\em A classification of SL$(n)$ invariant valuations}, Ann. of Math. (2) {\bf 172} (2010), 1219--1267. \end{thebibliography}
The Mathematics of Deep Learning
Prof. Gitta Kutyniok, Ludwig Maximilian University of Munich
Despite the outstanding success of deep neural networks in real-world applications, ranging from science to public life, most of the related research is empirically driven and a comprehensive mathematical foundation is still missing. At the same time, these methods have already shown their impressive potential in mathematical research areas such as imaging sciences, inverse problems, or numerical analysis of partial differential equations, sometimes by far outperforming classical mathematical approaches for particular problem classes. The goal of this lecture is to first provide an introduction into this new vibrant research area. We will then survey recent advances in two directions, namely the development of a mathematical foundation of deep learning and the introduction of novel deep learning-based approaches to solve inverse problems and partial differential equations.
Invited speakers
Subset products and derangements
Prof. Aner Shalev, Hebrew University of Jerusalem
In the past two decades there has been intense interest in products of subsets in finite groups. Two important examples are Gowers' theory of Quasi Random Groups and its applications by Nikolov, Pyber, Babai and others, and the theory of Approximate Groups and the Product Theorem of Breuillard-Green-Tao and Pyber-Szabo on growth in finite simple groups of Lie type of bounded rank, extending Helfgott's work. These deep theories yield strong results on products of three subsets (covering, growth). What can be said about products of two subsets? I will discuss a recent joint work with Michael Larsen and Pham Tiep on this challenging problem, focusing on products of two normal subsets of finite simple groups, and deriving a number of applications. Our main application concerns derangements (namely, fixed-point-free permutations), studied since the days of Jordan. We show that every element of a sufficiently large finite simple transitive permutation group is a product of two derangements. Related results and problems will also be discussed. The proofs combine group theory, algebraic geometry and representation theory; it applies the proof by Fulman and Guralnick of the Boston-Shalev conjecture on the proportion of derangements.
Regularization methods in inverse problems and machine learning
Prof. Martin Burger, FAU Erlangen-Nürnberg
Regularization methods are at the heart of the solution of inverse problems and are of increasing importance in modern machine learning. In this talk we will discuss the modern theory of (nonlinear) regularization methods and some applications. We will put a particular focus on variational and iterative regularization methods and their connection with learning problems: we discuss the use of such regularization methods for learning problems on the one hand, but also the current route of learning regularization methods from data.
Topology of symplectic fillings of contact $3$-manifolds
Prof. Burak Özbağcı, Koç University
Ever since Donaldson showed that every symplectic $4$-manifold admits a Lefschetz pencil and Giroux proved that every contact $3$-manifold admits an adapted open book decomposition, at the turn of the century, Lefschetz fibrations and open books have been used fruitfully to obtain interesting results about the topology of symplectic fillings of contact $3$-manifolds. In this talk, I will present my contribution to the subject at hand based on joint work with several coauthors during the past 20 years.
AAA-least squares rational approximation and solution of Laplace problems
Prof. Nick Trefethen, Oxford University
In the past five years, computation with rational functions has advanced greatly with the introduction of the AAA algorithm for barycentric rational approximation and of lightning least-squares solvers for Laplace, Stokes, and Helmholtz problems. Here we combine these methods into a two-step method for solving planar Laplace problems. First, complex rational approximations to the boundary data are determined by AAA approximation, locally near each corner or other singularity. The poles of these approximations outside the problem domain are then collected and used for a global least-squares fit to the solution. Typical problems are solved in a second of laptop time to 8-digit accuracy, all the way up to the corners. This is joint work with Stefano Costa.
Positive harmonic functions on the Heisenberg group
Mr. Yves Benoist, Université Paris-Sud
A harmonic function on a group is a function which is equal to the average of its translates, average with respect to a finitely supported measure. First, we will survey the history of this notion. Then we will describe the extremal non-negative harmonic functions on the Heisenberg group. We will see that the classical partition function occurs as such a function and that this function is the only one beyond harmonic characters.
Bayesian inverse problems, Gaussian processes, and PDEs
Prof. Richard Nickl, University of Cambridge
The Bayesian approach to inverse problems has become very popular in the last decade after seminal work by A. Stuart (2010). Particularly in nonlinear applications with PDEs and when using Gaussian process priors, this can leverage powerful MCMC algorithms to tackle difficult high dimensional and nonconvex inference problems, with associated \textit{uncertainty quantification methodology}. We review the main ideas and then discuss recent progress on rigorous mathematical performance guarantees for such algorithms. We will touch upon issues such as how to prove posterior consistency theorems, how to objectively validate posterior based statistical uncertainty quantification, as well as the polynomial time computability of posterior measures in some nonconvex model examples arising with PDEs.
Looking at Euler flows through a contact mirror: Universality and Turing completeness
Prof. Eva Miranda, Universitat Politecnica de Catalunya
The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao \cite{T1, T2, tao} launched a programme to address the global existence problem for the Euler and Navier-Stokes equations based on the concept of universality. Inspired by this proposal, we show that the stationary Euler equations exhibit several universality features, in the sense that, any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension \cite{cmpparxiv}. A key point in the proof is looking at the h-principle in contact geometry through a contact mirror, unveiled by Etnyre and Ghrist in \cite{EG} more than two decades ago. We end up this talk addressing a question raised by Moore in \cite{Mo} : \emph{\lq\lq Is hydrodynamics capable of performing computations?"}. The universality result above yields the Turing completeness of the steady Euler flows on a 17-dimensional sphere. Can this result be improved? In \cite{cmppPNAS} we construct a Turing complete Euler flow in dimension 3. Time permitting, we discuss this and other generalizations contained in \cite{cmparxiv}. This talk is based on several joint works with Cardona, Peralta-Salas and Presas. \begin{thebibliography}{99} \bibitem{cmpparxiv} R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. \emph{Universality of Euler flows and flexibility of Reeb embeddings}, arXiv:1911.01963. \bibitem{cmppPNAS} R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. \emph{Constructing Turing complete Euler flows in dimension $3$}. PNAS May 11, 2021 118 (19) e2026818118; https://doi.org/10.1073/pnas.2026818118. \bibitem{cmparxiv} R. Cardona, E. Miranda and D. Peralta-Salas, \emph{Turing universality of the incompressible Euler equations and a conjecture of Moore}, arXiv:2104.04356 . \bibitem{EG} J. Etnyre, R. Ghrist. \emph{Contact topology and hydrodynamics I. Beltrami fields and the Seifert conjecture}. Nonlinearity 13 (2000) 441--458. \bibitem{Mo} C. Moore. \emph{Generalized shifts: unpredictability and undecidability in dynamical systems}. Nonlinearity 4 (1991) 199--230. \bibitem{T1} T. Tao. \emph{On the universality of potential well dynamics}. Dyn. PDE 14 (2017) 219--238. \bibitem{T2} T. Tao. \emph{On the universality of the incompressible Euler equation on compact manifolds}. Discrete Cont. Dyn. Sys. A 38 (2018) 1553--1565. \bibitem{tao} T. Tao. \emph{Searching for singularities in the Navier-Stokes equations}. Nature Rev. Phys. 1 (2019) 418--419. \end{thebibliography}
Recent Results on Lieb-Thirring Inequalities
Prof. Rupert Frank, Caltech / LMU Munich
Lieb-Thirring inequalities are a mathematical expression of the uncertainty and exclusion principles in quantum mechanics. Since their discovery in 1975 they have played an important role in several areas of analysis and mathematical physics. We provide a gentle introduction to classical aspects of this subject and present some recent progress. Finally, we discuss extensions, in the spirit of Lieb-Thirring inequalities, of several inequalities in harmonic analysis to the setting of families of orthonormal functions.
Nonstandard finite elements for wave problems
Prof. Ilaria Perugia, University of Vienna
Finite elements are a powerful, flexible, and robust class of methods for the numerical approximation of solutions to partial differential equations. In their standard version, they are based on piecewise polynomial functions on a partition of the domain of interest. Continuity requirements are possibly dictated by the regularity of the exact solutions. By breaking these constraints, new methods that are specifically tailored to the problem at hand have been developed in order to better reproduce physical properties of the exact solutions, to enhance stability, and to improve accuracy vs. computational cost. Nonstandard finite element approximations of wave propagation problems based on the space-time paradigm will be the focus of this talk.
Finite groups of birational transformations
Prof. Yuri Prokhorov, Steklov Mathematical Institute
I survey the classification theory of finite groups of birational transformations of higher-dimensional algebraic varieties. This theory has been significantly developed during the last 10 years due to the success of the minimal model program. I concentrate on certain properties of these groups in arbitrary dimension. Also, I am going to discuss the three-dimensional case in more details.
The dawn of formalized mathematics
Prof. Andrej Bauer, University of Ljubljana
When I was a student of mathematics I was told that someone had formalized an entire book on analysis just to put to rest the question whether mathematics could be completely formalized, so that mathematicians could proceed with business as usual. I subsequently learned that the book was Landau's “Grundlagen”~\cite{landau65:_grund_analy}, the someone was L.~S.~van Benthem Jutting~\cite{benthem79:_check_landau_grund_autom}, the tool of choice was Automath~\cite{daalen94:_descr_autom_some_aspec_languag_theor}, and that popular accounts of history are rarely correct. Formalized mathematics did not die out. Computer scientists spent many years developing proof assistants~\cite{nuprl,coq-manual,agda,lean} -- programs that help create and verify formal proofs and constructions -- until they became good enough to attract the attention of mathematicians who felt that formalization had a place in mathematical practice. The initial successes came slowly and took a great deal of effort. In the last decade, the complete formalizations of the odd-order theorem~\cite{gonthier13:_machin_check_proof_odd_order_theor} and the solution of Kepler's conjecture~\cite{hales17:_formal_proof_kepler} sparked an interest and provided further evidence of viability of formalized mathematics. The essential role of proof assistants in the development of homotopy type theory~\cite{hottbook} and univalent mathematics~\cite{UniMath} showed that formalization can be an inspiration rather than an afterthought to traditional mathematics. % Today the community gathered around Lean~\cite{lean}, the newcomer among proof assistants, has tens of thousands of members and is growing very rapidly thanks to the miracle of social networking. % The new generation is ushering in a new era of mathematics. Formalized mathematics, in tandem with other forms of computerized mathematics~\cite{CarFarKohRab:bmobb19}, provides better management of mathematical knowledge, an opportunity to carry out ever more complex and larger projects, and hitherto unseen levels of precision. However, its transformative power runs still deeper. % The practice of formalization teaches us that formal constructions and proofs are much more than pointless transliteration of mathematical ideas into dry symbolic form. Formal proofs have rich structure, worthy of attention by a mathematician as well as a logician; contrary to popular belief, they can directly and elegantly express mathematical insights and ideas; and by striving to make them slicker and more elegant, new mathematics can be discovered. Formalized mathematics is changing the role of foundations of mathematics, too. A good century ago, a philosophical crisis necessitated the development of logic and set theory, which served as the bedrock upon which the 20th century mathematics was built safely. However, most proof assistants shun logic and set theory in favor of type theory, the original resolution of the crisis given by Bertrand Russell~\cite{russell25} and reformulated into its modern form by Per Martin-Löf~\cite{Martin-Lof-1972}. The reasons for this phenomenon are yet to be fully understood, but we can speculate that type theory captures mathematical practice more faithfully because it directly expresses the structure and constructions of mathematical objects, whereas set theory provides plentiful raw material with little guidance on how mathematical objects are to be molded out if it. \begin{thebibliography}{10} \bibitem{nuprl} S.F. Allen, M.~Bickford, R.L. Constable, R.~Eaton, C.~Kreitz, L.~Lorigo, and E.~Moran. Innovations in computational type theory using {Nuprl}. {\em Journal of Applied Logic}, 4(4):428--469, 2006. \bibitem{CarFarKohRab:bmobb19} Jacques Carette, William~M. Farmer, Michael Kohlhase, and Florian Rabe. Big math and the one-brain barrier -- the tetrapod model of mathematical knowledge. {\em Mathematical Intelligencer}, 43(1):78--87, 2021. \bibitem{lean} Leo de~Moura, Sebastian Ullrich, and Dany Fabian. Lean theorem prover. \bibitem{gonthier13:_machin_check_proof_odd_order_theor} Georges Gonthier, Andrea Asperti, Jeremy Avigad, Yves Bertot, Cyril Cohen, François Garillot, Stéphane~Le Roux, Assia Mahboubi, Russell O'Connor, Sidi~Ould Biha, Ioana Pasca, Laurence Rideau, Alexey Solovyev, Enrico Tassi, and Laurent Théry. A machine-checked proof of the odd order theorem. In Sandrine Blazy, Christine Paulin{-}Mohring, and David Pichardie, editors, {\em Interactive Theorem Proving - 4th International Conference, {ITP} 2013, Rennes, France, July 22-26, 2013. Proceedings}, volume 7998 of {\em Lecture Notes in Computer Science}, pages 163--179. Springer, 2013. \bibitem{hales17:_formal_proof_kepler} Thomas Hales, Mark Adams, Gertrud Bauer, Tat~Dat Dang, John Harrison, Le~Truong Hoang, Cezary Kaliszyk, Victor Magron, Sean McLaughlin, Tat~Thang Nguyen, Quang~Truong Nguyen, Tobias Nipkow, Steven Obua, Joseph Pleso, Jason Rute, Alexey Solovyev, Thi Hoai~An Ta, Nam~Trung Tran, Thi~Diep Trieu, Josef Urban, Ky~Vu, and Roland Zumkeller. A formal proof of the {K}epler conjecture. {\em Forum of Mathematics, Pi}, 5, 2017. \bibitem{landau65:_grund_analy} E.~G. H.~Y. Landau. {\em Grundlagen der Analysis}. Chelsea Pub. Co., New York, NY, USA, 1965. \bibitem{Martin-Lof-1972} Per Martin-Löf. An intuitionistic theory of types. In Giovanni Sambin and Jan~M. Smith, editors, {\em Twenty-five years of constructive type theory ({V}enice, 1995)}, volume~36 of {\em Oxford Logic Guides}, pages 127--172. Oxford University Press, 1998. \bibitem{agda} Ulf Norell. {\em Towards a practical programming language based on dependent type theory}. PhD thesis, Chalmers University of Technology, 2007. \bibitem{hottbook} The Univalent~Foundations Program. {\em Homotopy Type Theory: Univalent Foundations for Mathematics} , Institute for Advanced Study, 2013. \bibitem{coq-manual} Coq~Development Team. The {Coq} proof assistant reference manual, version 8.7. \bibitem{benthem79:_check_landau_grund_autom} L.~S. van Benthem~Jutting. Checking {Landau’s} “{Grundlagen}” in the {Automath} system. In {\em Mathematical Centre Tracts}, volume~83. Mathematisch Centrum, The Netherlands, 1979. \bibitem{daalen94:_descr_autom_some_aspec_languag_theor} D.~T. van Daalen. A description of automath and some aspects of its language theory. In {\em Selected Papers on Automath}, pages 101--126. North-Holland Pub. Co., Amsterdam, The Netherlands, 1994. \bibitem{UniMath} Vladimir Voevodsky, Benedikt Ahrens, Daniel Grayson, et~al. {\em UniMath}: {Univalent} {Mathematics}, 2016. \bibitem{russell25} Alfred~North Whitehead and Bertrand Russell. {\em Principia Mathematica}. Cambridge University Press, 1925--1927. \end{thebibliography}
Exponential sums over finite fields
Prof. Emmanuel Kowalski, ETH Zürich
Exponential sums are among the simplest mathematical objects that one can imagine, but also among the most remarkably useful and versatile in number theory. This talk will survey the history, the mysteries and the surprises of such sums over finite fields, with a focus on questions related to the distribution of values of families of exponential sums. General principles and applications will be illustrated by concrete examples, where sums of two squares, Sidon sets, Larsen's Alternative, the variance of arithmetic functions over function fields and the lines on cubic threefolds will make appearances. (Based on joint work with A. Forey and J. Fresán)
Some Recent Developments on the Geometry of Random Spherical Eigenfunctions
Prof. Domenico Marinucci, University of Rome Tor Vergata
A lot of efforts have been devoted in the last decade to the investigation of the high-frequency behaviour of geometric functionals for the excursion sets of random spherical harmonics, i.e., Gaussian eigenfunctions for the spherical Laplacian $\Delta_{S^2}$. In this talk we shall review some of these results, with particular reference to the asymptotic behaviour of variances, phase transitions in the nodal case (the Berry's Cancellation Phenomenon), the distribution of the fluctuations around the expected values, and the asymptotic correlation among different functionals. We shall also discuss some connections with the Gaussian Kinematic Formula, with Wiener-Chaos expansions and with recent developments in the derivation of Quantitative Central Limit Theorems (the so-called Stein-Malliavin approach).
Laplacians on infinite graphs
Prof. Aleksey Kostenko, University of Ljubljana
There are two different notions of a Laplacian operator associated with graphs: discrete graph Laplacians and continuous Laplacians on metric graphs (widely known as quantum graphs). Both objects have a venerable history as they are related to several diverse branches of mathematics and mathematical physics. The existing literature usually treats these two Laplacian operators separately. In this talk, I will focus on the relationship between them (spectral, parabolic and geometric properties). One of our main conceptual messages is that these two settings should be regarded as complementary (rather than opposite) and exactly their interplay leads to important further insight on both sides. Based on joint work with Noema Nicolussi.
HMS categorical symmetries and hypergeometric systems
Špela Špenko, Université Libré de Bruxelles
Hilbert's 21st problem asks about the existence of Fuchsian linear differential equations with a prescribed "monodromy representation" of the fundamental group. The first (slightly erroneous) solution was proposed by a Slovenian mathematician Plemelj. A suitably adapted version of this problem was solved, depending on the context, by Deligne, Mebkhout, Kashiwara-Kawai, Beilinson-Bernstein, ... The solution is now known as the Riemann-Hilbert correspondence. Homological mirror symmetry predicts the existence of an action of the fundamental group of the "stringy Kähler moduli space" (SKMS) on the derived category of an algebraic variety. This prediction was established by Halpern-Leistner and Sam for certain toric varieties. The decategorification of the action found by HLS yields a representation of the fundamental group of the SKMS and in joint work with Michel Van den Bergh we show that it is given by the monodromy of an explicit hypergeometric system of differential equations.
Fast algorithms from low-rank updates
Prof. Daniel Kressner, EPFL
The development of efficient numerical algorithms for solving large-scale linear systems is one of the success stories of numerical linear algebra that has had a tremendous impact on our ability to perform complex numerical simulations and large-scale statistical computations. Many of these developments are based on multilevel and domain decomposition techniques, which are closely linked to Schur complements and low-rank updates of matrices. In this talk, we explain how these tools carry over to other important linear algebra problems, including matrix functions and matrix equations. Fast algorithms are derived from combining divide-and-conquer strategies with low-rank updates of matrix functions. The convergence analysis of these algorithms is built on a multivariate extension of the celebrated Crouzeix-Palencia result. The newly developed algorithms are capable of addressing a wide variety of matrix functions and matrix structures, including sparse matrices as well as matrices with hierarchical low rank and Toeplitz-like structures. Their versatility will be demonstrated with several applications and extensions. This talk is based on joint work with Bernhard Beckermann, Alice Cortinovis, Leonardo Robol, Stefano Massei, and Marcel Schweitzer.
EMS Prize Winners
Elliptic curves and modularity
Jack Thorne, University of Cambridge
The modularity conjecture for elliptic curves is most famous in its formulation for elliptic curves over the rational numbers -- indeed, Wiles proved the modularity of semistable elliptic curves over the rationals as part of his proof of Fermat's Last Theorem. Recently it has become possible to attack the problem of modularity of elliptic curves over more general number fields. I will explain what this means, what we know, and what kinds of consequences we can expect for the solution of Diophantine equations.
Excitation spectrum of dilute trapped Bose gases
Phan Thanh Nam, LMU Munich
We will discuss the low-lying eigenvalues of trapped Bose gases in the Gross-Pitaevskii regime. In particular, we will derive a nonlinear correction to the Bogoliubov approximation, thus capturing precisely the two-body scattering process of the particles. This extends a result of Boccato, Brennecke, Cenatiempo and Schlein on the homogeneous Bose gas. The talk is based on joint work with Arnaud Triay.
Zero sets of Laplace eigenfunctions
Alexandr Logunov, Princeton University
In the beginning of 19th century Napoleon set a prize for the best mathematical explanation of Chladni’s resonance experiments. Nodal geometry studies the zeroes of solutions to elliptic differential equations such as the visible curves that appear in Chladni's nodal portraits. We will discuss the geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interesting relation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. Zero sets of Laplace eigenfunctions on surfaces are unions of smooth curves with equiangular intersections. Topology of the zero set could be quite complicated, but Yau conjectured that the total length of the zero set is comparable to the square root of the eigenvalue for all eigenfunctions. We will start with open questions about spherical harmonics and explain some methods to study nodal sets, which are zero sets of solutions of elliptic PDE.
On primes, almost primes and the Möbius function in short intervals
Kaisa Matomäki, University of Turku
In this talk I will introduce some fundamental concepts in analytic number theory such as primes, the Riemann zeta function and the Möbius function. I will discuss classical results concerning them and connecting them to each other. Then I will move on to discussing the distribution of primes, almost primes and the Möbius function in short intervals. In particular I will describe some recent progress on the topic.
Felix Klein Prize Winner
Overcoming the curse of dimensionality: from nonlinear Monte Carlo to deep learning
Prof. Arnulf Jentzen, University of Münster & The Chinese University of Hong Kong, Shenzhen
Partial differential equations (PDEs) are among the most universal tools used in modelling problems in nature and man-made complex systems. For example, stochastic PDEs are a fundamental ingredient in models for nonlinear filtering problems in chemical engineering and weather forecasting, deterministic Schroedinger PDEs describe the wave function in a quantum physical system, deterministic Hamiltonian-Jacobi-Bellman PDEs are employed in operations research to describe optimal control problems where companys aim to minimise their costs, and deterministic Black-Scholes-type PDEs are highly employed in portfolio optimization models as well as in state-of-the-art pricing and hedging models for financial derivatives. The PDEs appearing in such models are often high-dimensional as the number of dimensions, roughly speaking, corresponds to the number of all involved interacting substances, particles, resources, agents, or assets in the model. For instance, in the case of the above mentioned financial engineering models the dimensionality of the PDE often corresponds to the number of financial assets in the involved hedging portfolio. Such PDEs can typically not be solved explicitly and it is one of the most challenging tasks in applied mathematics to develop approximation algorithms which are able to approximatively compute solutions of high-dimensional PDEs. Nearly all approximation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality" in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE. With such algorithms it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used. In the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of dimensionality can be overcome by means of Monte Carlo approximation algorithms and the Feynman-Kac formula. In this talk we prove that suitable deep neural network approximations do indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs and we thereby prove, for the first time, that a general semilinear parabolic PDE can be solved approximatively without the curse of dimensionality.
Public speakers
Topological explorations in neuroscience
Prof. Kathryn Hess, EPFL
The brain of each and every one of us is composed of hundreds of billions of neurons – often called nerve cells – connected by hundreds of trillions of synapses, which transmit electrical signals from one neuron to another. In reaction to stimulus, waves of electrical activity traverse the network of neurons, processing the incoming information. Tools provided by the field of mathematics called algebraic topology enable us to detect and describe the rich structure hidden in this dynamic tapestry. During this talk, I will guide you on a mathematical mystery tour of what topology has revealed about how the brain processes information.
Hirzebruch Lecture
A mathematical journey through scales
Prof. Martin Hairer, Imperial College London
The tiny world of particles and atoms and the gigantic world of the entire universe are separated by scales spanning about forty orders of magnitudes. As we move from one to the other, the laws of nature can behave in drastically different ways, going from quantum physics to general relativity through Newton’s classical mechanics, not to mention other intermediate "ad hoc" theories. Understanding the way in which the behaviour of mathematical models changes as we move from one scale to another is one of the great classical questions in mathematics and theoretical physics. The aim of this talk is to explore how these questions still inform and motivate interesting problems in probability theory and why so-called toy models, despite their superficially playful character, can sometimes lead to useful quantitative (and not just qualitative) predictions.
A game theory and its applications (MS - ID 56)
Probability
Nash Equilibria in certain two-choice multi-player games played on the ladder graph
Ms. Victoria Sánchez Muñoz, National University of Ireland Galway
We compute analytically the number of Nash Equilibria (NE) for a two-choice game played on a (circular) ladder graph with $2n$ players. We consider a set of games with generic payoff parameters, with the only requirement that a NE occurs if the players choose opposite strategies (anti-coordination game). The results show that for both, the ladder and circular ladder, the number of NE grows exponentially with (half) the number of players $n$, as $N_{NE}(2n)\sim C(\varphi)^n$, where $\varphi=1.618...$ is the golden ratio and $C_{circ}>C_{ladder}$. In addition, the value of the scaling factor $C_{ladder}$ depends on the value of the payoff parameters. However, that is no longer true for the circular ladder (3-degree graph), that is $C_{circ}$ is constant, which might suggest that the topology of the graph indeed plays an important role for setting the number of NE.
A journey from pure to applied mathematics (MS - ID 53)
General Topics
Very weak solutions to PDEs in inhomogeneous and anisotropic spaces
Prof. Iwona Chlebicka, University of Warsaw
I will discuss well-posedness and regularity to nonlinear PDEs of simple divergent form \[ - {\rm div} A(x,\nabla u)= f\quad \text{or}\quad \partial_t u- {\rm div} A(t,x,\nabla u)= f,\] where the datum $f$ is merely integrable or even is a measure, whereas the growth of $A$ is governed by an inhomogeneous and fully anisotropic $N$-function $M(x,\nabla u)$. Inhomogeneity means space-dependence and anisotropy yields dependence of $M$ on $\nabla u$ not necessarily via its length $|\nabla u|$. The datum is too poorly regular for weak solutions to exist, thus a more delicate notion of very weak solutions is necessary. Besides existence and uniqueness they share some regularity properties of the weak ones. The generality we admit covers classical linear and polynomial growth operators possibly with measurable coefficients, generalizations of $p$-Laplacian involving log-H\"older continuous variable exponent, as well as problems posed in fully anisotropic Orlicz spaces (under no growth conditions), and double-phase spaces within the range of parameters sharp for density of smooth functions. The conditions for the density and their importance will be stressed. The talk will be based on series of joint papers with: Youssuf Ahmida, Angela Alberico, Andrea Cianchi, Piotr Gwiazda, Ahmed Youssfi, and Anna Zatorska-Goldstein.
Flows of nonsmooth vector fields: new results on non uniqueness and commutativity
Prof. Maria Colombo, EPFL Lausanne
Given a vector field in $R^d$, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow (namely, the solution X(t) of the ODE $X'(t)=b(t,X(t))$ from any initial datum $x\in R^d$) provided the vector field is sufficiently smooth. The theorem looses its validity as soon as v is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields. The talk presents an overview and new results in the context of the celebrated DiPerna-Lions and Ambrosio’s theory on flows of Sobolev vector fields, including a negative answer to the following long-standing open question: are the trajectories of the ODE unique for a.e. initial datum in $R^d$ for vector fields as in Di Perna and Lions theorem? We will exploit the connection between the notion of flow and an associated PDE, the transport equation, and combine ingredients from probability theory, harmonic analysis, and the “convex integration” method for the construction of nonunique solutions to certain PDEs.
The fast p-Laplacian evolution equation. Global Harnack principle and fine asymptotic behaviour
Dr. Diana Stan, Universidad de Cantabria
We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast p-Laplacian evolution equation on the whole Euclidean space, in the so-called "good fast diffusion range". It is well-known that non-negative solutions behave for large times as B, the Barenblatt (or fundamental) solution, which has an explicit expression. We prove the so-called Global Harnack Principle (GHP), that is, precise global pointwise upper and lower estimates of nonnegative solutions in terms of B. This can be considered the nonlinear counterpart of the celebrated Gaussian estimates for the linear heat equation. To the best of our knowledge, analogous issues for the linear heat equation, do not possess such clear answers, only partial results are known. Also, we characterize the maximal (hence optimal) class of initial data such that the GHP holds, by means of an integral tail condition, easy to check. Finally, we derive sharp global quantitative upper bounds of the modulus of the gradient of the solution, and, when data are radially decreasing, we show uniform convergence in relative error for the gradients.
Algorithmic Graph Theory (MS - ID 54)
Combinatorics and Discrete Mathematics
The Slow-Coloring Game on a Graph
Prof. Douglas West, Zhejiang Normal University and University of Illinois
The {\it slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a non-empty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it; the score under optimal play is the {\it cost} of the graph. A greedy strategy for Painter keeps the cost of $G$ to at most $\chi(G)n$ when $G$ has $n$ vertices, which is asyptotically sharp for Tur\'an graphs. On various classes Painter can do better. For $n$-vertex trees the maximum cost is $\lfloor 3n/2\rfloor$. There is a linear-time algorithm and inductive formula to compute the cost on trees, and we know all the extremal $n$-vertex trees. Also, Painter can keep the cost to at most $(1+3k/4)n$ when $G$ is $k$-degenerate, $7n/3$ when $G$ is outerplanar, $3.9857n$ when $G$ is acyclically $5$-colorable, and $3.4n$ when $G$ is planar. These bounds are not believed to be sharp. We will discuss strategies (algorithms) for Lister and Painter that establish various lower and upper bounds. The results appear in three papers with various subsets of Grzegorz Gutowski, Tomasz Krawczyk, Thomas Mahoney, Krzysztof Maziarz, Gregory J. Puleo, Michal Zajac, and Xuding Zhu.
Shifting any path to an avoidable one
Matjaž Krnc, FAMNIT (University of Primorska)
A vertex $v$ in a graph $G$ is avoidable if every induced path on three vertices with middle vertex $v$ is contained in an induced cycle. Dirac's classical result from 1961 on the existence of simplicial vertices in chordal graphs is equivalent to the statement that every chordal graph has an avoidable vertex. Beisegel, Chudovsky, Gurvich, Milani\v{c}, and Servatius (2019) generalized the notion of avoidable vertices to \emph{avoidable paths}, and conjectured that every graph that contains an induced $k$-vertex path also contains an avoidable induced $k$-vertex path; they proved the result for $k = 2$. The case $k = 1$ was known much earlier, due to a work of Ohtsuki, Cheung, and Fujisawa in 1976. The conjecture was proved for all $k$ in 2020 by Bonamy, Defrain, Hatzel, and Thiebaut. This result generalizes the mentioned results regarding avoidable vertices, as well as a result by Chv\'atal, Rusu, and Sritharan (2002) suggested by West on the existence of simplicial $k$-vertex paths in graphs excluding induced cycles of length at least $k+3$. In this talk we discuss an adaptation of the approach by Bonamy et al.~from a reconfiguration point of view. We say that two induced $k$-vertex paths are \emph{shifts} of each other if their union is an induced path with $k+1$ vertices and show that in every graph, every induced path can be shifted to an avoidable one. We also present an analogous result about paths that are not necessarily induced, where an efficient reconfiguration sequence relies on the properties of depth-first search trees.
Circularly compatible ones, $D$-circularity, and proper circular-arc bigraphs
Dr. Martín Safe, Universidad Nacional del Sur
In 1969, Alan Tucker characterized proper circular-arc graphs as those graphs whose augmented adjacency matrices have the circularly compatible ones property. Moreover, he also found a polynomial-time algorithm for deciding whether any given augmented adjacency matrix has the circularly compatible ones property. These results allowed him to devise the first polynomial-time recognition algorithm for proper circular-arc graphs. However, as Tucker himself remarks, he did not solve the problems of finding a structure theorem and an efficient recognition algorithm for the circularly compatible ones property in arbitrary matrices (i.e., not restricted to augmented adjacency matrices only). In this work, we solve these problems. More precisely, we give a minimal forbidden submatrix characterization for the circularly compatible ones property in arbitrary matrices and a linear-time recognition algorithm for the same property. We derive these results from analogous ones for the related $D$-circular property. Interestingly, these results lead to a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for proper circular-arc bigraphs, solving a problem first posed by Basu, Das, Ghosh, and Sen [J. Graph Theory, 73(4):361--376, 2013]. Our findings generalize some known results about $D$-interval hypergraphs and proper interval bigraphs.
Beyond treewidth: the tree-independence number
Prof. Martin Milanič, University of Primorska
We introduce a new graph invariant called \emph{tree-independence number}. This is a common generalization of treewidth and independence number, in the sense that bounded treewidth or bounded independence number implies bounded tree-independence number. The tree-independence number of a graph $G$ is defined as the smallest positive integer $k$ such that $G$ has a tree decomposition whose bags induce subgraphs of $G$ with independence number at most $k$. While for $k = 1$ we obtain the well-known class of chordal graphs, we show that the problem of computing the tree-independence number of a graph is NP-hard in general. We consider six graph containment relations (the subgraph, topological minor, and minor relations, as well as their induced variants) and for each of them completely characterize graph classes of bounded tree-independence number defined by a single forbidden graph with respect to the relation. In each of these bounded cases, a tree decomposition with small independence number can be computed efficiently, which implies polynomial-time solvability of the Maximum Weight Independent Set (MWIS) problem. This in particular applies to an infinite family of generalizations of the class of chordal graphs, for which a polynomial-time algorithm for the MWIS problem was given by Frank in 1976.
Clique-Width: Harnessing the Power of Atoms
Dr. Konrad Dabrowski, University of Leeds
Many {\sf NP}-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class ${\cal G}$ if they are so on the atoms (graphs with no clique cut-set) of ${\cal G}$. Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph $G$ is {\it $H$-free} if $H$ is not an induced subgraph of $G$, and it is {\it $(H_1,H_2)$-free} if it is both $H_1$-free and $H_2$-free. A class of $H$-free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for $(H_1,H_2)$-free graphs, as evidenced by one known example. We prove the existence of another such pair $(H_1,H_2)$ and classify the boundedness of clique-width on $(H_1,H_2)$-free atoms for all but 18 cases. \begin{thebibliography}{1} \bibitem{DMNPR20} Konrad K. Dabrowski, Tom\'a\v{s} Masa\v{r}\'ik, Jana Novotn\'a, Dani\"el Paulusma, and Pawe{\l} Rzazewski. Clique-width: Harnessing the power of atoms. {\em Proc. WG 2020, LNCS}, 12301:119--133, 2020. Full version: arXiv:2006.03578. \end{thebibliography}
Width parameters and graph classes: the case of mim-width
Dr. Andrea Munaro, Queen's University Belfast
A large number of $\mathsf{NP}$-hard graph problems become polynomial-time solvable on graph classes where the mim-width is bounded and quickly computable. Hence, when solving such problems on special graph classes, it is helpful to know whether the graph class under consideration has bounded mim-width. We extend the toolkit for proving (un)boundedness of mim-width of graph classes and initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes. We present summary theorems of the current state of the art for the boundedness of mim-width for $(H_1,H_2)$-free graphs and observe several interesting consequences. We also study the mim-width of generalized convex graphs. This allows us to re-prove and strengthen a large number of known results.
Colourful components in $k$-caterpillars and planar graphs
Mr. Clément Dallard, University of Primorska
A connected component of a vertex-coloured graph is said to be \emph{colourful} if all its vertices have different colours. By extension, a graph is colourful if all its connected components are colourful. Given a vertex-coloured graph and an integer $p$, the \textsc{Colourful Components} problem asks whether there exist at most $p$ edges whose removal makes the graph colourful. Bulteau, Dabrowski, Fertin, Johnson, Paulusma, and Vialette (2019) proved that \textsc{Colourful Components} is NP-complete on trees with maximum degree at most $6$ and asked whether the same would hold if the maximum degree is at most~$5$. We show that the problem remains NP-complete on binary $4$-caterpillars, ternary $3$-caterpillars and quaternary $2$-caterpillars, where a \emph{$k$-caterpillar} is a tree containing a path $P$ such that every vertex is at distance at most $k$ from~$P$. On the other hand, we provide a linear-time algorithm for $1$-caterpillars (without restriction on the maximum degree), and thus almost settle the complexity dichotomy on $k$-caterpillars with respect to the maximum degree. \textsc{Colourful Components} has also been studied in vertex-coloured graphs with a bounded number of colours. Bruckner, H\"{u}ffner, Komusiewicz, Niedermeier, Thiel, and Uhlmann (2012) showed that the problem is NP-complete on $3$-coloured graphs with maximum degree $6$, and Bulteau et al. asked whether the problem would remain NP-complete on graphs with bounded number of colors and maximum degree~$3$. We answer their question in the affirmative by showing that the problem is NP-complete on $5$-coloured planar graphs with maximum degree $4$ and on $12$-coloured planar graphs with maximum degree~$3$.
Analysis of PDEs on Networks (MS - ID 26)
Partial Differential Equations and Applications
The quintic NLS on the tadpole graph
Prof. Diego Noja, Università di Milano Bicocca
The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schr\"{o}dinger equation with quintic power nonlinearity and Kirchhoff boundary conditions at the vertex. The profile of a standing wave with frequency $\omega\in (-\infty,0)$ is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $L^6$. The set of standing waves so defined strictly includes the set of ground states, i.e. the global minimizers of the energy at constant mass ($L^2$-norm), but it is actually wider. While ground states exist only for a certain interval of masses, the above standing waves exist for every $\omega \in (-\infty,0)$ and correspond to a bigger interval of masses. It is proven that there exist critical frequencies $\omega_1$ and $\omega_0$ with $-\infty < \omega_1 < \omega_0 < 0$ such that the standing waves are the ground state for $\omega \in [\omega_0,0)$, local constrained minima of the energy for $\omega \in (\omega_1,\omega_0)$ and saddle points of the energy at constant mass for $\omega \in (-\infty,\omega_1)$.\\ Joint work with D.E. Pelinovsky.
First order Mean Field Games on networks
Prof. Claudio Marchi, University of Padova
The theory of Mean Field Games studies the asymptotic behaviour of differential games (mainly in terms of their Nash equilibria) as the number of players tends to infinity. In these games, the players are rational and indistinguishable: each player aims at choosing its trajectory so to minimize a cost which depends on the trajectory itself and on the distribution of the whole population of agents. We focus our attention on deterministic Mean Field Games with finite horizon in which the states of the players are constrained in a network (in our setting, a network is given by a finite collection of vertices connected by continuous edges which cannot self-intersect). In these games, an agent can control its dynamics and has to pay a cost formed by a running cost depending on the evolution of the distribution of all agents and a terminal cost depending on the distribution of all agents at terminal time. As in the Lagrangian approach, we introduce a relaxed notion of Mean Field Games equilibria and we shall deal with probability measures on trajectories on the network instead of probability measures on the network. This is a joint work with: Y. Achdou (Univ. of Paris), P. Mannucci (Univ. of Padova) and N. Tchou (Univ. of Rennes).
Discontinuous ground states for the NLSE on $\mathbb{R}$ with a \emph{F\"{u}l\"{o}p-Tsutsui $\boldsymbol\delta$} interaction
Ms. Alice Ruighi, Politecnico di Torino
We analyse the existence and the stability of the ground states of the one-dimensional nonlinear Schr\"{o}dinger equation with a focusing power nonlinearity and a defect located at the origin. A ground state is intended as a global minimizer of the action functional on the Nehari's manifold and the defect considered is a F\"{u}l\"{o}p-Tsutsui $\delta$ type, namely a $\delta$ condition that allows discontinuities. The existence of ground states is proved by variational techniques, while the stability results follow from the Grillakis-Shatah-Strauss' theory. \\ This is a joint work with Riccardo Adami.
Analysis on Graphs (MS - ID 48)
Mathematical Physics
Effects of time-reversal asymmetry in the vertex coupling of quantum graphs
Prof. Pavel Exner, Czech Academy of Sciences, Nuclear Physics Institute
The talk concerns the effects coming from a violation of time-reversal symmetry in the vertex coupling of quantum graph, focusing on the situation when the asymmetry is maximal at a fixed energy. It is shown that such a coupling has a topological property: the transport behaviour of such a vertex in the high-energy regime depend substantially on the vertex parity. We explore consequences of this fact for several classes of graphs, both finite and infinite periodic ones. The results come from a common work with Marzieh Baradaran, Ji\v{r}\'{\i} Lipovsk\'{y}, and Milo\v{s} Tater.
Multifractal eigenfunctions in a singular quantum billiard
Prof. Jon Keating, University of Oxford and London Mathematical Society
Spectral statistics and questions relating to quantum ergodicity in star graphs are closely related to those of Šeba billiards (rectangular billiards with a singular point scatterer) and other intermediate systems. In intermediate systems, it has been suggested in the Physics literature that quantum eigenfunctions should exhibit mulifractal properties. I shall discuss the proof that this is the case for Šeba billiards. The work I shall report on was done jointly with Henrik Ueberschaer.
Stochastic completeness and uniqueness class for graphs
Dr. Xueping Huang, Nanjing University of Information Science and Technology
Uniqueness class for the heat equation on a weighted graph is closely related to stochastic completeness of the corresponding minimal continuous time random walk. For a class of so called globally local graphs, we obtain essentially sharp criteria which are in the same form as for manifolds. Sharp volume growth type criteria for stochastic completeness then follow, after a reduction to the globally local case. This talk is based on joint work with M. Keller and M. Schmidt.
Neumann domains on metric graphs
Prof. Ram Band, Technion - Israel Institute of Technology
The Neumann points of an eigenfunction $f$ on a quantum (metric) graph are the interior zeros of $f'$. The Neumann domains of $f$ are the sub-graphs bounded by the Neumann points. Neumann points and Neumann domains are the counterparts of the well-studied nodal points and nodal domains. We present the following three main properties of Neumann domains: their count, wavelength capacity and spectral position. We study their bounds and probability distributions and use those to investigate inverse spectral problems. The relevant probability distributions are rigorously defined in terms of selected random variables for quantum graphs. To this end, we provide conditions for considering spectral functions of quantum graphs as random variables with respect to the natural density on $\mathbb{N}$. The talk is based on joint work with Lior Alon.
Analysis, Control and Inverse Problems for Partial Differential Equations (MS - ID 22)
Optimization and Control
The Calderón problem with corrupted data
Dr. Pedro Caro, Basque Center for Applied Mathematics
The inverse Calderón problem consists in determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the Dirichlet-to-Neumann map and, therefore, one usually assumes the data to be given by such map. This situation corresponds to having access to infinite-precision measurements, which is unrealistic. In this talk, I will consider the Calderón problem assuming data to contain measurement errors and provide formulas to reconstruct the conductivity and its normal derivative on the surface (joint work with Andoni García). I will also present similar results for Maxwell's equations (joint work with Ru-Yu Lai, Yi-Hsuan Lin, Ting Zhou ). When modelling errors in these two different frameworks, one realizes the existence of certain freedom that yields different reconstruction formulas. To understand the whole picture of what is going on, we will rewrite the problem in a different setting, which will bring us to analyse the observational limit of wave packets with noisy measurements (joint work with Cristóbal J. Meroño).
Strong unique continuation at the boundary in linear elasticity and its connection with optimal stability in the determination of unknown boundaries
Prof. Edi Rosset, Università di Trieste
\date{} \textbf{Abstract:} Quantitative estimates of Strong Unique Continuation at the boundary for solutions to the isotropic Kirchhoff-Love plates subject to Dirichlet conditions, and for solutions to the Generalized plane stress problem subject to Neumann conditions are presented. These results have been applied to prove optimal stability estimates for the inverse problem of determining unknown boundaries. \medskip \noindent References: [1] G. Alessandrini, E. Rosset, S. Vessella, Optimal three spheres inequality at the boundary for the Kirchhoff-Love plate's equation with Dirichlet conditions, Arch. Rational Mech. Anal., 231 (2019), 1455--1486. [2] A. Morassi, E. Rosset, S. Vessella, Optimal stability in the identification of a rigid inclusion in an isotropic Kirchhoff-Love plate, SIAM J. Math. Anal., 51 (2019), 731--747. [3] A. Morassi, E. Rosset, S. Vessella, Optimal identification of a cavity in the Generalized Plane Stress problem in linear elasticity, J. Eur. Math. Soc., to appear. [4] A. Morassi, E. Rosset, S. Vessella, Doubling Inequality at the Boundary for the Kirchhoff-Love Plate's Equation with Dirichlet Conditions, Le Matematiche, LXXV (2020), 27--55, Open access.
Asymptotic behavior of dispersive electromagnetic waves in bounded domains
Prof. Cristina Pignotti, University of L'Aquila
We analyze the stability of Maxwell equations in bounded domains taking into account electric and magnetization effects. Well-posedness of the model is obtained by means of semigroup theory. Then, a passitivity assumption guarantees the boundedness of the associated semigroup. Under suitable sufficient conditions, we prove the exponential or polynomial decay of the energy. Moreover, some illustrative examples are presented.
Weak Solutions for an Implicit, Degenerate Poro-elastic Plate System
Dr. Justin Webster, University of Maryland, Baltimore County
We consider a recent plate model obtained as a scaled limit of the three dimensional quasi-static Biot system of poro-elasticity. The result is a ``2.5" dimensional linear system that couples traditional Euler-Bernoulli plate dynamics to a pressure equation in three dimensions, where diffusion acts only transversely. Motived by application, we allow the permeability function to be time-dependent, making the problem non-autonomous and disqualifying much of the standard theory. Weak solutions are defined and the problem is framed abstractly as an implicit, degenerate evolution problem: $$[Bp]_t+A(t)p=S.$$ Existence is obtained, and uniqueness follows under additional hypotheses on the temporal regularity of the permeability. Time permitting, we address the inertial case with constant permeability by way of semigroup theory.
Optimal control problem for a repulsive chemotaxis system
Dr. María Angeles Rodríguez-Bellido, Universidad de Sevilla
The chemotaxis phenomenon can be understood as the directed movement of living organisms in response to chemical gradients. Keller and Segel \cite{keller-segel} proposed a mathematical model that describes the chemotactic aggregation of cellular slime molds. These molds move preferentially towards relatively high concentrations of a chemical substance secreted by the amoebae themselves. Such mechanism is called \textit{chemo-attraction} with production. However, when the regions of high chemical concentration generate a repulsive effect on the organisms, the phenomenon is called \textit{chemo-repulsion}. In this work, we want to study an optimal control problem for the (repulsive) Keller-Segel model and a bilinear control acting on the chemical equation in a $2D$ and $3D$ domains. The system can be written as: \begin{equation}\label{KS} \left\{\begin{array}{rcll} \partial_t u -\Delta u - \nabla \cdot (u \, \nabla v) &=& 0 & \mbox{in $\Omega \times (0,T)$,}\\ \noalign{\vspace{-1ex}}\\ \partial_t v -\Delta v + v &=& u + f \, v\, 1_{\Omega_c}& \mbox{in $\Omega \times (0,T)$,}\\ \noalign{\vspace{-1ex}}\\ \partial_{\textbf{n}} u= \partial_{\textbf{n}} v &=&0 & \mbox{on $\partial\Omega \times (0,T)$,}\\ \noalign{\vspace{-1ex}}\\ u(0,\cdot)=u_0\ge 0, \quad v(0,\cdot)&=&v_0\ge 0 & \mbox{in $\Omega$,} \end{array}\right. \end{equation} being $f:Q_c:=(0,T)\times \Omega_c\to \mathbb{R}$ (the control) with $\Omega_c\subset \Omega\subset \mathbb{R}^n$ ($n=2,3$) the control domain, and the state $u,v:Q:=(0,T)\times \Omega_c\to \mathbb{R}_+$ the celular density and chemical concentration, respectively. Here, $\textbf{n}$ is the outward unit normal vector to $\partial\Omega$. \medskip The existence and uniqueness of global in time weak solution $(u,v)$ for the uncontrolled system is known (see for instance \cite{cieslak,Diego4}). \medskip In this work we study an optimal control problem subject to a chemo-repulsion system with linear production term, and in which a bilinear control acts injecting or extracting chemical substance on a subdomain of control $\Omega_c\subset\Omega$. Existence of weak solutions are stablished (in the $3D$ case by using a regularity criterion), and, as a consequence, a global optimal solution together with first-order optimality conditions for local optimal solutions are deduced. \medskip The results presented in this talk are based on \cite{Exequiel1,Exequiel2}. \begin{thebibliography}{99} \markboth{}{} \bibitem{cieslak} Cieslak, T., Lauren\c cot, P., Morales-Rodrigo, C.: Global existence and convergence to steady states in a chemorepulsion system. Parabolic and Navier-Stokes equations. Part 1, 105-117, Banach Center Publ., 81, Part 1, Polish Acad. Sci. Inst. Math., Warsaw, 2008. \bibitem{Exequiel1} Guill\'en-Gonz\'alez, F.; Mallea-Zepeda, E.; Rodr\'{\i}guez-Bellido, M.A. {Optimal bilinear control problem related to a chemo-repulsion system in 2D domains.} ESAIM Control Optim. Calc. Var. 26 (2020), Paper No. 29, 21 pp. \bibitem{Exequiel2} Guill\'en-Gonz\'alez, F.; Mallea-Zepeda, E.; Rodr\'{\i}guez-Bellido, M.A. {A regularity criterion for a 3D chemo-repulsion system and its application to a bilinear optimal control problem.} SIAM J. Control Optim. 58 (2020), no. 3, 1457--1490. \bibitem{Diego4} Guill\'en-Gonz\'alez, F.; Rodr\'{\i}guez-Bellido, M. A.; Rueda-G\'omez, D. A. {Unconditionally energy stable fully discrete schemes for a chemo-repulsion model.} Math. Comp. 88 (2019), no. 319, 2069--2099. \bibitem{keller-segel} Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theo. Biol. Vol. 26, 399-415 (1970). \end{thebibliography}
Applied Combinatorial and Geometric Topology (MS - ID 34)
Combinatorics and Discrete Mathematics
On the Connectivity of Branch Loci of Spaces of Curves
Prof. Milagros Izquierdo, Linköping University
Since the 19th century the theory of Riemann surfaces has a central place in mathematics putting together complex analysis, algebraic and hyperbolic geometry, group theory and combinatorial methods. Since Riemann, Klein and Poincar{'e} among others, we know that a compact Riemann surface is a complex curve, and also the quotient of the hyperbolic plane by a Fuchsian group. \smallskip In this talk we study the connectivity of the moduli spaces of Riemann surfaces (i.e in spaces of Fuchsian groups). Spaces of Fuchsian groups are orbifolds where the singular locus is formed by Riemann surfaces with automorphisms: {\it the branch loci}: With a few exceptions the branch loci is disconnected and consists of several connected components. This talk is a survey of the different methods and topics playing together in the theory of Riemann surfaces. \smallskip Joint wotk with Antonio F. Costa
Classifying compact PL 4-manifolds according to generalized regular genus and G-degree
Prof. Paola Cristofori, University of Modena and Reggio Emilia
{\it (d+1)-colored graphs}, that is $(d+1)$-regular graphs endowed with a proper edge-coloration, are the objects of a long-studied representation theory for closed PL $d$-manifolds, which has been recently extended to the whole class of compact PL $d$-manifolds. In this context, combinatorially defined PL invariants play a relevant role; in this talk we will focus on two of them: the {\it generalized regular genus} and the {\it G-degree}. The former extends to higher dimension the classical notion of Heegaard genus for 3-manifolds; the latter has arisen in connection with {\it Colored Tensor Models} (CTM), a particular kind of tensor models, that have been intensively studied in the last years, mainly as an approach to quantum gravity in dimension greater than two. CTMs established a link between colored graphs and tensor models, since the Feynman graphs of a $d$-dimensional CTM are precisely $(d+1)$-colored graphs. Furthermore, the G-degree of a colored graph is a crucial quantity driving the 1/N expansion of the free energy of a CTM. This talk will mainly concern recent results achieved in dimension 4: in particular, the classification of all compact PL 4-manifolds with generalized regular genus at most one or with G-degree at most 18. Furthermore, we will discuss interesting classes of 5-colored graphs ({\it semi-simple and weak semi-simple crystallizations}), representing compact PL 4-manifolds with empty or connected boundary and minimizing the invariants. In the simply-connected case they also belong to a wider class of 5-colored graphs which are proved to induce handle-decompositions of the represented 4-manifold lacking in 1- and/or 3-handles; therefore their study is strictly related to the problem, posed by Kirby, of the existence of special handle-decompositions for any simply-connected closed PL 4-manifold.
Explicit computation of some families of Hurwitz numbers
Prof. Carlo Petronio, Università di Pisa
I will describe the computation (based on dessins d'enfant) of the number of equivalence classes of surface branched covers matching a given branch datum belonging to a certain very specific family.
Complexity of graph manifolds
Prof. Alessia Cattabriga, University of Bologna
The notion of complexity for compact 3-dimensional manifolds, was introduced by Matveev in the nineties as a way to measure how complicated a manifold is; indeed, for closed orientable irreducible manifolds, the complexity coincides with the minimum number of tetrahedra needed to construct the manifold, with the only exceptions of the 3-sphere, the projective space and the lens space L(3, 1) all having complexity zero. There exists a census of manifolds according to increasing complexity that, for the orientable case, is Recognizer catalogue and includes all manifolds up to complexity 12 (see http://matlas.math.csu.ru/?page=search). In this talk, after recalling some general result about complexity, I will focus on an important class of manifolds: graph manifolds. These manifolds have been introduced and classified by Waldhausen in the sixties and are defined as compact 3-manifolds obtained by gluing Seifert fibre spaces along toric boundary components. I will present an upper bound for their complexity obtained in a joint work with Michele Mulazzani (University of Bologna) that is sharp for all the 14502 graph manifolds included in the Recognizer catalogue
Partitioning the projective plane and the dunce hat
Andrés David Santamaría-Galvis, University of Primorska
The faces of a simplicial complex induce a partial order by inclusion in a natural way. We say that the complex is partitionable if its poset can be partitioned into boolean intervals, with a maximal face at the top of each. In this work we show that all the triangulations of the real projective plane, the dunce hat, and the open Möbius strip are partitionable. To prove that, we introduce simple yet useful gluing tools that allow us to abridge the discussion about partitionability of a given complex in terms of smaller constituent relative subcomplexes. The gluing process generates partitioning schemes with a distinctive shelling-like flavor.
Universality classes of triangulations in dimensions greater than 2
Prof. Valentin Bonzom, Universite Sorbonne Paris Nord
Combinatorial maps are cellular topological models for surfaces, which include triangulations, quadrangulations, etc. of all genera. It is well-known that at fixed topology, all models lie in the same universality class. It means in particular that they have the same critical exponents in the enumeration formula and the same limit object at large scales. For instance, the universality class for planar maps is also known as the universality class of 2D pure quantum gravity and its scaling limit is known as the Brownian sphere. It is important to generalize this framework to greater dimensions, with motivations from combinatorics, topology, mathematical physics and quantum gravity. However, it has been a challenge due to the intricate nature of combinatorics and topology in dimensions 3 and greater. Here I will present results based on the model of colored triangulations for PL-manifolds. They admit a representation as edge-colored graphs which is amenable to combinatorial analysis. I will focus on a combinatorial generalization of Euler's relation for combinatorial maps, which is to bound the number of $(d-2)$-simplices linearly in the number of $d$-simplices, and identifying the triangulations which maximize the bound for different colored building blocks. In 3d, I prove that those triangulations, for any set of colored building blocks homeomorphic to the 3-ball, are in bijection with trees. In even dimensions, we have proved that several universality classes can be achieved. Both those results are in sharp contrast with the case of surfaces.
Approximation Theory and Applications (MS - ID 78)
Analysis and its Applications
Gauss-Lucas theorem in polynomial dynamics
Dr. Margaret Stawiska-Friedland, American Mathematical Society/MathSciNet
Using versions of the Gauss-Lucas theorem adapted to dynamics, we prove that for every complex polynomial $p$ of degree $d \geq 2$ the convex hull $H_p$ of the Julia set $J_p$ of $p$ satisfies $p^{-1}(H_p) \subset H_p$. This settles positively a conjecture by P. Alexandersson. We also characterize the families of polynomials for which the equality $p^{-1}(H_p) = H_p$ is achieved.
A complex funtion theory for Mellin Analysis and applications to sampling
Prof. Carlo Bardaro, University of Perugia
The aim of this research is to extend in a simple way the well-known Paley--Wiener theorem of Fourier Analysis, which characterizes the so-called bandlimited functions, to the setting of Mellin transform. In order to do that, we introduced in these last years a notion of polar analytic function [1], which provides a simple way of describing functions that are analytic on a part of the Riemann surface of the logarithm. Applications to various topics of Mellin Analysis are developed, in particular sampling type theorems and quadrature formulae (for a survey see [2]). \vskip0,3cm \noindent [1] Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Development of a new concept of polar analytic functions useful in Mellin analysis. Complex Var. Elliptic Equ. {\bf 64}(12), 2040--2062 (2019) \\ \noindent [2] Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Polar-analytic functions: old and new results, applications. Pre-print submitted, (2021)
Metric Fourier approximation of set-valued functions of bounded variation
Dr. Elena Berdysheva, Justus Liebig University Giessen
We study set-valued functions (SVFs) mapping a real interval to compact sets in~${\mathbb R}^d$. Older approaches to the approximation investigated almost exclusively SVFs with convex images (values), the standard methods suffer from convexification. In this talk I will describe a new construction that adopts the trigonometric Fourier series to set-valued functions with general (not necessarily convex) compact images. Our main result is analogous to the classical Dirichlet-Jordan Theorem for real functions. It states the pointwise convergence in the Hausdorff metric of the metric Fourier partial sums of a multifunction of bounded variation to a set determined by the values of the metric selections of the function. In particular, if the multifunction $F$ is of bounded variation and continuous at a point $x$, then the metric Fourier partial sums of it at $x$ converge to $F(x)$.
A Hermite-Hadamard type inequality with applications to the estimation of moments of convex functions of random variables
Mrs. PAȘCA RALUCA IOANA , TECHNICAL UNIVERSITY CLUJ-NAPOCA
In this paper we present a Hermite-Hadamard type inequality with applications to the estimation of moments for convex functions of several random variables. A particular case of the derived result can be applied to estimate the sum of moments for identically distributed random variables. We were motivated in our research by the applications of moment estimation in fields such as probability theory, mathematical statistics and statistical learning.
A generalization of a local form of the classical Markov inequality
Dr. Tomasz Beberok, University of Applied Sciences in Tarnow
In this talk we introduce a generalization to compact subsets of certain algebraic varieties of the classical Markov inequality on the derivatives of a polynomial in terms of its own values. We also introduce an extension to such sets of a local form of the classical Markov inequality, and show the equivalence of introduced Markov and local Markov inequalities.
Arithmetic and Geometry of Algebraic Surfaces (MS - ID 45)
Algebraic and Complex Geometry
Generalized Kummer surfaces and a configuration of conics in the plane
Mrs. Alessandra Sarti, University of Poitiers
A generalized Kummer surface is the minimal resolution of the quotient of an abelian surface by an automorphism of order three. The quotient surface contains nine cusps and this is a special example of a singular K3 surface. The converse is also true : the minimal resolution of a K3 surface containing nine cusps is a generalized Kummer surface. In this paper we study several configurations of nine cusps on the same K3 surface, which is the double cover of the plane ramified on a sextic with nine cusps. We get the new configurations by studying conics through the singular points of the sextic. The aim of the construction is to investigate the following question: is it possible that the K3 surface is the generalized Kummer surface associated to two non-isomorphic abelian surfaces ? This is a joint work with D. Kohl and X. Roulleau.
Infinitesimal Torelli for elliptic surfaces revisited
Prof. Remke Kloosterman, University of Padova
In this talk we consider the infinitesimal Torelli problem for elliptic surfaces without multiple fibers. We give a new proof for the case of elliptic surfaces without multiple fibers with Euler number at least 24 and nonconstant j-invariant and with Euler number at least 72 and constant j-invariant. For all of the remaining cases we will indicate whether infinitesimal Torelli holds, does not hold or our methods are insufficient to decide. This solve an issue raised by Atsushi Ikeda in a recent paper, in connection with his construction a counterexample to infinitesimal Torelli with $p_g=q=1$.
New families of surfaces with canonical map of high degree
Prof. Roberto Pignatelli, University of Trento
By a celebrated result of A. Beauville, if the dimension of the image of the canonical map of a complex surface is two, then the image is either of genus zero or a canonical surface. In both cases the degree of the map is uniformely bounded from above. Until a few years ago, very few examples of surfaces with canonical map of high degree were known. Thanks to the work of some collegues, new examples have been obtained in recent years, opening up new research questions and perspectives. In this seminar we will present a new method for constructing surfaces with canonical map of high degree and some new examples produced with this method. Finally, we will discuss the contribution our examples make to the aforementioned issues.
Rigid complex manifolds via product quotients
Dr. Christian Gleissner, University of Bayreuth
Product quotients give rise to a large class of varieties equipped with a lot of symmetry. They are particularly useful as a tool to provide examples of complex manifolds with special properties. In this talk I will explain how to construct in each dimension dimension n (at least three) an example of a rigid n-manifold of Kodaira dimension one, as a resolution of a singular product quotient. Their existence was conjectured by I. Bauer and F. Catanese and constructed in a joint project with I. Bauer.
CA18232: Variational Methods and Equations on Graphs (MS - ID 40)
Partial Differential Equations and Applications
Spectral geometry of quantum graphs via surgery principles
Dr. James Kennedy, University of Lisbon
``Surgery" on a (metric) graph means making a small, generally local, change to its structure: for example, joining two vertices, lengthening an edge, or maybe removing an edge and reinserting it somewhere else. We will introduce a number of sharp new surgery principles which allow one to control the eigenvalues of the Laplacian on a metric graph with any of the usual vertex conditions (natural, Dirichlet or delta). We will illustrate how these principles can be used to give new proofs and sharper versions of existing ``isoperimetric"-type eigenvalue estimates by sketching a result which interpolates between the theorems of Nicaise and Band--L\'evy for the first non-trivial eigenvalue of the Laplacian with natural vertex conditions. This is based on joint work with Gregory Berkolaiko, Pavel Kurasov and Delio Mugnolo.
Uniqueness and non-uniqueness of prescribed mass NLS ground states on metric graphs
Dr. Simone Dovetta, Università degli Studi di Roma "La Sapienza"
This talk addresses the problem of uniqueness of ground states of prescribed mass for the Nonlinear Schr\"odinger Energy with power nonlinearity on noncompact metric graphs with half--lines. We first show that, up to an at most countable set of masses, all ground states at given mass solve the same equation, that is the Lagrange multiplier appearing in the NLS equation is constant on the set of ground states of mass $\mu$. On the one hand, we apply this result to prove uniqueness of ground states on two specific families of noncompact graphs. On the other hand, we construct a graph that admits at least two ground states with the same mass having different Lagrange multipliers. This shows that the result for Lagrange multipliers is sharp in general, in the sense that one cannot get rid of the at most countable set of masses where it may fail without further assumptions. Our proofs are based on careful variational arguments and rearrangement techniques, and hold both for the subcritical regime $p\in (2,6)$ and in the critical case $p=6$. This is a joint work with Enrico Serra and Paolo Tilli.
Stability and asymptotic properties of dissipative equations coupled with ordinary differential equations
Prof. Serge Nicaise, Université Polytechnique Hauts-de-France
In this talk, we will present some stability results of a system corresponding to the coupling between a dissipative equation (set in an infinite dimensional space) and an ordinary differential equation. Namely we consider $U, P$ solution of the system \begin{equation}\label{1} \left\{\begin{array}{ll}U_t=\mathcal{A} U +M P, \hbox{ in } H,\\ P_t=BP+ N U, \hbox{ in } X,\\ U(0)=U_0,P(0)=P_0, \end{array}\right. \end{equation} where $\mathcal{A}$ is the generator of a $C_0$ semigroup in the Hilbert space $H$, $B$ is a bounded operator from another Hilbert space $X$, and $M$, $N$ are supposed to be bounded operators. Many problems from physics enter in this framework, let us mention dispersive medium models, generalized telegraph equations, Volterra integro-differential equations, and cascades of {ODE}-hyperbolic systems. The goal is to find sufficient (and necessary) conditions on the involved operators $\mathcal{A}$, $B$, $M$ and $N$ that garantee stability properties of system (\ref{1}), i.e., strong stability, exponential stability or polynomial one. We will illustrate our general results by an example of generalized telegraph equations set on networks.
Kinetic and macroscopic diffusion models for gas mixtures in the context of respiration
Dr. Bérénice GREC, University of Paris
In this talk, I will first discuss shortly the context of respiration, and in particular the need to describe accurately the diffusion of respiratory gases in the lower part of the lung. At the macroscopic level, diffusion processes for mixtures are often modelled using cross-diffusion models. In order both to determine the regime in which such models are valid, and to compute the binary diffusion coefficients, it is of particular interest to derive these models from a description at the mesoscopic level by means of kinetic equations. More precisely, we consider the Boltzmann equations for mixtures with general cross-sections (i.e. for any kind of molecules interactions), and obtain the so-called Maxwell-Stefan equations by performing a Hilbert asymptotic expansion at low Knudsen and Mach numbers. This allows us to compute the values of the Maxwell-Stefan diffusion coefficients with explicit formulae with respect to the cross-sections. We also justify the specific ansatz we use thanks to the so-called moment method. This is a joint work with Laurent Boudin and Vincent Pavan.
Flows in infinite networks
Prof. Marjeta Kramar Fijavž, University of Ljubljana
We consider linear transport processes in infinite metric graphs in the $L^{\infty}$-setting. We apply the theory of bi-continuous operator semigroups to obtain well-posedness of the problem under different assumptions on the velocities and for general stochastic matrices appearing in the boundary conditions.
Combinatorial Designs (MS - ID 16)
Combinatorics and Discrete Mathematics
Testing Arrays for Fault Localization
Prof. Charles Colbourn, Arizona State University
A {\sl separating hash family} ${\sf SHF}_\lambda(N; k,v,\{w_1,\dots,w_s\})$ is an $N \times k$ array on $v$ symbols, with the property that no matter how we choose disjoint sets $C_1, \dots, C_s$ of columns with $|C_i| = w_i$, there are at least $\lambda$ rows in which, for every $1 \leq i < j \leq s$, no entry in a column of $C_i$ equals that in a column of $C_j$. (That is, there are $\lambda$ rows in which sets $\{C_1,\dots,C_s\}$ are {\sl separated}.) Separating hash families have numerous applications in combinatorial cryptography and in the construction of various combinatorial arrays; typically, one only considers whether two symbols are the same or different. We instead employ symbols that have algebraic significance. We consider an ${\sf SHF}_\lambda(N; k,q^s,\{w_1,\dots,w_s\})$ whose symbols are column vectors from ${\mathbb F}_q^s$. The entry in row $r$ and column $c$ of the ${\sf SHF}$ is denoted by ${\bf v}_{r,c}$. Suppose that $C_1, \dots, C_s$ is a set of disjoint sets of columns. Row $r$ is {\sl covering} for $\{C_1, \dots, C_s\}$ if, whenever we choose $s$ columns $\{ \gamma_i \in C_i : 1 \leq i \leq s\}$, the $s \times s$ matrix $[ {\bf v}_{r,\gamma_1} \cdots {\bf v}_{r,\gamma_s} ]$ is nonsingular over ${\mathbb F}_q$. Then the ${\sf SHF}_\lambda(N; k,q^s,\{w_1,\dots,w_s\})$ is {\sl covering} if, for every way to choose $\{C_1, \dots, C_s\}$, there are at least $\lambda$ covering rows. We establish that covering separating hash families of type $1^t d^1$ give an effective construction for detecting arrays, which are useful in screening complex systems to find interactions among $t$ or fewer factors without being masked by $d$ or fewer other interactions. This connection easily accommodates outlier and missing responses in the screening. We explore asymptotic existence results and explicit constructions using finite geometries for covering separating hash families. We develop randomized and derandomized construction algorithms and discuss consequences for detecting arrays. This is joint work with Violet R. Syrotiuk (ASU).
A reduction of the spectrum problem for sun systems
Prof. Anita Pasotti, University of Brescia
A $k$-cycle with a pendant edge attached to each vertex is called a $k$-sun. When we approached the existence problem for $k$-sun systems of order $v$, complete solutions were known only for $k=3,4,5,6,8,10,14$ and for $k=2^t$. Here, we reduce this problem to the orders $v$ in the range $2k< v < 6k$ satisfying the obvious necessary conditions. Thanks to this result, we provide a complete solution whenever $k$ is an odd prime, and some partial results whenever $k$ is twice a prime. \begin{itemize} \item[1)] M. Buratti, A. Pasotti, T. Traetta, \emph{A reduction of the spectrum problem for odd sun systems and the prime case}, J. Combin. Des. \textbf{29} (2021), 5--37. \item[2)] A. Pasotti, T. Traetta, \emph{Even sun systems of the complete graph}, in preparation. \end{itemize}
Heffter Arrays and Biembbedings of Cycle Systems on Orientable Surfaces
Prof. E. Şule Yazıcı , Koç University
In this talk we will review the recent developments on square Heffter arrays, $H(n;k)$, and their applications on face $2$-colourable embeddings of the complete graph $K_{2nk+1}$ on an orientable surfaces. Square Heffter arrays, $H(n;k)$, are $n\times n$ arrays such that each row and each column contains $k$ filled cells, each row and column sum is divisible by $2nk+1$ and either $x$ or $-x$ appears in the array for each integer $1\leq x\leq nk$. Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face $2$-colourable embedding of the complete graph $K_{2nk+1}$ on an orientable surface, where for each colour, the faces give a $k$-cycle system. These necessary conditions pertain to cyclic orderings of the entries in each row and each column of the Heffter array and are: (1) for each row and each column the sequential partial sums determined by the cyclic ordering must be distinct modulo $2nk+1$; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. We construct Heffter arrays that satisfy condition (1) whenever (a) $k\equiv 0\ mod\ 4$; or (b) $n\equiv 1\ mod\ 4$ and $k\equiv 3\ mod\ 4$; or (c) $n\equiv 0\ mod\ 4$, $k\equiv 3\ mod\ 4$ and $n\gg k$. As a corollary to the above we obtain pairs of orthogonal $k$-cycle decompositions of $K_{2nk+1}$. Furthermore we study when these arrays satisfy condition (2). We show the existence of face $2$-colourable embeddings of cycle decompositions of the complete graph when $n\equiv 1\ mod\ 4$ and $k\equiv 3\ mod\ 4$, $n\gg k\geq 7$ (provided that when $n\equiv 0\ mod\ 3$ then $k\equiv 7\ mod\ 12$).
Bipartite 2-factorizations of complete multigraphs via layering
Dr. Mateja Sajna, University of Ottawa
Layering is in principle a simple method that allows us to obtain a type-specific 2-factorization of a complete multigraph (or complete multigraph minus a 1-factor) from existing 2-factorizations of complete multigraphs and complete multigraphs minus a 1-factor. This technique is particularly effective when constructing bipartite 2-factorizations; that is, 2-factorizations with all cycles of even length. In this talk, we shall give a thorough introduction to layering, and then describe new bipartite 2-factorizations of complete multigraphs obtained by layering. In particular, for complete multigraphs and bipartite 2-factors with no 2-cycles, we obtain a complete solution to the Oberwolfach Problem and an almost complete solution to the Hamilton-Waterloo Problem. This is joint work with Amin Bahmanian.
Strictly additive 2-designs
Dr. Anamari Nakic, Faculty of Electrical Engineering and Computing, University of Zagreb
This work draws inspiration from an interesting theory developed in [1]. A design $(V,{\cal B})$ is said to be {\it additive} if $V$ is a subset of an abelian group $G$ and the elements of any block $B\in{\cal B}$ sum up to zero. We propose to speak of a {\it strictly additive} design when $V$ coincides with $G$. Up to last year, apart from the obvious examples of the $2-(q^n,q,1)$ designs associated with the affine geometry AG$(n,q)$, all known strictly additive 2-designs had a quite ``big" $\lambda$. Very recently, a strictly additive $2-(81,6,2)$ design has been found in [3]. This design, besides being {\it simple} (the only design with these parameters previously known [2] has sixteen pairs of repeated blocks), has the property that every block is union of two parallel lines of AG$(4,3)$. In the attempt of getting other strictly additive designs with this property we found some infinite series of 2-designs whose parameter-sets are probably new. In this talk, besides presenting the above series, I will try to outline a proof that for every odd $k$, there are infinitely many values of $v$ for which a strictly additive $2-(v,k,1)$ design exists. \bigskip \noindent [1] A. Caggegi, G. Falcone, M. Pavone,\textit{ On the additivity of block designs}, J. Algebr. Comb. 45, 271--294 (2017). \noindent [2] H. Hanani, {\it Balanced incomplete block designs and related designs}, Discrete Math. 11 (1975), 255--369. \noindent [3] A. Nakic, \textit{The first example of a simple $2-(81,6,2)$ design}, Examples and Counterexamples 1 (2021).
A few new triplanes
Prof. Sanja Rukavina, University of Rijeka
An incidence structure ${\cal D}=({\cal P},{\cal B},I)$, with point set ${\cal P}$, block set ${\cal B}$ and incidence $I$ is a $t$-$(v,k,\lambda)$ design, if $|{\cal P}|=v$, every block $B \in {\cal B}$ is incident with precisely $k$ points, and every $t$ distinct points are together incident with precisely $\lambda$ blocks. We consider triplanes, i.e. symmetric block designs with $\lambda = 3$. Triplanes of order $12$, i.e. symmetric (71,15,3) designs, have the greatest number of points among all known triplanes and it is not known if a triplane $(v,k,3)$ exists for $v>71$. \\ In this talk, in addition to reviewing previously known results, we give the first example of a triplane of order 12 that doesn't admit an automorphism of order 3. \begin{thebibliography}{12} \bibitem{CDDDR} D. Crnkovi\'{c}, D. Dumi\v{c}i\'{c} Danilovi\'{c}, S. Rukavina, Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms, J. Algebra Comb. Discrete Struct. Appl. 3 (2016), 145--154. \bibitem{geocr} D. Crnkovi\'{c}, S. Rukavina, L. Sim\v{c}i\'{c}, On triplanes of order twelve admitting an automorphism of order six and their binary and ternary codes, Util. Math. 103 (2017), 23--40. \end{thebibliography}
Universal sequences and Euler tours in hypergraphs
Prof. Deryk Osthus, University of Birmingham
We show that a quasirandom $k$-uniform hypergraph $G$ has a tight Euler tour subject to the necessary condition that $k$ divides all vertex degrees. The case when $G$ is complete confirms a \$100 conjecture of Chung, Diaconis and Graham from 1989 on the existence of universal cycles for the $k$-subsets of an $n$-set. Our proof is based on random walks and the proof of the existence of $H$-designs (by Glock, K\"uhn, Lo and Osthus), i.e. decompositions of a complete hypergraph into copies of an arbitrary hypergraph $H$ (subject to divisibility conditions).
A lower bound on permutation codes of distance $n-1$
Dr. Peter Dukes, University of Victoria
A construction for mutually orthogonal latin squares (MOLS) inspired by pairwise balanced designs is shown to hold more generally for a class of permutation codes of length $n$ and minimum distance $n-1$. Using ingredients when $n$ equals a prime or a prime plus one, and applying a number sieve, we obtain a general lower bound $M(n,n-1) \ge n^{1.0797}$ on the size of such codes for large $n$. This represents a small improvement on the guarantee given from MOLS.
Factorizations of infinite graphs
Dr. Simone Costa, University of Brescia
Let $\mathcal{F}=\{F_{\alpha}: \alpha\in \mathcal{A}\}$ be a family of infinite graphs. The Factorization Problem $FP(\mathcal{F}, \Lambda)$ asks whether $\mathcal{F}$ can be realized as a factorization of a given infinite graph $\Lambda$, namely, whether there is a factorization $\mathcal{G}=\{\Gamma_{\alpha}: \alpha\in \mathcal{A}\}$ of $\Lambda$ such that each $\Gamma_{\alpha}$ is a copy of $F_{\alpha}$. Inspired by the results on regular $1$-factorizations of infinite complete graphs $[1]$ and on the resolvability of infinite designs $[4]$, we study this problem when $\Lambda$ is either the Rado graph $R$ or the complete graph $K_\aleph$ of infinite order $\aleph$. When $\mathcal{F}$ is a countable family, we show that $FP(\mathcal{F}, R)$ is solvable if and only if each graph in $\mathcal{F}$ has no finite dominating set. Generalizing the existence result of $[2]$, we also prove that $FP(\mathcal{F}, K_\aleph)$ admits a solution whenever the cardinality $\mathcal{F}$ coincides with the order and the domination numbers of its graphs. Finally, in the case of countable complete graphs, we show some non-existence results when the domination numbers of the graphs in $\mathcal{F}$ are finite. \vskip0.4cm \begin{large} \textbf{References} \end{large} \begin{enumerate}[{[1]}] \item S. Bonvicini and G. Mazzuoccolo. Abelian $1$-factorizations in infinite graphs, \textit{European J. Combin.}, 31 (2010), 1847--1852. \item S. Costa. A complete solution to the infinite Oberwolfach problem, \textit{J. Combin. Des.}, 28 (2020), 366--383. \item S. Costa and T. Traetta. Factorizing the Rado graph and infinite complete graphs, \textit{Submitted (arXiv:2103.11992)}. \item P.~Danziger, D.~Horsley and B.~S.~Webb. Resolvability of infinite designs, \textit{J. Combin. Theory Ser. A}, 123 (2014), 73--85. \end{enumerate}
An Evans-style result for block designs
Dr. Daniel Horsley, Monash University
A now-proven conjecture of Evans states that any partial latin square with at most $n-1$ filled cells can be completed to a latin square. This is sharp: there are uncompletable partial latin squares with $n$ filled cells. This talk will discuss the analogous problem for block designs. An \textit{$(n,k,1)$-design} is a collection of $k$-subsets (\textit{blocks}) of a set of $n$ \textit{points} such that each pair of points occur together in exactly one block. If this restriction is relaxed to require only that each pair of points occur together in at most one block we instead have a \textit{partial $(n,k,1)$-design}. I will outline a proof that any partial $(n,k,1)$-design with at most $\frac{n-1}{k-1}-k+1$ blocks is completable to a $(n,k,1)$-design provided that $n$ is sufficiently large and obeys the obvious necessary conditions for an $(n,k,1)$-design to exist. This result is sharp for all $k$. I will also mention some related results concerning edge decompositions of almost complete graphs into copies of $K_k$.
Characterizing isomorphism classes of Latin squares by fractal dimensions of image patterns
Dr. Raul M. Falcon, Universidad de Sevilla
Based on the construction of pseudo-random sequences arisen from a given Latin square, Dimitrova and Markovski [1] described in 2007 a graphical representation of quasigroups by means of fractal image patterns. The recognition and analysis of such patterns have recently arisen [2,3] as an efficient new approach for classifying Latin squares into isomorphism classes. This talk delves into this topic by focusing on the use of the differential box-counting method for determining the mean fractal dimension of the homogenized standard sets associated to these fractal image patterns. It constitutes a new Latin square isomorphism invariant which is analyzed in this talk for characterizing isomorphism classes of non-idempotent Latin squares in an efficient computational way. \medskip {\bf References}: \begin{enumerate} \item[1)] V. Dimitrova, S. Markovski, {\em Classification of quasigroups by image patterns}. In: Proceedings of the Fifth International Conference for Informatics and Information Technology, Bitola, Macedonia, 2007; 152--160. \item[2)] R. M. Falc\'on, {\em Recognition and analysis of image patterns based on Latin squares by means of Computational Algebraic Geometry}, Mathematics {\bf 9} (2021), paper 666, 26 pp. \item[3)] R. M. Falc\'on, V. \'Alvarez, F. Gudiel, {\em A Computational Algebraic Geometry approach to analyze pseudo-random sequences based on Latin squares}, Adv. Comput. Math. {\bf 45} (2019), 1769--1792. \end{enumerate}
On the existence of large set of partitioned incomplete Latin squares
Ms. Cong Shen, Nanjing Normal University
In this talk, we survey the existence of large sets of partitioned incomplete Latin squares (LSPILS). Algebraic and combinatorial methods are employed to construct the large sets of partitioned incomplete Latin squares of type $g^nu^1$. Furthermore, we prove that there exists a pair of orthogonal LSPILS$(1^pu^1)$s for any odd $u$ and some even values of $u$, where $p$ is a prime. Lastly, we propose some problems for further research.
Nov\'{a}k's conjecture on cyclic Steiner triple systems and its generalization
Prof. Tao Feng, Beijing Jiaotong University
Nov\'{a}k conjectured in 1974 that for any cyclic Steiner triple systems of order $v$ with $v\equiv 1\pmod{6}$, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. In this talk, we shall consider the generalization of this conjecture to cyclic $(v,k,\lambda)$-designs with $1 \leq \lambda \leq k-1$. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for all $v$, but we conjecture that it holds whenever $v$ is sufficiently large compared to $k$. We confirm that the generalization of the conjecture holds when $v$ is prime and $\lambda=1$ and also when $\lambda \leq (k-1)/2$ and $v$ is sufficiently large compared to $k$. As a corollary, we show that for any $k \geq 3$, with the possible exception of finitely many composite orders $v$, every cyclic $(v,k,1)$-design without short orbits is generated by a $(v,k,1)$-disjoint difference family.
On the classification of unitals on 28 points of low rank
Prof. Alfred Wassermann, University of Bayreuth
Unitals are combinatorial $2$-$(q^3 + 1, q + 1, 1)$ designs. In 1981, Brouwer constructed $138$ nonisomorphic unitals for $q=3$, i.e.~$2$-$(28, 4, 1)$ designs. He observed that the $2$-rank of the constructed unitals is at least $19$, where the $p$-rank of a design is defined as the rank of the incidence matrix between points and blocks of the design over the finite field GF($p$). In 1998, McGuire, Tonchev and Ward proved that indeed the $2$-rank of a unital on $28$ points is between $19$ and $27$ and that there is a unique $2$-$(28, 4, 1)$ design, the Ree unital $R(3)$, with $2$-rank 19. In the same year, Jaffe and Tonchev showed that there is no unital on $28$ points of $2$-rank $20$ and there are exactly $4$ isomorphism classes of unitals of rank $21$. Here, we present the complete classification by computer of unitals of $2$-rank $22$, $23$ and $24$. There are $12$ isomorphism classes of unitals of $2$-rank $22$, $78$ isomorphism classes of unitals of $2$-rank $23$, and $298$ isomorphism classes of unitals of $2$-rank $24$.
Dual incidences and $t$-designs in elementary abelian groups
Dr. Kristijan Tabak, Rochester Institute of Technology - Croatia
Let $q$ be a prime and $E_{q^n}$ is an elementary abelian group of order $q^n.$ Let $\mathcal{H}$ be a collection of some subgroups of $E_{q^n}$ of order $q^k$. A pair $(E_{q^n}, \mathcal{H})$ is a $t-(n,k,\lambda)_q$ design if every subgroup of $E_{q^n}$ of order $q^t$ is contained in exactly $\lambda$ groups from $\mathcal{H}.$ This definition corresponds to the classical definition of a $q$-analog design. We introduce two incidence structures denoted by $\mathcal{D}_{max}$ and $\mathcal{D}_{min}$ with $\mathcal{H}$ as set of points. The blocks of $\mathcal{D}_{max}$ are labeled by maximal subgroups of $E_{q^n}$, while the blocks of $\mathcal{D}_{min}$ are labeled by groups of order $q.$ We fully describe a duality between $\mathcal{D}_{max}$ and $\mathcal{D}_{min}$ by proving some identities over group rings. The proven results are used to provide a full description of incidence matrices of $\mathcal{D}_{max}$ and $\mathcal{D}_{min}$ and their mutual dependance.
Resolving sets and identifying codes in finite geometries
Prof. György Kiss, Eötvös Loránd University, Budapest & University of Primorska, Koper
Let $\Gamma=(V,E)$ be a finite, simple, undirected graph. A vertex $v \in V$ is resolved by $S=\{v_1,\ldots,v_n\}\subset V$ if the list of distances $(d(v,v_1),d(v,v_2),\ldots,$ $d(v,v_n))$ is unique. $S$ is a \emph{resolving set} for $\Gamma$ if it resolves all the elements of $V$. A subset $D\subset V$ is a \emph{dominating set} if each vertex is either in $D$ or adjacent to a vertex in $D.$ A vertex $s$ \emph{separates} $u$ and $v$ if exactly one of $u$ and $v$ is in $N[s].$ A subset $S\subset V$ is a \emph{separating set} if it separates every pair of vertices of $G.$ Finally, a subset $C\subset V$ is an \emph{identifying code} for $V$ if it is both a dominating and separating set. \smallskip In this talk resolving sets and identifying codes for graphs arising from finite geometries (e.g. Levi graphs of projective and affine planes and spaces, generalized quadrangles) are considered. We present several constructions and give estimates on the sizes of these objects.
Complex Analysis and Geometry (MS - ID 17)
Algebraic and Complex Geometry
Seiberg-Witten equations and pseudoholomorphic curves
Prof. Armen Sergeev, Steklov Mathematical Institute
%\documentclass{article} %\begin{document} \begin{center}SEIBERG--WITTEN EQUATIONS AND PSEUDOHOLOMORPHIC CURVES\\ Armen SERGEEV\end{center} %\address{Steklov Mathematical Institute, Moscow} \bigskip % \maketitle Seiberg--Witten equations (SW-equations for short) were proposed in order to produce a new kind of invariant for smooth 4-dimensional manifolds. These equations, opposite to the conformally invariant Yang--Mills equations, are not invariant under scale transformations. So to draw a useful information from these equations one should plug the scale parameter $\lambda$ into them and take the limit $\lambda\to\infty$. If we consider such limit in the case of 4-dimensional symplectic manifolds solutions of SW-equations will concentrate in a neighborhood of some pseudoholomorphic curve (more precisely, pseudoholomorphic divisor) while SW-equations reduce to some vortex equations in normal planes of the curve. The vortex equations are in fact static Ginzburg--Landau equations known in the superconductivity theory. So solutions of the limiting adiabatic SW-equations are given by families of vortices in the complex plane parameterized by the point z running along the limiting pseudoholomorphic curve. This parameter plays the role of complex time while the adiabatic SW-equations coincide with a nonlinear $\bar\partial$-equation with respect to this parameter.
Images of CR manifolds
Prof. Jiri Lebl, Oklahoma State University
We study CR singular submanifolds that have a removable CR singularity, that is, manifolds that are singular, but whose CR structure extends through the singularity. Bishop surfaces are trivial examples, but in higher CR dimension, generically the CR singularity is not removable. In particular, we study real codimension 2 submanifolds in ${\mathbb{C}}^3$. This is joint work with Alan Noell and Sivaguru Ravisankar.
Equivalence of neighborhoods of embedded compact complex manifolds and higher codimension foliations
Mr. Laurent Stolovitch, CNRS-Université Côte d'Azur
We consider an embedded $n$-dimensional compact complex manifold in $n+d$ dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert's formal principle program. We will give conditions ensuring that a neighborhood of $C_n$ in $M_{n+d}$ is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold's result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in $M_{n+d}$ having $C_n$ as a compact leaf, extending Ueda's theory to the high codimension case. Both problems appear as a kind linearization problem involving {\it small divisors condition} arising from solutions to their cohomological equations.
Computational aspects of commutative and noncommutative positive polynomials (MS - ID 77)
Algebra
Non-commutative polynomial in quantum physics
Prof. Antonio Acin, ICFO
Quantum physics is a natural source of problems involving the optimization of polynomials over non-commuting variables. The talk first introduces some of the most relevant problems, explains how they can be tackled using semi-definite programming hierarchies and concludes with some of the open problems. References: [1] S. Pironio, M. Navascues, A. Acin, Convergent relaxations of polynomial optimization problems with non-commuting variables, SIAM J. Optim. 20 (5), 2157 (2010). [2] E. Wolfe, A. Pozas-Kerstjens, M. Grinberg, D. Rosset, A. Acín, M. Navascues, Quantum Inflation: A General Approach to Quantum Causal Compatibility, to appear in Phys. Rev. X. [3] A. Pozas-Kerstjens, R. Rabelo, L. Rudnicki, R. Chaves, D. Cavalcanti, M. Navascues, A. Acín, Bounding the sets of classical and quantum correlations in networks, Phys. Rev. Lett. 123, 140503 (2019). [4] F. Baccari, C. Gogolin, P. Wittek, A. Acín, Verifying the output of quantum optimizers with ground-state energy lower bounds, Phys. Rev. Research 2, 043163 (2020).
Quantum de Finetti theorems and Reznick’s Positivstellensatz
Ion Nechita, CNRS, Toulouse University
We present a proof of Reznick’s quantitative Positivstellensatz using ideas from Quantum Information Theory. This result gives tractable conditions for a positive polynomial to be written as a sum of squares. We relate such results to de Finetti theorems in Quantum Information.
Optimizing conditional entropies for quantum correlations
Dr. Omar Fawzi, Inria
The rates of quantum cryptographic protocols are usually expressed in terms of a conditional entropy minimized over a certain set of quantum states. In the so-called device-independent setting, the minimization is over all the quantum states of arbitrary dimension jointly held by the adversary and the parties that are consistent with the statistics that are seen by the parties. We introduce new quantum divergences and use techniques from noncommutative polynomial optimization to approximate such entropic quantities. Based on https://arxiv.org/abs/2007.12575
Optimization over trace polynomials
Dr. Victor Magron, LAAS CNRS
Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascu\'es and Ac\'in scheme [New J. Phys., 2008] for optimization of noncommutative polynomials. The Gelfand-Naimark-Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.
Efficient noncommutative polynomial optimization by exploiting sparsity
Dr. Jie Wang, LAAS-CNRS
Many problems arising from quantum information can be modelled as noncommutative polynomial optimization problems. The moment-SOHS hierarchy approximates the optimum of noncommutative polynomial optimization problems by solving a sequence of semidefinite programming relaxations with increasing sizes. In this talk, I will show how to exploit various sparsity patterns encoded in the problem data to improve scalability of the moment-SOHS hierarchy for eigenvalue and trace optimization problems.
Quantum polynomial optimisation problems for dimension $d$ variables, with symmetries
Dr. Marc-Olivier Renou, ICFO - The Institute of Photonic Sciences
Quantum Information Theory (QIT) involves quantum states and measurements, mathematically represented as non-commutative positive operators. A typical problem in QIT is to find the minimum of a polynomial expression in these operators. For instance, finding the maximum violation of a Bell inequality, or the ground state energy of a Hamiltonian are polynomial optimisation problems in non-commuting variables. The NPA hierarchy [New J. Phys. 10, 073013 (2008)], which can be viewed as the “eigenvalue” version of Lasserre’s hierarchy, provides a converging hierarchy of SDP relaxation of a non-commutative polynomial optimisation problem involving variables of unbounded dimension. This hierarchy converges, it is one of QIT main technical tool. Importantly, some QIT problems concern operators of bounded dimension $d$. The NPA hierarchy was extended into the NV hierarchy [Phys. Rev. Lett.115, 020501(2015)] to tackle this case. In this method, one first sample at random many dimension $d$ operators satisfying the constraint, and compute the associated moment matrix. This first step discovers the moment matrix vector space, over which the relaxed SDP problem is solved in a second step. In this talk, we will first review this method. Then, based on [Phys. Rev. Lett. 122, 070501], we will show how one can reduce the computational requirements by several orders of magnitude, exploiting the eventual symmetries present in the optimization problem.
Sum-of-Squares proofs of logarithmic Sobolev inequalities on finite Markov chains
Dr. Hamza Fawzi, University of Cambridge
Logarithmic Sobolev inequalities play an important role in understanding the mixing times of Markov chains on finite state spaces. It is typically not easy to determine, or indeed approximate, the optimal constant for which such inequalities hold. In this paper, we describe a semidefinite programming relaxation for the logarithmic Sobolev constant of a finite Markov chain. This relaxation gives certified lower bounds on the logarithmic Sobolev constant. Numerical experiments show that the solution to this relaxation is often very close to the true constant. Finally, we use this relaxation to obtain a sum-of-squares proof that the logarithmic Sobolev constant is equal to half the Poincaré constant for the specific case of a simple random walk on the odd n-cycle, with n in {5,7,…,21}. Previously this was known only for n=5 and even n.
Quantum Isomorphism of Graphs: an Overview
Dr. Laura Mančinska, University of Copenhagen
In this talk I will introduce a quantum version of graph isomorphism. Our point of departure will be an interactive protocol (nonlocal game) where two provers try to convince a verifier that two graphs are isomorphic. Allowing provers to take advantage of shared quantum resources will then allow us to define quantum isomorphism as the ability of quantum players to win the corresponding game with certainty. We will see that quantum isomorphism can be naturally reformulated in the languages of quantum groups and counting complexity. In particular, we will see that two graphs are quantum isomorphic if and only if they have the same homomorphism counts from all planar graphs.
Positive maps and trace polynomials from the symmetric group
Dr. Felix Huber, Jagiellonian University
With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials $X_{\alpha_1} \cdots X_{\alpha_r}$ and their traces $\operatorname{tr}(X_{\alpha_1} \cdots X_{\alpha_r})$. Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley-Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.
Semidefinite relaxations in non-convex spaces
Dr. Alex Pozas-Kerstjens, Universidad Complutense de Madrid
In analogy with Lassere's and Parillo's hierarchies of semidefinite relaxations for polynomial optimization problems, a semidefinite hierarchy exisits for non-commutative polynomial optimization problems of the form \begin{equation*} \begin{split} p^* = &\underset{(\mathcal{H},\phi,X)}{\text{min}}\quad \langle \phi,p(X) \phi\rangle \\ &\,\,\,\,\text{s.t.}\quad\quad q_i(X)\succeq 0 \end{split}, \end{equation*} where $\phi$ is a normalized vector in the Hilbert space $\mathcal{H}$, $X=(X_1,\dots,X_n)$ are the non-commuting variables in the problem, and $p(X)$ and $q_i(X)$ are polynomials in the variables $X$. This hierarchy, known as the Navascu\'es-Pironio-Ac\'in hierarchy, has encountered important applications in physics, where it has become a central tool in quantum information theory and in the certification of quantum phenomena. Its success has motivated its application, within quantum information theory, in more complex scenarios where the relevant search spaces are not convex, and thus the characterizing constraints are in conflict with semidefinite formulations. The paradigmatic example is optimizing over probability distributions obtained by parties performing measurements on quantum systems where not all parties receive a share from every system available. The most illustrating consequence is that, in certain situations, marginalization over a selected number of parties makes the resulting probability distribution to factorize. In this talk we address the problem of non-commutative polynomial optimization in non-convex search spaces and present two means of providing monotonically increasing lower bounds on its solution, that retain the characteristic that the newly formulated problems remain being semidefinite programs. The first directly addresses factorization-type constraints by adding auxiliary commuting variables that allow to encode (relaxations of) polynomial trace constraints, while the second considers multiple copies of the problem variables and constrains them by requiring the satisfaction of invariance under suitable permutations. In addition to presenting the methods and exemplifying their applicability in simple situations, we highlight the fundamental questions that remain open, mostly regarding to their convergence.
Configurations (MS - ID 81)
Combinatorics and Discrete Mathematics
Connected $(n_{k})$ configurations exist for almost all $n$
Dr. Leah Berman, University of Alaska Fairbanks
A geometric $(n_{k})$ configuration is a collection of points and straight lines, typically in the Euclidean plane, so that each line passes through $k$ of the points and each of the points lies on $k$ of the lines. In a series of papers, Branko Gr\"unbaum showed that geometric $(n_{4})$ configurations exist for all $n \geq 24$, using a series of geometric constructions later called the ``Gr\"unbaum Calculus''. In this talk, we will show that for each $k > 4$, there exists an integer $N_{k}$ so that for \emph{all} $n \geq N_{k}$, there exists at least one $(n_{k})$ configuration, by generalizing the Gr\"unbaum Calculus operations to produce more highly incident configurations. This is joint work with G\'abor G\'evay and Toma\v{z} Pisanski.
Barycentric configurations in real space
Prof. Hendrik Van Maldeghem, Ghent University
Barycentric configurations are configurations with three points per line in real projective space such that (homogeneous) coordinates exist with the property that, for each line, the sum of the coordinate tuples of the three points on that line is the zero tuple. Such configurations turn up naturally and we develop some theory about them. In particular there exist universal barycentric embeddings and a general construction method if the geometry is self-polar. We apply these results to the Biggs-Smith geometry on 102 points, providing a (new) geometric construction of the Biggs-Smith graph making the full automorphism group apparent. We also mention a connection with ovoids.
Transitions between configurations
Prof. Gábor Gévay, University of Szeged
In this talk we briefly review (without aiming at completeness) various procedures by means of which new configurations can be obtained from old configurations. Among them, there are binary operations, like e.g.\ the \emph{Cartesian product} and the \emph{incidence sum}. Several other operations are collectively called the \emph{Gr\"unbaum calculus}. The \emph{incidence switch} operation can be defined on the level of incidence graphs (also called Levi graphs) of configurations, and in some cases it is in close connection with realization problems of configurations as well as with incidence theorems. Considering point-line, point-circle and point-conic configurations, there are interesting ad hoc constructions by which from a configuration of one of these geometric types another one can be derived, thus realizing \emph{transitions} between configurations in a very general sense. We also present interesting examples and applications. The most recent results mentioned in this talk are based on joint work with Toma\v{z} Pisanski and Leah Wrenn Berman.
Taxonomy of Three-Qubit Doilies
Dr. Metod Saniga, Slovak Academy of Sciences
We study doilies (i.\,e., $W(3,2)$'s) living in $W(5,2)$, when the points of the latter space are parametrized by canonical three-fold products of Pauli matrices and the associated identity matrix (i.\,e., by three-qubit observables). Key characteristics of such a doily are: the number of its negative lines, distribution of types of observables, character of the geometric hyperplane the doily shares with the distinguished (non-singular) quadric of $W(5,2)$ and the structure of its Veldkamp space. $W(5,2)$ is endowed with 90 negative lines of two types and its 1344 doilies fall into 13 types. 279 out of 480 doilies with three negative lines are composite, i.\,e. they all originate from the two-qubit doily by selecting in the latter a geometric hyperplane and formally adding to each two-qubit observable, at the same position, the identity matrix if an observable lies on the hyperplane and the same Pauli matrix for any other observable. Further, given a doily and any of its geometric hyperplanes, there are other three doilies possessing the same hyperplane. There is also a particular type of doilies a representative of which features a point each line through which is negative.
Hirzebruch-type inequalities and extreme point-line configurations
Prof. Piotr Pokora, Pedagogical University of Cracow
In my talk I would like to report on very recent developments devoted to extreme point-line configurations from an algebraic perspective of Hirzebruch-type inequalities. We recall some inequalities, especially the most powerful variant based on orbifolds, and then I will report on extreme combinatorial problems for which Hirzebruch-type inequalities played a decisive role. Time permitting, we will present a short proof of the Weak Dirac Conjecture.
Strongly regular configurations
Prof. Vedran Krčadinac, University of Zagreb Faculty of Science
We report on the recent study~[1] of combinatorial configurations with the associated point and line graphs being strongly regular. A prominent family of such configurations are the partial geometries, introduced by R.~C.~Bose in~1963. We focus on such configurations that are not partial geometries nor known generalisations such as semipartial geometries and strongly regular $(\alpha,\beta)$-geometries, neither are they elliptic semiplanes of P.~Dembowski. Several families are constructed, necessary existence conditions are proved, and a table of feasible parameters with at most $200$ points is presented. \begin{enumerate}[{[1]}] \item M.~Abreu, M.~Funk, V.~Kr\v{c}adinac, D.~Labbate, \emph{Strongly regular configurations}, preprint, 2021. \verb|https://arxiv.org/abs/2104.04880| \end{enumerate}
Configurations from strong deficient difference sets
Dr. Marién Abreu, Università degli Studi della Basilicata
In [1] we have studied combinatorial configurations with the associated point and line graphs being strongly regular, which we call \emph{strongly regular configurations}. In the talk ``Strongly regular configurations'' of this minisymposium, Vedran Kr\v{c}adinac will present existing known families of strongly regular configurations; constructions of several other families; necessary existence conditions and a table of feasible parameters with at most $200$ points. Let $G$ be a group of order $v$. A subset $D \subset G$ of size $k$ is a \emph{deficient difference set} the left differences $d^{-1}_1d_2$ are all distinct. Considering the elements of $G$ as points and the\emph{ development} $devD = \{gD | g \in G\}$ as lines a symmetric $(v_k)$ configuration is obtained and it has $G$ as its automorphism group acting regularly on the points and lines. Let $\Delta(D) = \{d^{-1}_1d_2 | d_1, d_2 \in D, d_1 \ne d_2\}$ be the set of left differences of $D$. For a group element $x \in G \setminus \{1\}$, denote by $n(x) = |\Delta(D) \cap x\Delta(D)|$. If $n(x) = \lambda$ for every $x \in \Delta(D)$, and $n(x) = \mu$ for every $x \notin \Delta(D)$, $D$ is said to be a \emph{strong deficient difference set} ($SDDS$) for $(v_k; \lambda, \mu)$. Here, we present one of the new families of strongly regular configurations constructed in [1], with parameters different from semipartial geometries and arising from strong deficient difference sets, as well as two examples arising from Hall's plane and its dual. Moreover, from the exhaustive search performed in groups of order $v \le 200$ further four examples corresponding to strong deficient difference sets, but not in the previous families, are obtained. \medskip \noindent [1] M.~Abreu, M.~Funk, V.~Kr\v{c}adinac, D.~Labbate, Strongly regular configurations, preprint, 2021. https://arxiv.org/abs/2104.04880
Highly symmetric configurations
Dr. Jurij Kovič, UP FAMNIT Koper and IMFM Ljubljana
In the talk some general methods and special techniques for the construction of highly symmetric configurations will be presented. As an application of these theoretical principles, some examples of highly symmetric spatial geometric configurations of points and lines with the symmetry of Platonic solids will be given.
4-lateral matroids induced by $n_3$-configurations
Dr. Michael Raney, Georgetown University
A \emph{4-lateral matroid} induced by an $n_3$-configuration is a rank-4 matroid whose ground set consists of the blocks (lines) of the configuration, and for which any 4 elements of the ground set are independent if and only if they do not determine a 4-lateral within the configuration. We characterize the $n_3$-configurations which induce 4-lateral matroids, and provide examples of small $n_3$-configurations which do so.
Convex bodies - approximation and sections (MS - ID 61)
Analysis and its Applications
On the complex hypothesis of Banach
Prof. Luis Montejano, Instituto de Matematicas
The following is known as the geometric hypothesis of Banach: let $V$ be an m-dimensional Banach space (over the real or the complex numbers) with unit ball $B$ and suppose all $n$-dimensional subspaces of $V$ are isometric (all the n-sections of B are affinely equivalent). In 1932, Banach conjectured that under this hypothesis $V$ is a Hilbert space (the boundary of $B$ is an ellipsoid). Gromow proved in 1967 that the conjecture is true for $n$=even and Dvoretzky and V. Milman derived the same conclusion under the hypothesis $n$=infinity. We prove this conjecture for $n=4k+1$, with the possible exception of $V$ a real Banach space and $n=133$. [G.Bor, L.Hernandez-Lamoneda, V. Jim\'enez and L. Montejano. To appear Geometry $\&$ Topology] for the real case and [J. Bracho, L. Montejano, submitted to J. of Convex Analysis] for the complex case. The ingredients of the proof are classical homotopic theory, irreducible representations of the orthogonal group and convex geometry. For the complex case, suppose B is a convex body contained in complex space $C^{n+1}$, with the property that all its complex n-sections through the origin are complex affinity equivalent to a fixed complex n-dimensional body $K$. Studying the topology of the complex fibre bundle $SU(n) -> SU(n-1)-> S^{2n+1}$, it is possible to prove that if $n$=even, then $K$ must be a ball and using homotopical properties of the irreducible representations we prove that if $n=4+1$ then $K$ must be a body of revolution. Finally, we prove, using convex geometry and topology that, if this is the case, then there must be a section of $B$ which is an complex ellipsoid and consequently $B$ must be also a complex ellipsoid.
The $L_p$ Minkowski problem and polytopal approximation
Prof. Karoly Boroczky, Alfred Renyi Institute of Mathematics
The $L_p$ Minkowski problem , a Monge-Ampere type equation on the sphere, is a recent version of the classical Minkowski problem. I will review cases when one can reduce the equation to properties of polytopes, and then use polytopal approximation to solve the PDE in general.
The Golden ratio and high dimensional mean inequalities
Dr. Bernardo González Merino, Universidad de Murcia
The classical inequalities between means state that \begin{equation}\label{eq:mean_inequalities} \min\{a,b\}\leq\left(\frac{a^{-1}+b^{-1}}{2}\right)^{-1}\leq \sqrt{ab}\leq\frac{a+b}{2}\leq\max\{a,b\}, \end{equation} for any $a,b>0$, with equality if and only if $a=b$. One can naturally extend \eqref{eq:mean_inequalities} considering means of $n$-dimensional compact, convex sets. Notice that means of $K$ and $-K$ are commonly known as symmetrizations of $K$. In this context, we will show that in even dimensions, if $K$ has a large Minkowski asymmetry, then the corresponding inequalities between the symmetrizations of $K$ can no longer be optimal. Especially in the planar case, we compute that the range of asymmetries of $K$ for which the inequalities between the symmetrizations of $K$ can be optimal is $[1,\phi]$, where $\phi$ is the Golden ratio. Indeed, we introduce the Golden House, (up to linear transformations) the only Minkowski centered set with asymmetry $\phi$ such that the symmetrizations of $K$ are successively optimally contained in each other.
A solution to some problems of Conway and Guy on monostable polyhedra
Dr. Zsolt Langi, Budapest University of Technology
A convex polyhedron is called monostable if it can rest in stable position only on one of its faces. In this talk we investigate three questions of Conway, regarding monostable polyhedra, from the open problem book of Croft, Falconer and Guy (Unsolved Problems in Geometry, Springer, New York, 1991), which first appeared in the literature in a 1969 paper. In this talk we answer two of these problems. The main tool of our proof is a general theorem describing approximations of smooth convex bodies by convex polyhedra in terms of their static equilibrium points.
Current topics in Complex Analysis (MS - ID 32)
Analysis and its Applications
Recent studies on the image domain of starlike functions
Ms. Sahsene Altinkaya, Beykent University
The geometric properties of image domain of starlike functions is of prominent significance to present a comprehensive study on starlike functions. For this purpose, by using a relation of subordination, we investigate a family of starlike functions in the open unit disc \begin{eqnarray*} \mathbf{U}=\left\{ z\in \mathbb{C}:\left\vert z\right\vert <1\right\} . \end{eqnarray*} Subsequently, we discuss some interesting geometric properties, radius problems, general coefficients for this class. Further, we point out several recent studies related to image domain of starlike functions.
Differential Geometry: Old and New (MS - ID 15)
Differential Geometry and Applications
Non-holonomic equations for sub-Riemannian extremals and metrizable parabolic geometries
Prof. Jan Slovak, Masaryk University
I will report several recent attempts linking sub-Riemannian geometries to the rich geometry of filtered manifolds, particularly the parabolic ones. After touching on some relation between the canonical Cartan geometries, I shall present an approach to sub-Riemannian extremals motivated by the tractor calculus. Finally, I will explore some implications of the BGG machinery to the (sub-Riemannian) metrizability of parabolic geometries. All that will be based on joined work with D. Alekseevsky, A. Medvedev, R. Gover, D. Calderbank, V. Soucek.
Contact CR submanifolds in odd-dimensional spheres: new examples
Prof. Marian Ioan MUNTEANU, University Alexandru Ioan Cuza of Iasi
The notion of $CR$-submanifold in K\"{a}hler manifolds was introduced by A. Bejancu in 70's, with the aim of unifying two existing notions, namely complex and totally real submanifolds in K\"{a}hler manifolds. Since then, the topic was rapidly developed, mainly in two directions: $\bullet$ Study $CR$-submanifolds in other almost Hermitian manifolds. $\bullet$ Find the odd dimensional analogue of $CR$-submanifolds. Thus, the notion of semi-invariant submanifold in Sasakian manifolds was introduced. Later on, the name was changed to contact CR-submanifolds. A huge interest in the last 20 years was focused on the study of $CR$-submanifolds of the nearly K\"{a}hler six dimensional unit sphere. Interesting and important properties of such submanifolds were discovered, for example, by M. Antic, M. Djoric, F. Dillen, L. Verstraelen, L. Vrancken. As the odd dimensional counterpart, contact $CR$-submanifolds in odd dimensional spheres were, recently, intensively studied. In this talk we focus on those proper contact $CR$-submanifolds, which are as closed as possible to totally geodesic ones in the seven dimensional spheres endowed with its canonical structure of a Sasakian space form. We give a complete classification for such a submanifold having dimension $4$ and describe the techniques of the study. We present also some very recent developments concerning dimension $5$ and $6$ and propose further problems in this direction. This presentation is based on some papers in collaboration with M. Djoric and L. Vrancken. {\bf Keywords:} (contact) CR-submanifold, Sasakian manifolds, minimal submanifolds, mixed and nearly totally geodesic CR-submanifolds {\bf References:} [1] M. Djoric, M.I. Munteanu, L. Vrancken: Four-dimensional contact $CR$-submanifolds in ${\mathbb{S}}^7(1)$, Math. Nachr. {\bf 290} (2017) 16, 2585--2596. \smallskip [2] M. Djoric, M.I. Munteanu, On certain contact $CR$-submanifolds in ${\mathbb{S}}^7$, Contemporary Mathematics, {\it Geometry of submanifolds}, Eds. (J. van der Veken et al.) {\bf 756} (2020) 111-120. \smallskip [3] M. Djoric, M.I. Munteanu, Five-dimensional contact $CR$-submanifolds in ${\mathbb{S}}^7(1)$, Mathematics, Special Issue {\it Riemannian Geometry of Submanifolds}, Guest Editor: Luc Vrancken, {\bf 8} (2020) 8, art. 1278.
On the Bishop frame of a partially null curve in Minkowski spacetime
Prof. Emilija Nešović, University of Kragujevac Faculty of Science
The Bishop frame $\{T,N_1,N_2\}$ (relatively parallel adapted frame) of a regular curve in Euclidean space ${E}^{3}$ contains the tangent vector field $T$ of the curve and two relatively parallel vector fields $N_1$ and $N_2$ whose derivatives in arc length parameter $s$ make minimal rotations along the curve. In Minkowski spaces $E^3_1$ and $E^4_1$, the Bishop frame of a non-null curve and a null Cartan curve has analogous property. In this talk, we present a method for obtaining the Bishop frame (rotation minimizing frame) of a partially null curve $\alpha$ lying in the lightlike hyperplane of Minkowski spacetime. We show that $\alpha$ has two possible Bishop frames, one of which coincides with its Frenet frame. By using spacetime geometric algebra, we derive the Darboux bivectors of Frenet and Bishop frame and give geometric interpretation of the Frenet and the Bishop curvatures in terms of areas obtained by projecting the Darboux bivector onto a spacelike or a lightlike plane.
$J$-trajectories in $\mathrm{Sol}_0^4$
Prof. Zlatko Erjavec, University of Zagreb
$J$-trajectories are arc length parameterized curves in almost Hermitian manifold which satisfy the equation $\nabla_{\dot{\gamma}}\dot{\gamma}=q J \dot{\gamma}$. $J$-trajectories are 4-dim analogon of 3-dim magnetic trajectories, curves which satisfy the Lorentz equation $\nabla_{\dot{\gamma}}\dot{\gamma}=q \phi \dot{\gamma}.$ In this talk $J$-trajectories in the 4-dimensional solvable Lie group $\mathrm{Sol}_0^4$ are considered. Moreover, the first and the second curvature of a non-geodesic $J$-trajectory in an arbitrary 4-dimensional LCK manifold whose anti Lee field has constant length are examined. In particular, the curvatures of non-geodesic $J$-trajectories in $\mathrm{Sol}_0^4$ are characterized.
Characterization of manifolds of constant curvature by spherical curves and ruled surfaces
Dr. Luiz C. B. da Silva, Weizmann Institute of Science
Space forms, i.e., Riemannian manifolds of constant sectional curvature, play a prominent role in geometry and an important problem consists of finding properties that characterize them. In this talk, we report results from [1], where we show that the validity of some theorems concerning curves and surfaces can be used for this purpose. For example, it is known that the so-called rotation minimizing (RM) frames allow for a characterization of geodesic spherical curves in Euclidean, hyperbolic, and spherical spaces through a linear equation involving the coefficients that dictate the RM frame motion [2]. Here, we shall prove the converse, i.e., if all geodesic spherical curves on a manifold are characterized by a certain linear equation, then all the geodesic spheres with a sufficiently small radius are totally umbilical, and consequently, the ambient manifold is a space form. (We also present an alternative proof, in terms of RM frames, for space forms as the only manifolds where all geodesic spheres are totally umbilical [3].) In addition, we furnish two other characterizations in terms of (i) an inequality involving the mean curvature of a geodesic sphere and the curvature function of their curves and (ii) the vanishing of the total torsion of closed spherical curves in the case of 3d manifolds. (These are the converse of previous results [4].) Finally, we introduce ruled surfaces and show that if all extrinsically flat surfaces in a 3d manifold are ruled, then the manifold is a space form. \newline \newline {[1] Da Silva, L.C.B. and Da Silva, J.D.: ``Characterization of manifolds of constant curvature by spherical curves". Annali di Matematica \textbf{199}, 217 (2020); Da Silva, L.C.B. and Da Silva, J.D.: ``Ruled and extrinsically flat surfaces in three-dimensional manifolds of constant curvature". Unpublished manuscript 2021. \newline [2] Bishop, R.L.: ``There is more than one way to frame a curve". Am. Math. Mon. \textbf{82}, 246 (1975); Da Silva, L.C.B. and Da Silva, J.D.: ``Characterization of curves that lie on a geodesic sphere or on a totally geodesic hypersurface in a hyperbolic space or in a sphere". Mediterr. J. Math. \textbf{15}, 70 (2018) \newline [3] Kulkarni, R.S.: ``A finite version of Schur’s theorem". Proc. Am. Math. Soc. \textbf{53}, 440 (1975); Vanhecke, L. and Willmore, T.J.: ``Jacobi fields and geodesic spheres". Proc. R. Soc. Edinb. A \textbf{82}, 233 (1979); Chen, B.Y. and Vanhecke, L.: ``Differential geometry of geodesic spheres". J. Reine Angew. Math. \textbf{325}, 28 (1981). \newline [4] Baek, J., Kim, D.S., and Kim, Y.H.: ``A characterization of the unit sphere". Am. Math. Mon. \textbf{110}, 830 (2003); Pansonato, C.C. and Costa, S.I.R., ``Total torsion of curves in three-dimensional manifolds". Geom. Dedicata \textbf{136}, 111 (2008).}
Differential equations, dynamical systems and applications (MS - ID 52)
Analysis and its Applications
A boundary value problem for system of differential equations with piecewise-constant argument of generalized type
Prof. Anar Assanova, Institute of mathematics and mathematical modeling
In present communication, at the interval $[0, T]$ we consider the two-point boundary value problem for the system of differential equations with piecewise-constant argument of generalized type $$ \frac{dx}{dt}= A(t)x(t) + D(t)x(\gamma(t)) + f(t), \eqno (1)$$ $$ Bx(0) + C x(T)=d, \qquad d \in R^n, \eqno (2)$$ where $x(t)=col(x_1(t),x_2(t),...,x_n(t))$ is unknown vector function, the $(n\times n)$ matrices $A(t)$, $D(t)$ and $n$ vector function $f(t)$ are continuous on $ [0,T]$; $\gamma(t)=\zeta_j\ $ if $\ t\in [\theta_j, \theta_{j+1})$, $\ j=\overline{0,N-1}$; $\ \theta_j \leq \zeta_j\leq \theta_{j+1}\ $ for all $\ j=0,1,...,N-1$; $ 0=\theta_0<\theta_1 <...<\theta_{N-1}<\theta_N=T$; $ B$, $C$ are constant matrices, and $d$ is constant vector. A solution to problem (1), (2) is a vector function $ x(t)$ is continuously differentiable on $ [0,T]$, it satisfies the system (1) and boundary condition (2). Differential equations with piecewise-constant argument of generalized type (DEPCAG) are more suitable for modeling and solving various application problems, including areas of neural networks, discontinuous dynamical systems, hybrid systems, etc.[1-2]. To date, the theory of DEPCAG on the entire axis has been developed and their applications have been implemented. However, the question of solvability of boundary value problems for systems of DEPCAG on a finite interval still remains open. We are investigated the questions of existence and uniqueness of the solution to problem (1), (2). The parametrization method [3] is applied for the solving to problem (1), (2) and based on the construction and solving system of linear algebraic equations in arbitrary vectors of new general solutions [4]. By introducing additional parameters we reduce the original problem (1), (2) to an equivalent boundary value problem for system of differential equations with parameters. The algorithm is offered of findings of approximate solution studying problem also it is proved its convergence. Conditions of unique solvability to problem (1), (2) are established in the terms of initial data. The results can be used in the numerical solving of application problems. \vskip 0.1cm {\bf Acknowledgement.} The work is partially supported by grant of the Ministry of Education and Science of the Republic of Kazakhstan No AP 08855726. \vskip 0.1cm %\noindent {\centerline {\bf References}} 1. {\it Akhmet M.~U.} On the reduction principle for differential equations with piecewise-constant argument of generalized type, {\it J. Math. Anal. Appl.}, \textbf{336} (2007), 646-663. 2. {\it Akhmet M.~U.} Integral manifolds of differential equations with piecewise-constant argument of generalized type, {\it Nonlinear Analysis}, \textbf{66} (2007), 367-383. 3. {\it Dzhumabayev D.~S.} Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation, {\it U.S.S.R. Comput. Math. and Math. Phys.}, \textbf{29} (1989), 34-46. 4. {\it Dzhumabaev D.~S.} New general solutions to linear Fredholm integro-differential equations and their applications on solving the boundary value problems. {\it J. Comput. Appl. Math.}, \textbf{336} (2018), 79-108.
Materials with memory: an overview on admissible kernels in the integro-differential model equations
Prof. Sandra Carillo, Universita` "La Sapienza", Rome, Italy
Materials with memory are modelled via integro-differential equations in which the integral term is introduced to take into account the past {\it history} of the material. The {\it classical} regularity requirements the kernel, which in viscoelasticity represents the relaxation function, is assumed to satisfy are considered. Aiming to model wider classes of materials, less restrictive assumptions are adopted. The results presented are part of a joint research program with M. Chipot, V. Valente and G. Vergara Caffarelli.
Analytic properties of the complete formal normal form for the Bogdanov--Takens singularity
Dr. Ewa Stróżyna, Warsaw University of Technology
%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%%%%%%%%%%%% In our previous paper a complete formal normal forms for germs of 2--dimensional holomorphic vector fields with nilpotent singularity was obtained. That classification is quite nontrivial (7 cases), but it can be divided into general types like in the case of the elementary singularities. One could expect that also the analytic properties of the normal forms for the nilpotent singularities are analogous to the case of the elementary singularities. This is really true. In the cases analogous to the focus and the node the normal form is analytic. In the case analogous to the nonresonant saddle the normal form is often nonanalytic due to the small divisors phenomenon. In the cases analogous to the resonant saddles (including saddle--nodes) the normal form is nonanalytic due to properties of some homological operators associated with the first nontrivial term in the orbital normal form.
EU-MATHS-IN: mathematics for industry in Europe (MS - ID 66)
Numerical Analysis and Scientific Computing
Quantitative Comparison of Deep Learning-Based Image Reconstruction Methods for Low-Dose and Sparse-Angle CT Applications
Mr. Johannes Leuschner, University of Bremen
Over the last years, deep learning methods have significantly pushed the state-of-the-art results in applications like imaging, speech recognition and time series forecasting. This development also starts to apply to the field of computed tomography (CT). In medical CT, one of the main goals lies in the reduction of the potentially harmful radiation dose a patient is exposed to during the scan. Depending on the reduction strategy, such low-dose measurements can be more noisy or starkly under-sampled. Hence, achieving high quality reconstructions with classical methods like filtered back-projection (FBP) can be challenging, which motivated the investigation of a number of deep learning approaches for this task. With the purpose of comparing such approaches fairly, we evaluate their performance on fixed benchmark setups. In particular, two CT applications are considered, for both of which large datasets of 2D training images and corresponding simulated projection data are publicly available: a) reconstruction of human chest CT images from low-intensity data, and b) reconstruction of apple CT images from sparse-angle data. In order to include a large variety of methods in the comparison, we organized open challenges for either task. Our current study comprises results obtained with popular deep learning approaches from various categories, like post-processing, learned iterative schemes and fully learned inversion. The test covers image quality of the reconstructions, but also aspects such as the required data or model knowledge and generalizability to other setups are considered. For reproducibility, both source code and reconstructed images are made publicly available.
Simulation, optimal management and infrastructure planning of gas transmission networks
Dr. Ángel Manuel Rueda, University of A Coruña
In this talk we plan to present the results of an ongoing collaboration with a Spanish company in the gas industry, for which we have developed a software, GANESO, that simulates and optimizes gas transmission networks. A gas transmission network consists basically of emission and consumption points, compressors and valves that are connected via pipes. The natural gas flows along the pipes, but the friction with the walls of the pipes decreases the pressure of the gas. At the demand points, the gas has to be delivered with a certain pressure, so it is necessary to counterbalance the pressure loss in the pipes using compressor stations. Yet, compressor stations operate consuming part of the gas that flows through the pipes, so it is important to manage the gas network in an efficient way to minimize such consumption. In order to tackle this problem in the steady-state setting, the methodology we have followed consists of implementing a slight variation of the standard sequential linear programming algorithms that can easily accommodate integer variables, combined with a control theory approach. Further, we have developed a simulator for the transient case, based on well-balanced finite volume methods for general flows and finite element methods for isothermal models. It is worth mentioning that we have also recently added two new features to GANESO in order to deal with energy coupling issues. On the one hand, GANESO was extended to be able to manage heterogeneous gas mixtures in the same network, including Hydrogen-rich mixtures. On the other hand, in order to integrate gas and electricity energy systems, a new simulation/optimization framework was developed for assessing the interdependency of both networks guaranteeing the security of supply of the whole system. In our collaboration we have also studied other related problems to gas networks, such as the allocation of gas losses, the infrastructure planning under uncertainty and the computation of tariffs for networks access according to the different methodologies proposed in EU directives. The developed software, along with the support and consulting analysis provided by the research group, has allowed the company to improve its strategic positioning in the gas sector.
Suboptimal scheduling of a fleet of AGVs to serve online requests
Mr. Markó Horváth, Institute for Computer Science and Control
In the talk we consider an online problem in which a fleet of vehicles must serve transportation requests arriving over time. We propose a dynamic scheduling strategy, which continuously updates the running schedule each time a new request arrives, or a vehicle completes a request. The vehicles move on the edges of a mixed graph modeling the transportation network of a workshop. Our strategy is complete in the sense that deadlock avoidence is guaranteed, and all requests get served. The primary objective is to minimize the total tardiness of serving the requests, and the secondary objective is to minimize the total distance traveled. We provide qualitative results and a comparison to another complete strategy from the literature. The main novelty of our approach is that we optimize the schedule to get better results, and we measure solution quality which is very rare in the AGV literature.
Energy methods and their applications in material science (MS - ID 25)
Analysis and its Applications
Asymptotic analysis of rigidity constraints modeling fiber-reinforced composites
Dr. Antonella Ritorto, Katholische Universität Eichstätt-Ingolstadt
In this talk, we discuss recent analytical results describing the effective deformation behavior of composite materials. Our model involves a rigidity constraint acting on long fibers that are semi-periodically distributed in a material body. The main result characterizes the weak limits of sequences of Sobolev maps whose gradients on the fibers correspond to rotations. Such limits exhibit a restrictive anisotropic geometry in the sense that they locally preserve the length in the fiber direction. We illustrate the results with examples of attainable macroscopic deformations. Under additional regularization in terms of second-order derivatives in the directions orthogonal to the fibers, the model is less flexible, and only rigid body motions can occur. Joint work with Dominik Engl (Utrecht University) and Carolin Kreisbeck (KU Eichst\"att-Ingolstadt).
Extremal and Probabilistic Combinatorics (MS - ID 20)
Combinatorics and Discrete Mathematics
Erdos-Rademacher Problem
Prof. Oleg Pikhurko, University of Warwick
We will give an overview of recent results on the Erdos-Rademacher Problem which asks for the minimum number of r-cliques that a graph with n vertices and m edges can have.
On a problem of M. Talagrand
Dr. Jinyoung Park, Institute for Advanced Study
We will discuss some special cases of a conjecture of M.~Talagrand relating two notions of ``threshold'' for an increasing family $\mathcal F$ of subsets of a finite set $X$. The full conjecture implies equivalence of the ``Fractional Expectation-Threshold Conjecture,'' due to Talagrand and recently proved by Frankston, Kahn, Narayanan, and myself, and the (stronger) ``Expectation-Threshold Conjecture'' of Kahn and Kalai. The conjecture under discussion here says there is a fixed $J$ such that if, for a given increasing family $\mathcal F$, $p\in [0,1]$ admits $\lambda: 2^X\rightarrow \mathbb R^+$ with \[\sum_{S\subseteq F} \lambda_S \ge 1 \quad \forall F\in \mathcal F\] and \[\sum_S \lambda_S p^{|S|} \le 1/2,\] then $p/J$ admits such a $\lambda$ taking values in $\{0,1\}$. Talagrand showed this when $\lambda$ is supported on singletons and suggested a couple of more challenging test cases. In the talk, I will give more detailed descriptions of this problem, and some proof ideas if time allows.
On graph norms
Mr. Jan Hladky, Institute of Computer Science of the Czech Academy of Sciences
The concept of graph norms, introduced by Hatami in 2010 (following suggestions from Lovász and Szegedy), is a convenient tool to work with subgraph densities. The definition is real-analytic, making use of the framework of graphons. I will describe several results obtained by in collaboration with Doležal-Grebík-Rocha-Rozhoň and Garbe-Lee. This in particular includes a characterization norming graphs (and alike) using the `step-Sidorenko' property and a characterization of disconnected norming graphs.
Counting transversals in group multiplication tables
Dr. Rudi Mrazović, University of Zagreb
Hall and Paige conjectured in 1955 that the multiplication table of a finite group $G$ has a transversal (a set of $|G|$ cells in distinct rows and columns and having different symbols) if and only if $G$ satisfies a straightforward necessary condition. This was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. I will discuss joint work with Sean Eberhard and Freddie Manners in which we approach the problem in a more analytic way that enables us to asymptotically count transversals.
Geometric analysis and low-dimensional topology (MS - ID 59)
Differential Geometry and Applications
Recent progress in Lagrangian mean curvature flow of surfaces
Prof. Jason Lotay, University of Oxford
Lagrangian mean curvature flow is potentially a powerful tool for tackling several important open problems in symplectic topology. The first non-trivial and important case is the flow of Lagrangian surfaces in 4-manifolds. I will describe some recent progress in understanding the Lagrangian mean curvature flow of surfaces, including general results about ancient solutions and the study of the Thomas-Yau conjecture in explicit settings.
The Witten Conjecture for homology $S^1 \times S^3$
Dr. Nikolai Saveliev, University of Miami
The Witten conjecture (1994) poses that the Seiberg--Witten invariants contain all of the topological information of the Donaldson polynomials. The natural domain of this conjecture comprises closed simply connected oriented smooth 4-manifolds with $b_+ > 1$, where the Seiberg--Witten invariants are obtained by a straightforward count of irreducible solutions to the Seiberg--Witten equations. The Seiberg-Witten invariants have also been extended to manifolds with $b_+ = 1$ using wall-crossing formulas. In our work with Mrowka and Ruberman (2009) we defined the Seiberg--Witten invariant for a class of manifolds $X$ with $b_+ = 0$ having homology of $S^1 \times S^3$. The usual count of irreducible solutions in this case depends on metric and perturbation but we succeeded in countering this dependence by a correction term to obtain a diffeomorphism invariant of $X$. In the spirit of the Witten conjecture, we conjectured that the degree zero Donaldson polynomial of $X$ can be expressed in terms of this invariant. I will describe the special cases in which the conjecture has been verified, together with some applications. This is a joint project with Jianfeng Lin and Daniel Ruberman.
Alternating links, rational balls, and tilings
Brendan Owens, University of Glasgow
If an alternating knot is a slice of a knotted 2-sphere, then it follows from work of Greene and Jabuka, using Donaldson's diagonalisation combined with Heegaard Floer theory, that the flow lattice $L$ of its Tait graph admits an embedding in the integer lattice $\mathbb{Z}^n$ of the same rank which is ``cubiquitous”: every unit cube in $\mathbb{Z}^n$ contains an element of $L$. The same is true more generally for an alternating link whose double branched cover bounds a rational homology 4-ball. This results in an upper bound on the determinant of such links. In this talk I will describe recent joint work with Josh Greene in which we classify alternating links for which this upper bound on determinant is realised. This makes use of Minkowski’s conjecture, proved by Haj\'{o}s in 1941, which states that every lattice tiling of $\mathbb{R}^n$ by cubes has a pair of cubes which share a facet.
Geometric-functional inequalities and related topics (MS - ID 23)
Partial Differential Equations and Applications
Fractional Orlicz-Sobolev spaces
Prof. Andrea Cianchi, Università di Firenze
Optimal embeddings for fractional-order Orlicz–Sobolev spaces are presented. Related Hardy type inequalities are proposed as well. Versions for fractional Orlicz-Sobolev seminorms of the Bourgain-Brezis-Mironescu theorem on the limit as the order of smoothness tends to 1 and of the Maz’ya-Shaposhnikova theorem on the limit as the order of smoothness tends to 0 are established.
A liquid-solid phase transition in a simple model for swarming
Prof. Rupert Frank, Caltech / LMU Munich
We consider a non-local shape optimization problem, which is motivated by a simple model for swarming and other self-assembly/aggregation models, and prove the existence of different phases. In particular, we show that in the large mass regime the ground state density profile is the characteristic function of a round ball. An essential ingredient in our proof is a strict rearrangement inequality with a quantitative error estimate. The talk is based on joint work with E. Lieb.
Reverse superposition estimates, lifting over a compact covering and extensions of traces for fractional Sobolev mappings
Prof. Jean Van Schaftingen, UCLouvain
When \(u \in W^{1, p} (\mathbb{R}^m)\), then \(\lvert u\rvert \in W^{1, p} (\mathbb{R}^m)\) and \[ \int_{\mathbb{R}^m} \vert D u\vert^p = \int_{\mathbb{R}^m} \vert D \vert u\vert \vert^p; \] this provides an a priori control on \(u\) by \(\lvert u \rvert\) in first-order Sobolev spaces. For fractional Sobolev spaces when \(sp > 1\), we prove a reverse oscillation inequality that yields a control on \(u\) by \(\lvert u \rvert\) in \(W^{s, p} (\mathbb{R}^m)\). As another consequence of the reverse oscillation estimate, given a covering map \(\pi : \widetilde{\mathcal{N}} \to \mathcal{N}\), with \(\widetilde{\mathcal{N}}\) compact, we prove any \(u \in W^{s, p} (\mathbb{R}^m, \mathcal{N})\) has a lifting, that is, can be written as \(u = \pi \circ \widetilde{u}\), with \(\widetilde{u} \in W^{s, p} (\mathbb{R}^m, \widetilde{\mathcal{N}})\). This completes the picture for lifting of fractional Sobolev maps and implies the surjectivity of the trace operator on Sobolev spaces of mappings into a manifold when the fundamental group is finite.
Compactness of Sobolev embeddings with upper Ahlfors regular measures
Zdeněk Mihula, Charles University, Faculty of Mathematics and Physics
A lot of different Sobolev-type embeddings (e.g., classical Sobolev embeddings in the Euclidean setting, boundary trace embeddings, trace embeddings on manifolds, some weighted Sobolev embeddings), which are often treated separately, can be viewed as special instances of Sobolev embeddings with respect to upper Ahlfors regular measures (i.e., Borel measures whose decay on balls is bounded from above by a power of their radii). The aim of this talk is to present in some sense sharp compactness results for such embeddings in the general setting of rearrangement-invariant spaces.
Measure of noncompactness of Sobolev embeddings
Dr. Vít Musil, Masaryk University
A bounded set can be covered by a single ball of some radius. Sometimes several balls of a smaller radius can also cover the set. A compact set can be covered by finitely many balls of arbitrary small radius. The smallest radius that allows to cover the set with finitely many balls therefore describes sets laying in between boundedness and compactness. Such quantity is called a measure of non-compactness. Based on the property of images of the unit balls, linear mappings between Banach spaces are also classified as bounded or compact and to those staying in between, we can assign the measure of non-compactness as well. An important instance of an operator is a Sobolev embedding. Compactness of a Sobolev embedding can constitute a crucial step in many applications in partial differential equations, probability theory, calculus of variations, mathematical physics and other disciplines. In the non-compact case, more subtle techniques have to be developed and the measure of non-compactness plays an indispensable role here. We give a survey of some recent new results on measure of non-compactness of Sobolev embeddings and related mappings.
Geometries Defined by Differential Forms (MS - ID 44)
Differential Geometry and Applications
Closed $G_2$-structures on compact locally homogeneous spaces
Prof. Anna Fino, Università di Torino
$G_2$-structures defined by a closed positive 3-form constitute the starting point in various known and potential methods to obtain holonomy $G_2$-metrics on 7-manifolds. Although linear, the closed condition for a $G_2$-structure is very restrictive, and no general results on the existence of closed $G_2$-structures on compact 7-manifolds are known. In the seminar I will present some results for compact locally homogeneous spaces admitting closed $G_2$-structures.
$G_2$- and $Spin(7)$-structures by means of vector cross products
Prof. Hong Van Le, Academy of Sciences of Czech Republic
The notion of a multilinear vector cross product (VCP) has been introduced by Gray as a natural generalization of the notion of an almost complex structure. Gray also associated $G_2$-structures on 7-manifolds and $Spin(7)$-structures on 8-manifolds with VCPs on the underlying manifolds. In my talk I shall present recent results on $G_2$-structures and $Spin(7)$-structures using VCPs: 1) the correspondence between parallel VCPs on a Riemannian manifold $M$ and parallel almost complex structures on a higher dimensional knot space over $M$ endowed with a $L^2$-metric, which generalizes Brylinski's, LeBrun's, Henrich's and Verbitsky's results for the case that $S$ is a codimension 2 submanifold in $M$, and $S = S^1$ or $M$ is a torsion-free $G_2$-manifold, respectively; 2) the similarities between integrable complex structures on one hand and torsion-free $G_2$-and Spin(7)-structures on the other hand; 3) the construction of CR-twistor spaces over $G_2$-manifolds, due to Verbitsky, and its extension to $Spin(7)$-manifolds. My lecture is based on my joint works with Domenico Fiorenza, Kotaro Kawai, Lorenz Schwachh\"ofer and Luca Vitagliano.
The Obata first eigenvalue theorem on a seven dimensional quaternionic contact manifold
Dimiter Vassilev, University of New Mexico
We give an Obata type rigidity result for the first eigenvalue of the sub-Laplacian. on a compact seven dimensional quaternionic contact manifold which satisfies a Lichnerowicz-type bound on its quaternionic contact Ricci curvature and has a non-negative Paneitz P-function. In particular, under the stated conditions, the lowest possible eigenvalue of the sub-Laplacian is achieved if and only if the manifold is qc-equivalent to the standard 3-Sasakian sphere.
Graph polynomials (MS - ID 62)
Combinatorics and Discrete Mathematics
Normal Subgroup Based Power Graph of a Finite Group
Prof. Seyed Ali Reza Ashrafi Ghomroodi, University of Kashan
Let $G$ be a finite group and $N \unlhd G$. The normal subgroup based power graph $\Gamma_N(G)$ is an undirected graph with vertex set $(G \setminus N) \cup \{e\}$ in which two distinct vertices $a$ and $b$ are adjacent if and only if $aN = b^mN$ or $bN = a^nN$, for some positive integers $m$ and $n$. In this talk, we report on our recent results on the normal subgroup based power graph of a finite group. As a consequence of this talk, a characterization of all pairs $(G,N)$ of a finite group $G$ and a proper non-trivial normal subgroup $N$ of $G$ such that $\Gamma_{H}(G)$ is split, bisplit or $(n-1)-$bisplit are given. Moreover, all finite groups $G$ with $c-$cyclic normal subgroup based power graph, $c \leq 4$, will be determined.
Some new results on independent domination polynomial of a graph
Prof. Saeid Alikhani, Yazd University
An independent dominating set of the simple graph $G=(V,E)$ is a vertex subset that is both dominating and independent in $G$. The independent domination polynomial of a graph $G$ is the polynomial $D_i(G,x)=\sum_{A} x^{|A|}$, summed over all independent dominating subsets $A\subseteq V$. A root of $D_i(G,x)$ is called an independence domination root. We first enumerate independent dominating sets in several classes of graphs made by a linear or cyclic concatenation of basic building blocks. Explicit recurrences are derived for the number of independent dominating sets of these kind of graphs. We also investigate the independent domination polynomials of some generalized compound graphs. As consequence, construct graphs whose independence domination roots are real.
Degree Deviation Measure of Graphs
Mr. Ali Ghalavand, University of Kashan
Let $G$ be a simple graph with n vertices and $m$ edges. The degree deviation measure of $G$ is defined as $s(G)$ $=$ $\sum_{v\in V(G)}|deg_G(v)- \frac{2m}{n}|$. The aim of this talk is to report a positive answer to the Conjecture 4.2 of [J. A. de Oliveira, C. S. Oliveira, C. Justel and N. M. Maia de Abreu, Measures of irregularity of graphs, \textit{Pesq. Oper.} \textbf{33} (3) (2013) 383--398].
Graphs and Groups, Geometries and GAP - G2G2 (MS - ID 7)
Combinatorics and Discrete Mathematics
Groups acting with low fixity
Rebecca Waldecker, MLU Halle-Wittenberg
This research project started in 2012, joint with Kay Magaard. Stemming from questions about automorphisms of Riemann surfaces, it has lead to many interesting problems in permutation group theory. I will discuss the status of the project and some open questions.
Computing distance-regular graph and association scheme parameters in SageMath with {\tt sage-drg}
Dr. Janoš Vidali, University of Ljubljana
{ The {\tt sage-drg} package~\cite{Vdrg} for the SageMath computer algebra system has been originally developed for computation of parameters of distance-regular graphs, and its functionality has later been extended to handle general association schemes. The package has been used to obtain nonexistence results for both distance-regular graphs and $Q$-polynomial association schemes, mostly using the triple intersection numbers technique, see for example~\cite{GSV,GVW,V}. Recently, checks for two new feasibility conditions have been implemented. The first technique, developed by Kodalen and Martin~\cite{KPhD}, relies on Schön\-berg's theorem on positive semidefinite functions in $S^{m-1}$ and its application on the minimal idempotents of an association scheme. The implementation of the relevant checks in {\tt sage-drg} allows us to replicate their nonexistence results for several feasible parameter sets for $Q$-polynomial association schemes. The second technique derives from Terwilliger's work on $P$- and $Q$-poly\-no\-mi\-al association schemes~\cite{T} and has most recently been used by Gavrilyuk and Koolen~\cite{GK,GKbf} to obtain some nonexistence and uniqueness results for $Q$-polynomial distance-regular graphs. The implementation of the relevant procedures in {\tt sage-drg} allows us to generalize their approach and derive nonexistence for many feasible parameter sets of classical distance-regular graphs. \begin{thebibliography}{10} \bibitem{GK} A.~L. Gavrilyuk and J.~H. Koolen. The {T}erwilliger polynomial of a {$Q$}-polynomial distance-regular graph and its application to pseudo-partition graphs. {\em Linear Algebra Appl.}, 466(1):117--140, 2015. {\tt doi:10.1016/j.laa.2014.09.048}. \bibitem{GKbf} A.~L. Gavrilyuk and J.~H. Koolen. A characterization of the graphs of bilinear $(d\times d)$-forms over $\mathbb{F}_2$. {\em Combinatorica}, 39(2):289--321, 2019. {\tt doi:10.1007/s00493-017-3573-4}. \bibitem{GSV} A.~L.~Gavrilyuk, S.~Suda and J.~Vidali. On tight $4$-designs in Hamming association schemes. {\em Combinatorica}, 2020. {\tt doi:10.1007/s00493-019-4115-z}. \bibitem{GVW} A.~L.~Gavrilyuk, J.~Vidali and J.~S.~Williford. On few-class $Q$-polynomial association schemes: feasible parameters and nonexistence results. 2019. {\tt arXiv:1908.10081}. \bibitem{KPhD} B.~G. Kodalen. {\em Cometric Association Schemes}. PhD thesis, 2019. {\tt arXiv:1905.06959}. \bibitem{T} P.~Terwiliger. Lecture note on Terwilliger algebra. Edited by H.~Suzuki, 1993. \bibitem{V} J.~Vidali. Using symbolic computation to prove nonexistence of distance-regular graphs. {\em Electron. J. Combin.}, 25(4):P4.21, 2018. {\tt http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i4p21}. \bibitem{Vdrg} J.~Vidali. {\tt jaanos/sage-drg}: {\tt sage-drg} {S}age package v0.9, 2019. {\tt https://github.com/jaanos/sage-drg/}, {\tt doi:10.5281/zenodo.3350856}. \end{thebibliography} }
Closures of solvable permutation groups
Prof. Andrey Vasilyev, Sobolev Institute of Mathematics
Let $m$ be a positive integer and let $\Omega$ be a finite set. The {\it $m$-closure} $G^{(m)}$ of $G\le Sym(\Omega)$ is the largest permutation group on $\Omega$ having the same orbits as $G$ in its induced action on the Cartesian product~$\Omega^m$. Wielandt \cite{Wielandt1969} showed that \begin{equation} G^{(1)}\ge G^{(2)}\ge\cdots\ge G^{(m)}=G^{(m+1)}=\cdots =G, \end{equation} for some $m<|\Omega|$. (Since the stabilizer in $G$ of all but one point is always trivial, $G^{(n-1)}=G$ where $n = | \Omega |$.) In this sense, the $m$-closure can be considered as a natural approximation of~$G$. It was shown by Praeger and Saxl~\cite{PS1992} that for $m\ge6$, the $m$-closure $G^{(m)}$ of a primitive permutation group $G$ has the same socle as $G$. Furthermore, they classified explicitly primitive groups $G$ and $H$ with different socles having the same $m$-orbits for $m\le5$. Unfortunately, their results say very little about closures of solvable permutation groups. The main goal of this talk is to present the results of \cite{BPVV2021}, where we study such closures. The $1$-closure of $G$ is the direct product of symmetric groups $Sym(\Delta)$, where~$\Delta$ runs over the orbits of~$G$. Thus the $1$-closure of a solvable group is solvable if and only if each of its orbits has cardinality at most~$4$. The case of $2$-closure is more interesting. The $2$-closure of every (solvable) $2$-transitive group $G\le Sym(\Omega)$ is $ Sym(\Omega)$; other examples of solvable~$G$ and nonsolvable $G^{(2)}$ appear in~\cite{Skresanov2019}. But, as shown by Wielandt ~\cite{Wielandt1969}, each of the classes of finite $p$-groups and groups of odd order is closed with respect to taking the $2$-closure. Currently, no characterization of solvable groups having solvable $2$-closure is known. Seress~\cite{Sere1996} observed that if $G$ is a primitive solvable group, then $G^{(5)}=G$; so the $5$-closure of a primitive solvable group is solvable. Our main result is the following stronger statement. \vspace{\baselineskip} \noindent {\bf Theorem 1.} {\em The $3$-closure of a solvable permutation group is solvable.} \vspace{\baselineskip} The corollary below is an immediate consequence of Theorem~1 and the chain of inclusions~(1). \vspace{\baselineskip} \noindent {\bf Corollary 2.} {\em For every integer $m\ge 3$, the $m$-closure of a solvable permutation group is solvable.} \vspace{\baselineskip} The work of the speaker is supported by the Mathematical Center in Akademgorodok, the agreement with Ministry of Science and High Education of the Russian Federation number 075-15-2019-1613. \begin{thebibliography}{99} \bibitem{BPVV2021} E.~A.~O'Brien, I.~Ponomarenko, A.~V.~Vasil'ev, E.~Vdovin, {\em The $3$-closure of a solvable permutation group is solvable}, subm. to J. Algebra (2021), see also arXiv:2012.14166. \bibitem{PS1992} C. E. Praeger and J. Saxl, {\em Closures of finite primitive permutation groups}, Bull. London Math. Soc., {\bf 24}, 251--258 (1992). \bibitem{Sere1996} {\'A}.~Seress, \emph{The minimal base size of primitive solvable permutation groups}, J. London Math. Soc., \textbf{53}, 243--255 (1996). \bibitem{Skresanov2019} S.~Skresanov, {\em Counterexamples to two conjectures in the Kourovka notebook}, Algebra Logic {\bf 58}, no.~3, 249--253 (2019). \bibitem{Wielandt1969} H.~Wielandt, \emph{Permutation groups through invariant relations and invariant functions}, The Ohio State University (1969). \end{thebibliography}
Eigenfunctions of the Star graphs for all non-zero eigenvalues
Prof. Vladislav Kabanov, Krasovskii Institute of Mathematics and Mechanics
Let $G$ be a finite group and $S$ be a subset of $G$ which does not contain the identity element and is closed under inversion. The Cayley graph $\mathrm{Cay}(G,S)$ is a graph with the vertex set $G$ in which two vertices $x$, $y$ are adjacent if and only if $x y^{-1} \in S$. For $\Omega=\{1,\dots, n\},\,n\geqslant 2$, we consider the symmetric group $\mathrm{Sym}_{\Omega}$ and put $S=\{(1~i)~|~\,i\in\{2,\ldots,n\}\}$. The \emph{Star graph} $S_n=\mathrm{Cay}(\mathrm{Sym}_{\Omega}, S)$ is the Cayley graph over the symmetric group $\mathrm{Sym}_{\Omega}$ with the generating set $S$. A function $f:V(\Gamma)\rightarrow \mathbb{R}$ is called an {\it eigenfunction} of a graph $\Gamma$ corresponding to an eigenvalue $\theta$ if $f\not\equiv 0$ and the equality \begin{equation}\label{LocalCondition} \theta\cdot f(x)=\sum\limits_{y\in{N(x)}}f(y) \end{equation} holds for any its vertex $x$, where $N(x)$ is the neighborhood of $x$ in $\Gamma$. The Star graph $S_n,\,n\geq 2$, is known to be integral (see \cite{CF12}), and its spectrum consists of all integers in the range from $-(n-1)$ to $n-1$ (except $0$ when $n=2,3$). Despite of the fact that spectral properties of the Star graph were studied (see \cite{AV09,CF12,F2000,KM12}), no explicit construction for the eigenfunctions was known. In \cite{GKKSV}, an explicit construction of eigenfunctions of $S_n$, $n\geq 3$, for all eigenvalues $\theta$ with $\frac{n-2}{2} < \theta < n-1$ was presented. In this work, we generalize ideas from \cite{GKKSV} and present eigenfunctions of the Star graph $S_n$, $n\geq 3$, for all its non-zero eigenvalues. \medskip The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-167 with the Ministry of Science and Higher Education of the Russian Federation. \begin{thebibliography}{99} \bibitem{AV09} A.~Abdollahi, E.~Vatandoost, Which Cayley graphs are integral? \emph{The Electronic Journal of Combinatorics}, \textbf{16} (2009) 6--7. \bibitem{CF12} G.~Chapuy, V.~Feray, A note on a Cayley graph of $Sym_{n}$, \emph{arXiv:1202.4976v2} (2012) 1--3. \bibitem{F2000} J.~Friedman, On Cayley graphs on the symmetric group generated by transpositions, \emph{Combinatorica} \textbf{20}(4) (2000) 505--519. \bibitem{GKKSV} S.~Goryainov, V.~V.~Kabanov, E.~Konstantinova, L.~Shalaginov, A.~Valyuzhenich, $PI$-eigenfunctions of the Star graphs, Linear Algebra and its Applications \textbf{586} (2020 7--27. \bibitem{KM12} R.~Krakovski, B.~Mohar, Spectrum of Cayley graphs on the symmetric group generated by transposition, \emph{Linear Algebra and its Applications}, \textbf{437} (2012) 1033--1039. \end{thebibliography}
Minimum supports of eigenfunctions of graphs
Mr. Alexandr Valyuzhenich, Sobolev Institute of Mathematics
Let $G=(V,E)$ be a graph with the adjacency matrix $A(G)$. The set of neighbors of a vertex $x$ is denoted by $N(x)$. Let $\lambda$ be an eigenvalue of the matrix $A(G)$. A function $f:V\longrightarrow{\mathbb{R}}$ is called a {\em $\lambda$-eigenfunction} of $G$ if $f\not\equiv 0$ and the equality $$ \lambda\cdot f(x)=\sum_{y\in{N(x)}}f(y) $$ holds for any vertex $x\in V$. In this talk we focus on the following extremal problem for eigenfunctions of graphs. \medskip \noindent {\bf Problem 1 (MS-problem).} {\it Let $G$ be a graph and let $\lambda$ be an eigenvalue of $G$. Find the minimum cardinality of the support of a $\lambda$-eigenfunction of $G$.} \medskip MS-problem was first formulated by Krotov and Vorob'ev \cite{11} in 2014 (they considered MS-problem for the Hamming graph). During the last six years, MS-problem has been actively studied for various families of distance-regular graphs \cite{1,2,4,5,6,7,8,9,10,11,12} and Cayley graphs on the symmetric group \cite{3}. In particular, MS-problem is completely solved for all eigenvalues of the Hamming graph \cite{9,10} and asymptotically solved for all eigenvalues of the Johnson graph \cite{12}. In this talk we will discuss several new results on MS-problem. \begin{thebibliography}{99} \bibitem{1} E. A. Bespalov, On the minimum supports of some eigenfunctions in the Doob graphs, {\it Siberian Electronic Mathematical Reports} 15 (2018) 258--266. \bibitem{2} S. Goryainov, V. Kabanov, L. Shalaginov, A. Valyuzhenich, On eigenfunctions and maximal cliques of Paley graphs of square order, {\it Finite Fields and Their Applications} 52 (2018) 361--369. \bibitem{3} V. Kabanov, E. V. Konstantinova, L. Shalaginov, A. Valyuzhenich, Minimum supports of eigenfunctions with the second largest eigenvalue of the Star graph, {\it The Electronic Journal of Combinatorics} 27(2) (2020) \#P2.14. \bibitem{4} D. S. Krotov, Trades in the combinatorial configurations, XII International Seminar Discrete Mathematics and its Applications, Moscow, 20--25 June 2016, 84--96 (in Russian). \bibitem{5} {D. S. Krotov, I. Yu. Mogilnykh, V. N. Potapov, To the theory of $q$-ary Steiner and other-type trades, {\it Discrete Mathematics} 339(3) (2016) 1150--1157.} \bibitem{6} {E. V. Sotnikova, Eigenfunctions supports of minimum cardinality in cubical distance-regular graphs, {\it Siberian Electronic Mathematical Reports} 15 (2018) 223--245.} \bibitem{7} {E. V. Sotnikova, Minimum supports of eigenfunctions in bilinear forms graphs, {\it Siberian Electronic Mathematical Reports} 16 (2019) 501--515.} \bibitem{8} {A. Valyuzhenich, Minimum supports of eigenfunctions of Hamming graphs, {\it Discrete Mathematics} 340(5) (2017) 1064--1068.} \bibitem{9} {A. Valyuzhenich, Eigenfunctions and minimum $1$-perfect bitrades in the Hamming graph, {\it Discrete Mathematics} 344(3) (2021) 112228.} \bibitem{10} {A. Valyuzhenich, K. Vorob'ev, Minimum supports of functions on the Hamming graphs with spectral constraints, {\it Discrete Mathematics} 342(5) (2019) 1351--1360.} \bibitem{11} {K. V. Vorobev, D. S. Krotov, Bounds for the size of a minimal $1$-perfect bitrade in a Hamming graph, Journal of Applied and Industrial Mathematics 9(1) (2015) 141--146, translated from Discrete Analysis and Operations Research 21(6) (2014) 3--10.} \bibitem{12} {K. Vorob'ev, I. Mogilnykh, A. Valyuzhenich, Minimum supports of eigenfunctions of Johnson graphs, {\it Discrete Mathematics} 341(8) (2018) 2151--2158.} \end{thebibliography}
On maximal cliques in Paley graphs of square order
Dr. Sergey Goryainov, Chelyabinsk State University
In \cite{B84}, Blokhuis studied maximum cliques in Paley graphs of square order $P(q^2)$. It was shown that a clique of size $q$ in $P(q^2)$ is necessarily a quadratic line in the corresponding affine plane $A(2,q)$. Let $r(q)$ denote the reminder after division of $q$ by 4. In \cite{BEHW96}, for any odd prime power $q$, a maximal (but not maximum) clique in $P(q^2)$ of size $\frac{q+r(q)}{2}$ was constructed. In \cite{GKSV18}, for any odd prime power $q$, a maximal clique in $P(q^2)$ of the same size $\frac{q+r(q)}{2}$ was constructed. This clique was shown to have a remarkable connection with eigenfunctions of $P(q^2)$ that have minimum cardinality of support $q+1$. In this talk, we discuss the constructions of maximal cliques from \cite{BEHW96} and \cite{GKSV18} and establish a correspondence between them. \medskip \noindent {\bf Acknowledgments.} Sergey Goryainov and Leonid Shalaginov are supported by RFBR according to the research project 20-51-53023}. \begin{thebibliography}{99} \bibitem{B84} {A.~Blokhuis, On subsets of $GF(q^2)$ with square differences. {\it Indag. Math.} \textbf{46} (1984) 369--372.} \bibitem{BEHW96} {R.~D.~Baker, G.~L.~Ebert, J.~Hemmeter, A.~J.~Woldar, Maximal cliques in the Paley graph of square order. {\it J. Statist. Plann. Inference} \textbf{56} (1996) 33--38.} \bibitem{GKSV18} {S.~V.~Goryainov, V.~V.~Kabanov, L.~V.~Shalaginov, A.~A.~Valyuzhenich, On eigenfunctions and maximal cliques of Paley graphs of square order. {\it Finite Fields and Their Applications} \textbf{52} (2018) 361--369.} \bibitem{GMS21} {S.~V.~Goryainov, A.~V.~Masley, L.~V.~Shalaginov, On a correspondence between maximal cliques in Paley graphs of square order. arXiv:2102.03822 (2021).} \end{thebibliography}
Strongly Deza graphs
Prof. Elena Konstantinova, Sobolev Institute of Mathematics
A Deza graph $G$ with parameters $(n,k,b,a)$ is a $k$-regular graph of order $n$ for which the number of common neighbours of two vertices takes just two values, $b$ or $a$, where $b\geqslant a$. Moreover, $G$ is not the complete graph or the edgeless graph. Deza graphs were introduced in~\cite{D94}, and the name was given in~\cite{EFHHH}, where the basics of Deza graph theory were founded and different constructions of Deza graphs were presented. Strongly regular graphs are a particular case of Deza graphs. Deza graphs can be considered in terms of matrices. Let $G$ be a graph with $n$ vertices, and $M$ be its adjacency matrix. Then $G$ is a Deza graph with parameters $(n,k,b,a)$ if and only if $$M^2=aA+bB+kI$$ for some symmetric $(0,1)$-matrices $A$ and $B$ such that $A+B+I=J$, where $J$ is the all ones matrix and $I$ is the identity matrix. Graphs $G_A$ and $G_B$ with matrices $A$ and $B$ are called the children of $G$. \vspace{\baselineskip} \noindent {\bf Definition.} {\em A Deza graph is called a strongly Deza graph if its children are strongly regular graphs.} \vspace{\baselineskip} \noindent {\bf Theorem 1.}~\cite[Theorem~3.2]{AGHKKS} {\em Let $G$ be a Deza graph with parameters $(n,k,b,a)$, $b>a$. Let $M,A,B$ be the adjacency matrices of $G$ and its children, respectively. If $\theta_1=k,\theta_2,\ldots,\theta_n$ are the eigenvalues of $M$, then {\rm (i)} the eigenvalues of $A$ are $$\alpha = \cfrac{b(n-1)-k(k-1)}{b-a},\ \alpha_2 =\cfrac{k-b-\theta_2^2}{b-a},\ \ldots,\ \alpha_n =\cfrac{k-b-\theta_n^2}{b-a};$$ {\rm (ii)} the eigenvalues of $B$ are $$\beta = \cfrac{a(n-1)-k(k-1)}{a-b},\ \beta_2 = \cfrac{k-a-\theta_2^2}{a-b},\ \ldots,\ \beta_n = \cfrac{k-a-\theta_n^2}{a-b}.$$} By Theorem above, a strongly Deza graph has at most three distinct absolute values of its eigenvalues. %But we can be more precise. \vspace{\baselineskip} \noindent {\bf Theorem 2.} {\em Suppose $G$ is a strongly Deza graph with parameters $(n,k,b,a)$. Then {\rm (i)} $G$ has at most five distinct eigenvalues. {\rm (ii)} If $G$ has two distinct eigenvalues, then $a=0$, $b=k-1 \geqslant 1$, and $G$ is a disjoint union of cliques of order $k+1$. {\rm (iii)} If $G$ has three distinct eigenvalues, then $G$ is a strongly regular graph with parameters $(n,k,\lambda ,\mu)$, where $\{\lambda,\mu\}=\{a,b\}$, or $G$ is disconnected and each component is a strongly regular graph with parameters $(v,k,b,b)$, or each component is a complete bipartite graph $K_{k,k}$ with $k\geqslant 2$.} If $G$ is a bipartite graph, then the~\emph{halved} graphs of $G$ are two connected components of the graph on the same vertex set, where two vertices are adjacent whenever they are at distance two in $G$. The next theorem gives a spectral characterization of strongly Deza graphs. \vspace{\baselineskip} \noindent {\bf Theorem 3.} {\em Let $G$ be a connected Deza graph with parameters $(n,k,b,a)$, $b>a$, and it has at most three distinct absolute values of its eigenvalues. ${\rm (i)}$ If $G$ is a non-bipartite graph, then $G$ is a strongly Deza graph. ${\rm (ii)}$ If $G$ is a bipartite graph, then either $G$ is a strongly Deza graph or its halved graphs are strongly Deza graphs.} \vspace{\baselineskip} We also discuss some results on distance-regular strongly Deza graphs. The main results of the talk are presented in~\cite{AHHKKS}. The work of the speaker is supported by the Mathematical Center in Akademgorodok, the agreement with Ministry of Science and High Education of the Russian Federation number 075-15-2019-1613. \begin{thebibliography}{99} \bibitem{AGHKKS} S.~Akbari, A.~H.~Ghodrati, M.~A.~Hosseinzadeh, V.~V.~Kabanov, E.~V.~Konstantinova, L.~V.~Shalaginov, Spectra of Deza graphs, {\it Linear and Multilinear Algebra} (2020). https://doi.org/10.1080/03081087.2020.1723472 \bibitem{AHHKKS} S.~Akbari, W.~H.~Haemers, M.~A.~Hosseinzadeh, V.~V.~Kabanov, E.~V.~Konstantinova, L.~Shalaginov, Spectra of strongly Deza graphs. https://arxiv.org/abs/2101.06877 \bibitem{D94} A.~Deza., M.~Deza, The ridge graph of the metric polytope and some relatives. In: Bisztriczky T., McMullen P., Schneider R., Weiss A.I. (eds) {\it Polytopes: Abstract, Convex and Computational}. NATO ASI Series (Series C: Mathematical and Physical Sciences), Vol.~440 (1994) 359--372, Springer, Dordrecht. \bibitem{EFHHH} M.~Erickson, S.~Fernando, W.~H.~Haemers, D.~Hardy, J.~Hemmeter, Deza graphs: A generalization of strongly regular graphs, {\it J. Combinatorial Design}, 7 (1999) 359--405. \end{thebibliography}
On coverings and perfect colorings of hypergraphs
Ms. Anna Taranenko, Sobolev Institute of Mathematics
In this talk, we consider perfect colorings (also known as equitable partitions) of hypergraphs. A perfect $k$-coloring of a hypergraph is a coloring of its vertices into $k$ colors such that colors of hyperedges incident to a vertex is defined by its color. A transversal of a hypergraph is one of the simplest examples of perfect $2$-colorings. While perfect colorings of graphs are well known and extensively applied, similar objects for hypergraphs are hardly studied. The main aim of the talk is to show that perfect colorings of hypergraphs have most of the nice algebraic properties of colorings of graphs. Firstly, we define the multidimensional parameter matrix of a perfect coloring of a hypergraph, study its eigenvalues, and connect them with the parameters of the corresponding perfect coloring of the bipartite representation. Next, we introduce coverings of hypergraphs and show that for a vast class of hypergraphs there exist coverings that can be partitioned into perfect matchings. At last, we establish that if two hypergraphs have the same minimal perfect coloring then there is a hypergraph covering both of them.
Discrete Fuglede conjecture on cyclic groups
Mr. Gergely Kiss, Alfréd Rényi Institute of Mathematics
Fuglede in 1974 conjectured that a bounded domain $S \subset \mathbb{R}^d$ tiles the $d$-dimensional Euclidean space if and only if the set of functions in $L^2(S)$ admits an orthogonal basis of exponential functions. In my talk we focus on the discrete version of Fuglede’s conjecture that can be formulated as follows. A subset $S$ of a finite abelian group $G$ tiles $G$ if and only if the character table of $G$ has a submatrix, whose rows are indexed by the elements of $S$, which is a complex Hadamard matrix. Fuglede's original conjecture were disproved first by Tao and the proof is based on a counterexample on elementary abelian $p$-groups. %This led to the first counterexample for the original problem in the continuous case. On the other hand, it is still an open question whether the discrete Fuglede's conjecture is true on cyclic groups. In my talk I will summarize the known results concerning this question. In particular, I will present our recent result which shows that the conjecture holds on cyclic groups whose order is the product of at most 4 (not necessarily different) primes. I will introduce a geometric technique that we called 'cube-rule' and which is an essential tool of the proof.
Graphs, Polynomials, Surfaces, and Knots (MS - ID 49)
Combinatorics and Discrete Mathematics
A colored version of Brylawski's tensor product formula and its applications
Dr. Gabor Hetyei, University of North Carolina at Charlotte
The tensor product of a graph and a pointed graph is obtained by replacing each edge of the first graph with a copy of the second. In this talk we outline a simple proof of Brylawski's formula for the Tutte polynomial of the tensor product which can be generalized to the colored Tutte polynomials introduced by Bollob\'as and Riordan. Consequences include formulas for Jones polynomials of (virtual) knots and for invariants of composite networks in which some major links are identical subnetworks in themselves. All results presented are joint work with Yuanan Diao, some of them are also joint work with Kenneth Hinson.
A new enumerator polynomial with a smart derivative
Prof. Serge Lawrencenko, Russian State University of Tourism and Service
Let $S_n$ be a given set of $n$-vertex simplicial complexes; e.g., a set of $n$-vertex paths, cycles, trees, or 2-cell embeddings of graphs, etc. We solve the problem of determining the cardinality $|S_n|$ in a double sense: (1) the labeled sense; all $n$ vertices are mapped bijectively onto the set of labels $\{1, 2, \ldots, n \}$ where different maps (labelings) may produce different complexes, (2) the unlabeled sense, that is, up to isomorphism (labels removed). A new enumerator polynomial, $P_n (x)$, will be introduced. It has interesting properties: The value $P_n (1)$ is equal to $|S_n|$ in the labeled sense while the value of the derivative $P’_n (1)$ is equal to $n!$ times $|S_n|$ in the unlabeled sense. For example, for paths with $n$ vertices $P_n (x) = (n!/2) x^2$. More examples and properties of the enumerator polynomial will be presented.
Tutte characters for combinatorial coalgebras
Dr. Alex Fink, Queen Mary University of London
The Tutte polynomial is a favourite invariant of matroids and graphs. So when one is working in a generalisation of these settings, for example arithmetic matroids or ribbon graphs, it is a tempting question to find a counterpart of the Tutte polynomial; answers have been given in many cases. Work of Krajewski, Moffatt, and Tanasa found a framework unifying the Tutte-like polynomials arising from graphs in surfaces using Hopf algebras. Our contribution, besides adding some more examples and observing that the machinery is useful for producing convolution formulae in the style of Kook-Reiner-Stanton-Etienne-Las Vergnas, is a generalisation of the formalism using comonoids in linear species. Joint with Cl\'ement Dupont and Luca Moci.
Limits for embedding distributions
Prof. Yichao Chen, SuZhou University of Science and Technolgy
In this paper, we first establish a central limit theorem which is new in probability, then we find and prove that, under some conditions, the embedding distributions of $H$-linear family of graphs with spiders are asymptotic normal distributions. As corollaries, the asymptotic normality for the embedding distributions of path-like sequence of graphs with spiders and the genus distributions of ladder-like sequence of graphs are given. We also prove that the limit of Euler-genus distributions is the same as that of crosscap-number distributions. The results here can been seen a version of central limit theorem in topological graph theory.
Spanning bipartite subgraphs of triangulations of a surface
Mr. Kenta Noguchi, Tokyo University of Science
A triangulation (resp., a quadrangulation) of a surface $S$ is an embedded graph (possibly with multiple edges and loops) on $S$ with each face bounded by a closed walk of length $3$ (resp., $4$). This talk focuses on the relationship between triangulations and quadrangulations of a surface. (a) Extension of a graph $G$ is the construction of a new graph by adding edges to some pairs of vertices in $G$. Obviously, every quadrangulation $G$ of any surface can be extended to a triangulation by adding a diagonal to each face of $G$. If we require some properties for the resulting triangulation, the problem might be difficult and interesting. We prove that every quadrangulation of any surface can be extended to an Eulerian triangulation. Furthermore, we give the explicit formula for the number of distinct Eulerian triangulations extended from a given quadrangulation of a surface. These completely solves the problem raised by Zhang and He \cite{ZH}. (b) It is easy to see that every loopless triangulation $G$ of any surface has a quadrangulation as a spanning subgraph of $G$. As well as (a), if we require some properties for the resulting quadrangulation, the problem might be difficult and interesting. K\"{u}ndgen and Thomassen \cite{KT} proved that every loopless Eulerian triangulation $G$ of the torus has a spanning nonbipartite quadrangulation, and that if $G$ has sufficiently large face width, then $G$ also has a bipartite one. We prove that a loopless Eulerian triangulation $G$ of the torus has a spanning bipartite quadrangulation if and only if $G$ does not have $K_7$ as a subgraph. This talk is based on the papers \cite{NNO1, NNO2, NNO3}. \begin{thebibliography}{99} \bibitem{KT} A. K\"{u}ndgen, C. Thomassen, Spanning quadrangulations of triangulated surfaces, \textit{Abh. Math. Semin. Univ. Hambg} \textbf{87} (2017), 357--368. \bibitem{NNO1} A. Nakamoto, K. Noguchi, K. Ozeki, Extension to even triangulations, \textit{SIAM J. Discrete Math.} \textbf{29} (2015), 2075--2087. \bibitem{NNO2} A. Nakamoto, K. Noguchi, K. Ozeki, Spanning bipartite quadrangulations of even triangulations, \textit{J. Graph Theory} \textbf{90} (2019), 267--287. \bibitem{NNO3} A. Nakamoto, K. Noguchi, K. Ozeki, Extension to $3$-colorable triangulations, \textit{SIAM J. Discrete Math.} \textbf{33} (2019), 1390--1414. \bibitem{ZH} H. Zhang, X. He, On even triangulations of $2$-connected embedded graphs, \textit{SIAM J. Comput.} \textbf{34} (2005), 683--696. \end{thebibliography}
Ternary self-distributive cohomology and invariants of framed links and knotted surfaces with boundary
Emanuele Zappala, University of Tartu
In this talk I will describe how to construct certain state-sum invariants of framed links that utilize the cohomology groups of ternary self-distributive racks and quandles. I will argue that these invariants can be considered, in an appropriate sense, as quantum invariants and give examples from Hopf algebras and $3$-Lie algebras, such as ternary Nambu-Lie algebras. Finally, I will explain how these ideas generalize to the case of (compact and oriented) surfaces with boundary knotted in the $3$-space.
On the Gross-Mansour-Tucker conjecture.
Sergei Chmutov, The Ohio State University
In this presentation I will explain a proof of the Gross, Mansour, Tucker conjecture claiming that any ribbon graph, distinct from explicit family of special exceptions, has a partial dual graph of different genus. This is a joint work with Fabien Vignes-Tourneret based of the preprint arXiv:2101.09319v1 [math.CO].
Embeddings with Eulerian faces II: degree conditions
Mark Ellingham, Vanderbilt University
As a natural special case of edge-outer embeddability, we consider the problem of finding maximum genus orientable directed embeddings. We allow some faces to be specified in advance. Digraphs with directed embeddings are necessarily eulerian. If we are given an eulerian digraph and a decomposition of the arcs into edge-disjoint directed walks, then we can regard this as a partial embedding, with the walks as specified face boundaries. If we can complete this to an embedding by adding one more face bounded by an euler circuit, then the embedding will have maximum genus subject to containing the specified faces. We show that this is always possible provided the underlying simple graph of our $n$-vertex digraph has minimum degree at least $(8n+1)/9$. This is a broad generalization of results for regular tournaments due to Bonnington et al.~(2002) and Griggs, McCourt and \v{S}ir\'{a}\v{n} (2020).
The two-variable Bollob\'as--Riordan polynomial of a connected even delta-matroid is irreducible
Dr. Steven Noble, Birkbeck, University of London
One of the most striking results concerning the Tutte polynomial is that the Tutte polynomial of a matroid is irreducible if and only if the matroid is connected. The most natural analogue of the Tutte polynomial for an even delta-matroid is perhaps a normalized two variable specialization \[(x-1)^{w(D)/2} R_D(x,y-1,1/\sqrt{(x-1)(y-1)},1)\] of the Bollob\'as--Riordan polynomial. We show that for even delta-matroids this two-variable Bollob\'as--Riordan polynomial is irreducible if and only if the delta-matroid is connected.
Tutte's dichromate for signed graphs
Dr. Andrew Goodall, Charles University
A signed graph is a graph with signed edges (positive or negative). Two signed graphs are considered equivalent if their edge signs differ on a cutset of the graph. Proper colourings and nowhere-zero flows of signed graphs are defined analogously to those of graphs. For graphs, these are both enumerated by evaluations of the Tutte polynomial. For signed graphs, Zaslavsky enumerated proper colourings, and recently DeVos--Rollov\'a--\v{S}\'amal showed that the number of nowhere-zero flows satisfies a deletion-contraction recurrence, and, independently, Qian--Ren and Goodall--Litjens--Regts--Vena gave a subset expansion formula. We construct a trivariate polynomial invariant of signed graphs that contains both the number of proper colourings and the number of nowhere-zero flows as evaluations: for this three variables are needed, giving a ``trivariate Tutte polynomial" for signed graphs. Specializations include Zaslavsky’s bivariate rank-generating polynomial of the (frame matroid of the) signed graph and the Tutte polynomial of the (cycle matroid of the) underlying graph.
Framed- and Biframed Knotoids
Mr. Wout Moltmaker, University of Oxford
In this talk I will recall the definition of a spherical knotoid and modify this definition to include a framing, in analogy to framed knots. I also define a further modification that includes a secondary 'coframing' to obtain 'biframed' knotoids. Afterwards I will give topological spaces whose ambient isotopy classes are in one-to-one correspondence with framed- and biframed knotoids respectively. Finally I will mention how biframed knotoids allow for the construction of quantum invariants.
Eulerian and even-face ribbon graph minors
Mr. Metrose Metsidik, Xinjiang Normal University
In this talk, we introduce Eulerian and even-face ribbon graph minors. These minors preserve Eulerian and even-face properties of ribbon graphs, respectively. We then characterize Eulerian, even-face, plane Eulerian and plane even-face ribbon graphs using these minors.
Groups, Graphs and Networks (MS - ID 75)
Combinatorics and Discrete Mathematics
Symmetries of bi-Cayley graphs
Prof. Jinxin Zhou, Beijing Jiaotong University
A graph $\Gamma$ admitting a group $H$ of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a {\em bi-Cayley graph} over $H$. Bi-Cayley graph is a natural generalization of Cayley graph. A large body of research has been developed in recent years to explore the symmetries of bi-Cayley graphs. Much of the work has been focused on the classification and construction of bi-Cayley graphs with specific symmetry properties. In this lecture, I will survey some recent results in this area.
Trivalent dihedrants and bi-dihedrants
Mimi Zhang, Hebei Normal University
A Cayley (resp. bi-Cayley) graph on a dihedral group is called a {\em dihedrant} (resp. {\em bi-dihedrant}). In 2000, a classification of trivalent arc-transitive dihedrants was given by Maru\v si\v c and Pisanski, and several years later, trivalent non-arc-transitive dihedrants of order $4p$ or $8p$ ($p$ a prime) were classified by Feng et al. As a generalization of these results, our first result presents a classification of trivalent non-arc-transitive dihedrants. Using this, a complete classification of trivalent vertex-transitive non-Cayley bi-dihedrants is given. As a by-product, we generalize a theorem in [The Electronic Journal of Combinatorics 19 (2012) $\#$P53]. This is joint work with Jin-Xin Zhou.
Harmonic Analysis and Partial Differential Equations (MS - ID 28)
Partial Differential Equations and Applications
$L^p$ estimates for wave equations with specific Lipschitz coefficients
Dr. Pierre Portal, Australian National University
For the standard linear wave equation $\partial_{t}^{2} u= \Delta u$, the solution at time $t$ belongs to $L^p(\mathbb{R}^d)$ for initial data $u(0,.) \in W^{(d-1)|\frac{1}{p}-\frac{1}{2}|,p}$, $\partial_{t} u(0,.)=0$. This is a classical result of Peral/Myachi from the 1980’s, which motivated the development of Fourier Integral Operator theory. It is optimal in terms of the order of the Sobolev space of initial data. In this talk, we discuss an extension of this result for certain wave equations, such as $\partial_{t}^{2} u= \sum _{j=1} ^{d} \partial_{j}a_{j}\partial_{j}u$, where $a_{j}$ are $C^{0,1}$ functions bounded from above and below and depending only on the $j$-th variable. The fact that such a result holds is somewhat surprising, given that other space-time (Strichartz) estimates typically fail for coefficients rougher than $C^{1,1}$. The proof is based on an approach to FIO theory via phase space Hardy spaces (recently developed by Hassell, P., Rozendaal) combined with operator theoretic and harmonic analytic methods. The talk presents the scheme of the proof, focusing on the ideas behind each of the key steps.
Pointwise convergence for the Schr\"odinger equation with orthonormal initial data
Prof. Neal Bez, Saitama University
For the one-dimensional Schr\"odinger equation, we will explain how to obtain maximal-in-time estimates for systems of orthonormal initial data and, as a result, certain pointwise convergence results associated with systems of infinitely many fermions. Our argument proceeds by establishing a maximal-in-space estimate for the fractional Schr\"odinger equation which, at the same time, addresses an endpoint problem raised by Rupert Frank and Julien Sabin. The talk is based on joint work with Sanghyuk Lee and Shohei Nakamura.
Fractional Integrals with Measure in Grand Lebesgue and Morrey spaces
Prof. Alexander Meskhi, A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University
In the last two decades, the theory of grand Lebesgue spaces $L^{p)}$ introduced by T. Iwaniec and C. Sbordone [IS] is one of the intensively developing directions in modern analysis. The necessity to investigate these spaces emerged from their rather essential role in various fields, in particular, in the integrability problem of Jacobian under minimal hypotheses. It turns out that in the theory of PDEs, the generalized grand Lebesgue spaces $L^{p),\theta}$ introduced by Greco, Iwaniec, and Sbordone [GIS] are appropriate for treating the existence and uniqueness, as well as the regularity problems for various non-linear differential equations. The aim of our talk is to give a complete characterization of a class of measures $\mu$ governing the boundedness of fractional integral operators $I^{\gamma}$ defined on a quasi-metric measure space $(X, d, \mu)$ (non\-homo\-ge\-neous space) from one grand Lebesgue spaces $L^{p), \theta_1}_{\mu}(X)$ into another one $L^{q), \theta_2}_{\mu}(X)$. As a corollary, we have a generalization of the Sobolev inequality for potentials with measure. D. Adams trace inequality (i.e., $L^{p), \theta_1}_{\mu}(X) \mapsto L^{q), \theta_2}_{\nu}(X)$ boundedness) is also derived for these operators in grand Lebesgue spaces. Appropriate problems for grand Morrey spaces are also studied. In the case of Morrey spaces, we assume that the underlying sets of spaces might be of infinite measure. Under some additional conditions on a measure, we investigate the sharpness of the second parameter $\theta_2$ in the target space. {\bf Acknowledgement:} The work was supported by the Shota Rustaveli National Foundation grant of Georgia (Project No. DI-18-118). \vskip+0.1cm \begin{center} {\bf References} \end{center} [GrIwSb] L. Greco, T. Iwaniec and C. Sbordone, Inverting the $p$-harmonic operator, {\em Manuscripta Math.} \textbf{92} (1997), no. 2, 249--258. [IwSb] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, {\em Arch. Rational Mech. Anal.} \textbf{119}(1992), no. 2, 129--143.
Boundary unique continuation of Dini domains
Zihui Zhao, University of Chicago
Let $u$ be a harmonic function in $\Omega \subset \mathbb{R}^d$. It is known that in the interior, the singular set $\mathcal{S}(u) = \{u=|\nabla u|=0 \}$ is $(d-2)$-dimensional, and moreover $\mathcal{S}(u)$ is $(d-2)$-rectifiable and its Minkowski content is bounded (depending on the frequency of $u$). We prove the analogue near the boundary for $C^1$-Dini domains: If the harmonic function u vanishes on an open subset $E$ of the boundary, then near $E$ the singular set $\mathcal{S}(u) \cap \overline{\Omega}$ is $(d-2)$-rectifiable and has bounded Minkowski content. Dini domain is the optimal domain for which $\nabla u$ is continuous towards the boundary, and in particular every $C^{1,\alpha}$ domain is Dini. The main difficulty is the lack of monotonicity formula near the boundary of a Dini domain. This is joint work with Carlos Kenig.
Spectral analysis of a confinement model in relativistic quantum mechanics
Dr. Albert Mas, Universitat Politècnica de Catalunya
In this talk we will focus on the Dirac operator on domains of $\mathbb{R}^3$ with confining boundary conditions of scalar and electrostatic type. This operator is a generalization of the MIT-bag operator, which is used as a simplified model for the confinement of quarks in hadrons that has interested many scientists in the last decades. It is conjectured that, under a volume constraint, the ball is the domain which has the smallest first positive eigenvalue of the MIT-bag operator. I will describe our results ---in collaboration with N. Arrizabalaga (U. Pa\'is Vasco), T. Sanz-Perela (U. Edinburgh and BCAM), and L. Vega (U. Pa\'is Vasco and BCAM)--- on the spectral analysis of the generalized operator. I will discuss on the parameterization of the eigenvalues, their symmetry and monotonicity properties, the optimality of the ball for large values of the parameter, and the connection to boundary Hardy spaces.
Hardy--Littlewood--Sobolev inequality for $p=1$
Prof. Dmitrii Stoliarov, St. Petersburg State University
The Hardy--Littlewood--Sobolev inequality $\|I_{\alpha} f\|_{L_q(\mathbb{R}^d)} \lesssim \|f\|_{L_p(\mathbb{R}^d)}$, where $I_\alpha$ is the Riesz potential of order $\alpha$ and $1/p - 1/q = \alpha /d$, fails at the endpoint $p=1$. I will show two ways to make the inequality true in this case. One way is to impose further restrictions on $f$ (like $f$ is a divergence free vector field), this way is related to the so-called Bourgain--Brezis inequalities. Another way is to replace the $L_q$-norm on the left hand side by an expression that has the same homogeneity as the $L_q$-norm, but possesses additional cancellations. The phenomenon has an analog in the world of martingale transforms, whose consideration suggests the right way to install induction on scales that proves those HLS inequalities for $p=1$.
Weak $L^1$ inequalities for noncommutative singular integrals
Mr. Jose Manuel Conde Alonso, Universidad Autónoma de Madrid
The classical Calderón-Zygmund decomposition is a fundamental tool that helps one study endpoint estimates near $L^1$. In this talk, we shall study an extension of the decomposition to a particular operator valued setting where noncommutativity makes its appearance, allowing us to get rid of the (usually necessary) UMD property of the Banach space where functions take values. The noncommutative extension entails a number of applications. One that we shall discuss concerns weak $L^1$ estimates for Fourier multipliers on groups. Based on joint work with L. Cadilhac and J. Parcet.
On a control of the maximal truncated Riesz transform by the Riesz transform; dimension-free estimates.
Mr. Blazej Wrobel, University of Wroclaw
W prove a dimension-free estimate for the $L^2(\mathbb{R}^d)$ norm of the maximal truncated Riesz transform in terms of the $L^2(\mathbb{R}^d)$ norm of the Riesz transform. Consequently, the vector of maximal truncated Riesz transforms has a dimension-free estimate on $L^2(\mathbb{R}^d).$ We also show that the maximal function of the vector of truncated Riesz transforms has a dimension-free estimate on all $L^p(\mathbb{R}^d)$ spaces, $1<p<\infty.$ The talk is based on joint work with Maciej Kucharski.
Pointwise ergodic theorems for bilinear polynomial averages
Dr. Mariusz Mirek, Rutgers University
We shall discuss the proof of pointwise almost everywhere convergence for the non-conventional (in the sense of Furstenberg and Weiss) bilinear polynomial ergodic averages. This is joint work with Ben Krause and Terry Tao: arXiv:2008.00857. We will also talk about recent progress in this area.
Multiplicative inequalities on BMO
Prof. Pavel Zatitskii, St. Petersburg University (SPbU)
We will talk about the so-called multiplicative inequality for BMO functions: $$\|\varphi\|_{L^r}^r \leq C_{p,r}\|\varphi\|_{L^p}^{p} \|\varphi\|_{BMO}^{r-p},$$ where $1<p<r<\infty$. We will discuss how to find sharp constants in this inequality for the case of quadratic norm on BMO space based on a segment, circle or a real line. Talking about cases of segment and circle we assume the average of $\varphi$ to be equal to zero. Also, we prove this inequality with dimension-free constant for the Garsia-type norm on BMO. The talk is based on joint work with D. Stolyarov, V. Vasyunin and I. Zlotnikov.
Higher-order evolution equations (MS - ID 43)
Partial Differential Equations and Applications
Weak solutions to the stochastic thin-film equation with nonlinear noise in divergence form
Dr. Manuel Gnann, Delft University of Technology
We investigate a degenerate-parabolic fourth-order stochastic partial differential equation modelling the spreading of thin liquid droplets under the influence of thermal noise. Using a combination of entropy and energy estimates, we are able to control the formation of shocks caused by the nonlinear noise in divergence form. In conjunction with a tailor-made approximation and regularization of the equation, we are thus able to prove existence of weak (martingale) solutions through a sequence of compactness arguments.
Low-dimensional Topology (MS - ID 11)
Topology
Embedding spheres in knot traces
Dr. Arunima Ray, Max Planck Institute for Mathematics
We characterise when the generator of the second homotopy group of a knot trace can be represented by a locally flat embedded 2-sphere with abelian fundamental group of the complement, in terms of classical and computable invariants of the corresponding knot.
Complex vs convex Morse functions and geodesic flow
Prof. Burak Özbağcı, Koç University
Suppose that S is a closed and oriented surface equipped with a Riemannian metric. In the literature, there are four seemingly distinct constructions of open books on the unit (co)-tangent bundle of S, having complex, symplectic, contact and Riemannian geometric flavors, respectively. We observe that all of these constructions are based on a suitable choice of a Morse function on S and show that once such a Morse function is fixed, the resulting open books are equivalent. This is a joint work with Pierre Dehornoy.
Bonded knots: a topological model for knotted proteins
Dr. Boštjan Gabrovšek, University of Ljubljana
We introduce bonded knots, oriented knots together with a set of properly embedded coloured arcs. Such structures can be used to topologically model protein structures, where the knots correspond to closed protein backbone chains and the bonds correspond to non-local interactions between the amino acids. The bond colours encode the interaction type (disulphide bridges, ionic bonds,...) that may appear in the conformation of the protein. We will define the HOMFLYPT skein module of rigid and non-rigid coloured bonded knots and show that the rigid version if freely generated by coloured $\Theta$-curves and handcuff links, whereas the non-rigid version is generated by trivial coloured $\Theta$-curves. In other words, there exits a well-defined invariant of rigid and non-rigid coloured bonded knots that respects the HOMFLYPT skein relation.
Flattening knotted surfaces
Dr. Eva Horvat, University of Ljubljana, Faculty for Education
A knotted surface $\mathcal{K}$ in the 4-sphere admits a projection to a 2-sphere, whose set of critical points coincides with a hyperbolic diagram of $\mathcal{K}$. We apply such projections, called flattenings, to define three invariants of embedded surfaces: the width, the trunk and the partition number. These invariants are studied for satellite 2-knots.
Non-orientable slice surfaces and inscribed rectangles.
Mr. Peter Feller, ETH Zurich
We consider the complexity of non-orientable locally-flat surfaces in the four-ball $B^4$ and in $S^1\times B^3$ with boundary a prescribed torus knot and discuss differences between the locally-flat and smooth setup. Our investigation is motivated by the following old metric problem posed by Toeplitz over a hundred years ago: Does every Jordan curve (the image of a continuous injection from the circle to the Euclidean plane), contain four points that form the corners of a square. Based on joint work with M. Golla.
Mathematical analysis: the interaction of fluids/ viscoelastic materials and solids (MS - ID 36)
Partial Differential Equations and Applications
Description of contacts in fluid-beam systems
Prof. Matthieu Hillairet, Universite Montpellier
In this talk we consider a family of systems describing the interactions between a film of fluid deposited on a horizontal substrate and a beam delimiting the upper boundary of the film. The fluid motion is prescribed by solving the incompressible Navier Stokes equations. The beam is assumed to move vertically only, several models are proposed depending on whether damping/viscosity terms are included or not. The coupling between the fluid and the beam is imposed via the continuity of velocity-fields and normal stress. Such systems have been thorouhgly studied in the recent years especially to develop a Cauchy for strong/weak solutions up to the possible first time of contact between the moving beam and the substrate. In this talk, we will focus on the decription of beam/substrate contact. We will first discuss finite-time occurence and then provide a Cauchy theory that handle contacts. This talk is based on collaborations with C. Grandmont (INRIA Paris), J. Lequeurre (Univ. Lorraine) and J-J Casanova (Univ. Paris Dauphine)
Motion of a Rigid body in a Compressible Fluid
Dr. Arnab Roy, Institute of Mathematics of the Czech Academy of Sciences
In this talk, we discuss the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We consider the Navier-slip boundary condition at the interface as well as at the boundary of the domain. We prove existence of a weak solution of the fluid-structure system up to collision.
Regularity of a weak solution to a linear fluid-composite structure interaction problem
Dr. MARIJA GALIĆ, Faculty of Science, University of Zagreb
We deal with the regularity of a weak solution to the fluid-composite structure interaction problem. The problem describes a linear fluid-structure interaction between an incompressible, viscous fluid flow, and an elastic structure composed of a cylindrical shell supported by a mesh-like elastic structure. The fluid and the mesh-supported structure are coupled via the kinematic and dynamic boundary coupling conditions describing continuity of velocity and balance of contact forces at the fluid-structure interface. It has been shown that there exists a weak solution to the described problem. By using the standard techniques from the analysis of partial differential equations, we prove that such weak solution possesses an additional regularity in both time and space variables.
Mathematical challenges in insurance (MS - ID 41)
Statistics and Financial Mathematics
Optimal Drawdowns in Insurance
Prof. Hanspeter Schmidli, University of Cologne
In the last years, drawdown measures have attained attention in financial mathematics. Recently, these measures had been applied to actuarial mathematics. Not all of the measures considered make sense in an insurance context. We therefore consider the expected discounted time in the critical area for an infinite time horizon. More specifically, let $X_t$ be the surplus process of an insurance portfolio and denote by $\bar X_t = \max\{\bar x, \sup_{s \le t} X_s\}$ the running maximum. The drawdown process is then $D_t = \bar X_t - X_t$. The expected discounted time in drawdown is defined as $v^1(\bar x-x) = \mathbb{E}[\int_0^\infty {\rm e}^{-\delta t} \mathbb{I}_{D_t > d} \;{\rm d} t]$. Here, $\delta$ is a preference parameter meaning drawdowns tomorrow are preferred to drawdowns today. In addition, the insurer can buy proportional reinsurance. We aim to find the optimal reinsurance strategy minimising the expected discounted time in drawdown. We find an explicit solution for a diffusion approximation and also discuss the classical Cram\'er--Lundberg model.
On the validation of internal models
Prof. Michel Dacorogna, Prime Re Solutions
The development of risk models for managing portfolios of financial institutions and insurance companies requires, both from the regulatory and management points of view, a strong validation of the quality of the results provided by internal risk models. In Solvency II for instance, regulators ask for independent validation reports from companies who apply for the approval of their internal models. Unfortunately, the usual statistical techniques do not work for the validation of risk models as we lack sufficient data to significantly test the results of the models. Indeed, we will never have enough data to statistically estimate the significance of the VaR at a probability of 1 over 200 years, which is the risk measure required by Solvency II. Instead, we need to develop various indirect strategies to test the relevance of the model. These indirect methods comprise various steps that we list and discuss. In this presentation, we analyze various ways to enable management and regulators to gain confidence in the quality of models. It all starts by ensuring a good calibration of the risk models and the dependencies between the various risk drivers. Then by applying stress tests to the model and various empirical analysis, in particular the probability integral transform, we can build a full and credible framework to validate risk models.
Internal Model Approach for Risk Management II
Dr. Vincenzo Russo, Generali Group
In the context of risk management needs in the insurance industry, mathematical and statistical methods are widely used to evaluate, measure and manage risks that may arise from the insurance business. In particular, since Solvency II regulation entered in force in 2016 across the European Union, insurance companies have been entitled to develop and implement proprietary Internal Models for the evaluation and monitoring of their risk profile. Such Internal Models are prescribed by the regulation to produce in output the full multivariate statistical distribution of the losses that a company or group might experience over a 1-year time horizon, across all risks and all types of business. The development of an Internal Model represents several methodological and computational challenges, and the specific methodological and technical aspects that characterize the design of the Generali Internal Model will be presented, stressing also the ability to take business decisions from the selected approach. The modelling starts from an appropriate identification and representation of the main risks affecting the assets and liabilities. These risk factors need to be described by means of standalone theoretical or empirical distributions, and the dependency structure that delineates their mutual interactions has then to be established and modelled, typically by using a Copula approach. Finally, the functional relationship between the risk factors and the movement of the values of assets and liabilities detained by the company needs is determined in order to derive the correspondent distribution of the potential losses of the Own Fund. The model heavily relies on numerical procedures, like Monte Carlo simulations, that have been implemented at many levels to address the lack of closed-form solutions for most of the quantities that need to be evaluated. The techniques applied to overcome the computational challenges underlying these numerical methods will be described, focusing on the procedures adopted to ensure their computational efficiency, numerical stability and robust convergence.
Mathematics in Education (MS - ID 19)
Mathematics Education and History of Mathematics
Computational/algorithmic thinking in the school mathematics
Dr. Djordje Kadijevich, Institute for Educational Research
\begin{abstract} Due to the globalization and internationalization of the mathematics curriculum, there is, for example, a rapidly developing interest in including computational/algorithmic thinking in mathematics education. By focusing on this thinking, this lecture comprises two parts. In the first part I summarize what computational/algorithmic thinking is, discuss recent research outcomes regarding it, and examine a place for this thinking in school mathematics, with an emphasis on recent international trends and emerging implications for mathematics education. In the second part I present a way to cultivate computational thinking in a mathematical context through data practice based upon the use of interactive displays, which, embedded in another context, could contribute the learning of statistics or computer science (informatics), for example. The content of the first part is based upon a joint research with Dr Max Stephens, University of Melbourne, Australia, whereas the content of the second part is based upon my own recent research. \end{abstract} \begin{thebibliography}{3} \bibitem {ref1} Kadijevich, D. M. (2019a). Interactive displays: Use of interactive charts and dashboards in education. In Tatnall A. (Ed.), Encyclopedia of education and information technologies. Cham, Switzerland: Springer. \bibitem {ref2} Kadijevich, D. M. (2019b). Cultivating computational thinking through data practice. In Passey, D., Bottino, R., Lewin, C., \& Sanchez, E. (Eds.), Empowering learners for life in the digital age (pp. 24–33). Cham, Switzerland: Springer. \bibitem {ref3} Stephens, M., \& Kadijevich, D. M. (2019). Computational/algorithmic thinking. In Lerman S. (Ed.), Encyclopedia of mathematics education. Dordrecht, the Netherlands: Springer. \end{thebibliography}
Approaches of gifted pupils to solving algebraic word problems
Mrs. Irena Budínová, Masaryk University, Faculty of Education
Gifted students, and especially mathematically gifted students, differ from other students in their approach to solving mathematical problems. As the complexity of mathematical problems increases in pupils of the 2nd grade of primary school, it is difficult to monitor these differences in pupils only from the success in the test. There are even tasks in which gifted pupils may be less successful than other pupils. The paper will briefly discuss the issue of educating gifted students in mathematics and one algebraic word problem will illustrate how more complex thinking of gifted students can affect their success in the test task.
How sixth graders' represent some mathematics concepts with drawings
Prof. Alenka Lipovec, University of Maribor
Visual representations enable teachers and researchers to interpret the meanings of mathematical concepts, relations and processes; therefore, they play an important role in mathematics education. In the present study, we analysed the students' understanding of various mathematical concepts using drawings. Numerical expressions \( 17-9 \) , \( 3 \cdot \left( 4+5 \right) , \) \ \( \frac{3}{5}\text{ of 15} \) and \( 2^{3} \) were provided to sixth grade elementary students (\textit{N }= 1595). Students were asked to draw one picture for each numerical expression that describes it. We were interested in whether pictures adequately represent the concept underlying the expression. The data were analysed with a combination of qualitative and quantitative methods. The results show that the participants represented the concepts quite adequately. Expectedly, less abstract concepts were depicted more adequately. In the qualitative content analysis, two themes emerged. Those themes illustrate two ways of mathematical understanding (instrumental and relational) and two types of mathematical knowledge (procedural and conceptual). Procedurally oriented images predominated; they were also more prone to arithmetic errors. The research results can help researchers to create new research tools to measure students' mathematical understanding and help teachers to find new approaches that give them insight into students' mathematical understanding. Keywords: visualisation, representation, understanding, subtraction, parenthesis, fraction, exponentiation.
Students’ achievements in solving geometric problems using visual representations in a virtual learning environment
Dr. Amalija Zakelj, University of Primorska, Faculty of Education
The process of visualization is considered to be indispensable in mathematics learning, specifically in mathematical problem solving. The role of visual representations is becoming increasingly important in technology-based learning. Researchers have presented a variety of frameworks for the optimal use of visual representations in complex, multimedia environments over the last two decades. For the purpose of the study presented in this paper, we designed a model of learning geometry in a virtual learning environment with the use of different learning resources, dynamic geometry programs and applets, which fosters visualisation and the exploration of geometric concepts through the manipulation of interactive virtual representations. The instructional design incorporates Bruner's (1966) three-stage learning model, where learning activities are designed following a concrete-visual-abstract sequencing of instruction, starting with the concrete and progressing through visual to abstract representations. We present the results of an experimental study aimed at exploring (1) whether geometry learning in a virtual learning environment (rich in various teaching material and activities, including dynamics geometry activities, and instructions) is reflected in higher student achievements in solving geometric problems and (2) whether solving geometric problems with the aid of visual representations contributes to higher student achievements. Key words: mathematical education, geometry, multiple representations, visualization, virtual learning environment
The role of Fermi problems in the concept of developing mathematical literacy among students
Mrs. Vida Manfreda Kolar, University of Ljubljana, Faculty of Education
The article deals with mathematical literacy in relation to mathematical knowledge and mathematical problems. It presents also the Slovenian project NA-MA POTI, which started in 2018 and aims to develop mathematical literacy at the national level, from kindergarten to secondary education. The concept of mathematical literacy identifies two cornerstones of mathematical literacy: 1) mathematical thinking, that is, understanding and using mathematical concepts, procedures, strategies and communication as a basis for mathematical literacy; and 2) problem solving in different contexts (personal, social, professional, scientific) that enable a mathematical approach. The latter also highlights mathematical modelling, which generally involves interpreting real-world observations using conceptual (mathematically structured) language. Fermi problems represent a special type of mathematical modelling problems. They are generally defined as problems that are at first sight unsolvable, are authentic and are not structured in the same way as school problems, require reasoning about the necessary data for solving and evaluating them, require mathematical knowledge, and allow the development of problem-solving strategies. Fermi problems, when properly introduced into the learning process, have the potential to develop students' mathematical literacy on both cornerstones. Teachers have a key role to play in this process, as they need to acquire appropriate competencies in mathematical literacy themselves first in order to be able to organise and implement appropriate learning situations in their teaching. The aim of the empirical part was to develop a scheme for assessing the quality of Fermi problems created by prospective primary school teachers for their implementation in mathematics lessons. For this purpose, we analysed examples of Fermi problems for fifth graders designed by prospective primary school teachers The problems were evaluated according to the characteristics of the modelling process: the complexity of the mathematization, the complexity of determining the data needed for the solution, the number of cycles in modelling, and the linguistic relevance of the problem. We defined as qualitative those Fermi problems which can be identified as having a high complexity of determining the data needed for the solution, have a high level of mathematization, are linguistically appropriate, and have more than one cycle. We argue that Fermi problems with such characteristics have a great potential in the process of developing mathematical literacy in students at basically all levels. We want to encourage prospective primary school teachers, as well as other stakeholders, to create their own Fermi problems. We believe that a given scheme for evaluating the quality of Fermi problems can guide teachers in creating their own problems or in deciding which problems are appropriate for classroom implementation according to the objectives. Key Words: mathematical literacy, problems, Fermi problems, modelling, teaching mathematics, prospective primary school teachers
Lessons learned from pre-service teachers’ narratives of math failure
Mrs. Sonja Lutovac, University of Oulu
This presentation will discuss the findings of the ‘Narrated Failures’ project (Academy of Finland, project ID 307672), where pre-service teachers’ narratives of math failure were analysed. Several studies were conducted in which both pre-service elementary school and pre-service mathematics teachers were instructed to write an essay on the topic of ‘math failure and identity’. They were asked to recall their autobiographical experiences of past failure, to define math failure, and reflect on how their failure experiences have shaped them as students, and as future teachers. The narrative analysis of individual cases, as well as cross-case comparison were performed in attempt to deepen understanding of what math failure is from the perspective of the two mentioned cohorts of future teachers of mathematics and how failure shapes their identity development as teachers. In this presentation, the findings of four distinct yet entwined investigations are discussed to shed light on how narratives of math failure can inform teacher education pedagogies. The need to understand math failure as an autobiographical experience is highlighted to better prepare both generalist and specialist future teachers for the variety of ways failure will manifest in their classrooms. Keywords: math failure, identity, narrative, pre-service teachers, teacher education
Making a rhombic 1080-hedron
Dr. Andreja Klančar, University of Primorska, Faculty of Education
Paper models of polyhedra are attractive but difficult to make. That is why in schools we prefer to make polyhedra from plastic parts (Polydron, Zometool). We are interested in golden rhombic solids. There are only five convex golden polyhedra: prolate and oblate rhombohedron, rhombic triacontahedron (Kepler, 1611), rhombic icosahedron (Fedorov, 1885), and Bilinski dodecahedron (1960). Our project is to make a polyhedron that has 1080 golden rhombuses as faces and has the symmetry of an icosahedron. Such a polyhedron has 2160 edges and 1062 vertices. So we need 2160 sticks and 1062 Zometool balls. Theoretically, we could describe the fabrication by starting with Kepler’s triacontahedron. Each rhombus is divided into 36 smaller ones. We then do an inversion on the vertices of valence 3 until no two adjacent rhombuses are in same plane. The orthogonal projection of such polyhedron along an axes of fivefold rotation form a Penrose tiling.
Graph Theory Between The World Wars
Dr. Antonín Slavík, Charles University
We focus on the development of graph theory in the interwar period. In the 1930s, there were still many mathematicians who did not regard graph theory as a serious discipline, owing partly to the fact that many results in this field were originally inspired by problems of recreational mathematics. However, the publication of the first textbook of graph theory in 1936, which presented it as a completely rigorous discipline with applications in linear algebra and set theory, helped to change this attitude. Meanwhile, some important results that are now classified as belonging to graph theory were formulated in the language of topological structures. Finally, one should not forget that numerous developments in graph theory grew out of purely practical problems, such as the construction of electricity networks, or the enumeration of chemical compounds.
Interactive teaching tools and mathematical student competency: Pedagogical experiment in higher mathematics courses teaching
Dr. Daria Termenzhy, Vasyl Stus Donetsk National University
The talk is devoted to implementation of various interactive teaching tools into the up-to-date higher mathematical education. This research reports a six years long experiment which took place in Vasyl Stus Donetsk National University in Ukraine. The experiment was led by own research group of three academics (including author) of the Faculty of Mathematics and IT (now Faculty of Information and Applied Technologies). We worked with 14 students of the Faculty who specialized in Mathematics from 2013/2014 till 2018/2019 academic years. The selection of students was based on their interest and ownership of appropriate gadgets for studying (laptop, smartphone, PC, webcam, etc.). The learning was done by carrying out student activities with applying interactive tools into several mathematical courses (Analytical Geometry (2013/2014), Logic (2014/2015), Differential Geometry (2014/2015), Methodology of Teaching Mathematics (2015/2016), History of Mathematics (2016/2017), IT in Teaching Mathematics (2017/2018), Preparation of Master’s Work (2018/2019). The students used educational interactive portal and distance courses at LMS Moodle developed by author. It included syllabus; tutorial video lectures; diagnostic tests; links to external webpages with mathematical resources that explore the different contents of the course; additional tasks (so-called “extraproblems”); e-lectures with links for downloading; online quizzes and surveys; online consultation module; interactive mathematical games; library of students’ scientific projects; discussion forum panel. We have a long-standing experience of the applying technologies for enhancing the quality of higher mathematics learning. I started a pilot experiment with applying of educational interactive portal in freshmen course of Analytical Geometry for Mathematics students at the University in Fall 2012. This study was preceded our experiment, some results were presented in author’s PhD thesis and proved the efficiency of applying interactive teaching tools in mathematical course. The test procedure of the experiment consisted of 5 parts: pre-test (Math and IT literacy), intellectual test (intellectual lability, ambition test), post-test (Math and IT literacy), several questionnaires and interviews. We also carried out some electronic surveys designed with applying Google Forms. More detail of this experiment and its results will be covered in the talk.
Peano- and Hilbert curve
Dr. Jan Zeman, University of West Bohemia
In this presentation, we give a historical survey on Hilbert’s interpretation of the space-filling Peano-curve which Hilbert presented at the GDNÄ session in Bremen in 1890. Since Hilbert was systematically working in the field of number theory in that period, we try to explain the reason for his sudden interest and immediate reaction to Peano’s surprising result from the same year which proved the possibility of a continuous surjective mapping of a line to a square. We argue that by means of his address, Hilbert was willingly trying to express his positive view on Cantor’s set theory, an affinity which was also present by selecting the continuum hypothesis as the first problem on his list of the mathematical problems for the $20^\mathrm{th}$ century and which lead his thinking also long afterwards in the 1920s while working on his so-called Hilbert’s program in logic. We present also Hilbert’s correspondence with H.~Minkowski on this topic. Minkowski, H., \emph{Briefe an David Hilbert}. Editors L. Rüdenberg, H. Zassenhaus. Berlin, Springer 1973. Weyl, H., David Hilbert and His Mathematical Work. \emph{Bulletin of the AMS} (50), 1944, pp. 612--654.
Software generation of images using mathematics
Mr. Bogdan Soban, retired
I use my software developed in the Visual Basic programming language to generate images. An algorithmic approach is used, which enables a high level of diversity of results with relatively short but mathematically supported software solutions. At each point of the image, a programming algorithm is executed, consisting of mathematical formulas that enter pieces of information, which are constant for the current cycle (genetic code) and variables generated by the process itself. The result of the calculation is the color of the point or the position where to remove the color from the color palette, which plays the role of a key variable. The image coordinate system, the Cartesian and polar coordinate system, and the logic of the complex plane are used to control the image. The whole process is supported by a time-determined random number generator, which supplies the system with random values of various parameters and thus ensures unpredictability and non-repetition of the results. From the viewpoint of mathematics, the following notions appear in the algorithms: complex algebraic expressions, trigonometric and logarithmic functions, plane geometry, iterations and recursions, determination of random numbers in areas, conversion of coordinates between different coordinate systems, control of stochastic motion of a point in plane and space, circular and spiral movements, computation with complex numbers, original formulas for various calculations, number sequences (Fibonacci) and other mathematical operations. The resulting paintings are of the abstract type, which for the most part do not show their mathematical provenance and approach fine art.
Five Mathematicians from MacTutor History of Mathematics
Dr. Marko Razpet, Retired
We exhibit posters describing lives and work of Jurij Vega, Franc Mo\v{c}nik, Josip Plemelj, Ivo Lah, and Ivan Vidav. These five Slovenian mathematicians that are covered by MacTutor History of Mathematics significantly influenced the development of mathematics in Slovenia.
Mathematics in biology and medicine (MS - ID 35)
Mathematics in Science and Technology
Solving the selection-recombination equation
Prof. Ellen Baake, Bielefeld University
The deterministic selection-recombination equation describes the evolution of the genetic type composition of a population under selection and recombination in a law of large numbers regime. So far, only the special case of three sites with selection acting on one of them has been treated, but only approximately and without an obvious path to generalisation. We use both an analytical and a probabilistic, genealogical approach for the case of an \emph{arbitrary} number of neutral sites linked to one selected site. This leads to a recursive integral representation of the solution. Starting from a variant of the ancestral selection-recombination graph, we develop an efficient genealogical structure, which may, equivalently, be represented as a weighted partitioning process, a family of Yule processes with initiation and resetting, and a family of initiation processes. We prove them to be dual to the solution of the differential equation forward in time and thus obtain a stochastic representation of the deterministic solution, along with the Markov semigroup in closed form.
Stochastic Hepatitis C model - conditions for disease extinction
Mr. Vuk Vujović, Faculty of Sciences and Mathematics, University of Niš
Hepatitis C is an infectious liver disease caused by the Hepatitis C virus and transmitted exclusively by infected person's blood. There are many papers in the literature which consider spread of Hepatitis C. In some of them infected persons are divided in two classes: acute and chronic persons, but we construct our stochastic model on basis of the deterministic one which takes into account the isolation stage of infection, too. Hence, we obtain, five-state stochastic model by using system of stochastic differential equations of Ito type. This model better describes variability and uncertainty which may manifest through the contact between persons in population. Also stochastic model is constructed on such way, that inherits the disease free equilibrium point of deterministic model. In this presentation the conditions for coefficient of stochastic systems that provide stability in probability of disease free equilibrium state are shown. On the other words, it means, that under these conditions Hepatitis C will die out in population. Our theoretical results show that reduction of contact rate and isolation of those with disease symptoms are the best measures for suppression of spread of infection. These theoretical results are also confirmed by numerical simulation.
Mathematical Modelling of the impact of Quarantine and Isolation-based control interventions on the Transmission Dynamics of Lassa fever
Mrs. Chinwendu Emilian Madubueze, University of Agriculture , Makurdi, Nigeria
Lassa fever is an acute viral hemorrhagic fever caused by the Lassa virus. It was first discovered in Nigeria in 1969 when two missionary nurses died of Lassa fever. Lassa fever infection is endemic in West African countries such as Nigeria, Sierra Leone, Guinea, Ghana and other West Africa Countries. It infects 100,000 - 300,000 cases annually with approximately 5,000 deaths. About 15 - 20\% of people hospitalized for Lassa fever died from the illness. There are few studies done on the modelling of Lassa fever transmission and control dynamics so far but none has considered the effect of quarantine and Isolation as control interventions against the transmission dynamics of Lassa fever. This paper studied the deterministic model of Lassa fever transmission dynamics with quarantine and isolation as control measures. The model is shown to be mathematically well-posed and epidemiologically meaningful. Detailed analyses (both qualitative and quantitative) were carried out to determine the equilibrium points, their stabilities, and the basic reproduction number necessary to control the Lassa fever transmission in the population. Numerical simulations were carried out to illustrate the analytical results.
Mathematics in the Digital Age of Science (MS - ID 50)
General Topics
Digital Collections of Examples in Mathematical Sciences
Prof. James Davenport, University of Bath
Examples, positive or negative, are the lifeblood of mathematics. This is particularly important when there is generic behaviour, but also lots of special cases. For example, almost all polynomials are irreducible, but nevertheless greeat effort is spent on factoring algorithms. A single example can be represented in various ways, but collections of examples are more difficult. Not that many collections exist, and those that do are often not in machine-processable forms. This means that the sort of contest that has powered the SAT and SMT communities is not available in general computer algebra, or many other fields of computational mathematics. We will make some recommendations here.
zbMATH Open as a hub for the Global Digital Mathematics Library
Dr. Olaf Teschke, FIZ Karlsruhe
2021 marks the transition of zbMATH to an open platform, which is not just freely accessible , but provides open data and APIs that facilitate the connection of diverse facets of mathematical information. We report on the achievements so far, and outline possible next steps in the future.
Standard and Custom APIs for Mathematical Information Retrieval
Dr. Fabian Müller, FIZ Karlsruhe
Industry-standard APIs like OAI-PMH exist for harvesting metadata from repository providers like zbMATH Open. They contain a common baseline functionality, which can however be extended in ways specific to the provider. Going further in this direction, customized REST APIs enable users to access data in a more structured fashion, as well as depositing data instead of just reading. We give an overview of our vision for zbMATH Open and its current status.
Matrix Computations and Numerical (Multi)Linear Algebra with Applications (MS - ID 47)
Numerical Analysis and Scientific Computing
Core reduction: Necessary and sufficient information in linear approximation problems
Mr. Martin Plešinger, Technical University of Liberec
We focus on linear approximation problems $Ax\approx b$, where $A$ is a given matrix, $x$ an unknown vector, and the given right-hand side vector $b$ is not in the range of $A$, i.e., $b\not\in R(A)$. By solving of such problem we usually mean replacing it by some minimization. Typically the least squares (LS) techniques can be used. We focus on the so-called total least squares (TLS) minimization \[ \min\|[g,E]\|_F \quad \text{s.t.} \quad (b+g)\in R(A+E). \] TLS has been studied since the early eighties. The trouble there is, contrary to the standard LS, that the minimization may not have a solution for the given $(A,b)$. The theory of core problem introduced in 2006 by Paige and Strako\v{s} brings a concept of necessary and sufficient information for solving the TLS minimization. This concept allows us to distinguish cases having and not having the TLS solution. Moreover, core problem theory clearly explains why it happens. In recent years the core problems thoery has been applied on several other linear problems $A(X)=B$ where the linear mapping $A$ as well as the right-hand side $B$ can have some particular structure.
Inverses of $k$-Toeplitz matrices for resonator arrays with multiple receivers
Dr. Jose Brox, Centre for Mathematics of the University of Coimbra
Wireless power transfer systems allow to avoid electrical contact and transfer power in rough environments with water, dust or dirt. They are used in electrical vehicle and mobile devices charging, biomedical devices powering, etc. But they have a drawback: in case of misalignment or distance from the transmitter to the receiver, the efficiency and power transmitted can drop abruptly. To overcome this inconvenience, arrays of resonators arranged in a plane are used to transfer power over longer distances through magnetic coupling, with receivers placed over the array to absorb the power transmitted. In the literature, these arrays have been examined using magnetoinductive wave theory or through the circuit analysis of the array, however considering only arrays with one receiver placed over them. Here we present the study of arrays with multiple receivers for which an arbitrary pattern of receivers is repeated over every $k$ resonators. In this case, the impedance matrix representing the circuit is tridiagonal with equal upper and lower diagonals and periodic main diagonal of period $k$. We show how to invert those matrices, by computing their determinants through linear recurrence relations and then using them to compute the minors appearing in the cofactor matrix. In this way we are able to provide rational formulas for the currents, power transmission and efficiency of the system. Work published in Applied Mathematics and Computation 377 (2020).
Parallel Newton-Chebyshev Polynomial Preconditioners for the Conjugate Gradient method
Prof. Angeles Martinez, University of Trieste
Discretization of PDEs modeling different processes and constrained/uncon\-strained optimization problems often require the repeated solution of large and sparse linear systems $A {\mathbf x }= {\mathbf b}$. The size of these system can be of order $10^6 \div 10^9$ and this calls for the use of iterative methods, equipped with ad-hoc preconditioners as accelerators running on a parallel computing environment. In most cases, the huge size of the matrices involved prevents their complete storage. In these instances only the application of the matrix to a vector is available as a routine (\textit{matrix -free regime}). Differently from direct factorization methods, iterative methods do not need the explicit knowledge of the coefficient matrix. The issue is the construction of a preconditioner which also work in a matrix-free regime. Polynomial preconditioners, i.e. preconditioners that can be expressed as $P_k(A)$, are very attractive for several reasons i.e. their construction is only theoretical, namely only the coefficients of the polynomial are to be computed with negligible computational cost, the application of $P_k(A)$ requires a number, $k$, of matrix-vector products so that they can be implemented in a matrix-free regime, and the eigenvectors of the preconditioned matrix are the same as those of $A$. We consider polynomial preconditioners to accelerate the Conjugate Gradient method in the solution of large symmetric positive definite linear systems in massively parallel environments. We put in connection a specialized Newton method to solve the matrix equation $X^{-1} = A$ [1] and the Chebyshev polynomials for preconditioning. We propose a simple strategy to avoid clustering of the extremal eigenvalues in order to speed-up convergence. Numerical results on very large linear systems (up to 8 billion unknowns) in a parallel environment show the efficiency of the proposed class of preconditioners. \begin{thebibliography}{99} \bibitem{Bergamaschi2021} {\sc L.~Bergamaschi and A.~Mart\'{\i}nez}, {\em Parallel {N}ewton-{C}hebyshev polynomial preconditioners for the {C}onjugate {G}radient method}, Computational and Mathematical Methods, (2021). \newblock to appear. \end{thebibliography}
Infinite Tensor Rings
Dr. Lana Periša, Visage technologies
While creating a flexible power method for computing the leftmost, i.e., algebraically smallest, eigenvalue of an infinite dimensional tensor eigenvalue problem, $Hx = \lambda x$, where the infinite dimensional symmetric matrix H exhibits a translational invariant structure, we study the theory of infinite Tensor Rings (iTRs). Under the assumption that the smallest eigenvalue of H is simple, representing the eigenvector as a translational invariant iTR allows the use of power iteration of $e^{-H}$. In order to implement this power iteration, we use a small parameter $t$ so that the infinite matrix-vector operation $e^{-Htx}$ can efficiently be approximated by the Lie product formula, also known as Suzuki–Trotter splitting. In this talk we further explain the motivation for defining iTRs and present their derived and used mathematical properties.
A $\mu$-mode-based integrator for solving evolution equations in Kronecker form
Prof. Marco Caliari, University of Verona
In this talk, we propose a $\mu$-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a $d$-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show that the action of the exponentials, i.e.\ the corresponding matrix-vector products, can be implemented efficiently on modern computer systems. We further explain how $\mu$-mode products can be used to compute spectral transformations efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving three-dimensional Schr\"odinger equations, and we show that the $\mu$-mode integrator can significantly outperform numerical methods well established in the field.\\ This is a joint work with Fabio Cassini, Lukas Einkemmer, Alexander Ostermann, and Franco Zivcovich.
Modeling roughness and long-range dependence with fractional processes (MS - ID 18)
Probability
Self-stabilizing processes
Dr. Jacques Levy Vehel, Case Law Analytics
A self-stabilizing processes $\{Z(t), t\in [t_0,t_1)\}$ is a random process which when localized, that is scaled to a fine limit near a given $t\in [t_0,t_1)$, has the distribution of an $\alpha(Z(t))$-stable process, where $\alpha: \mathbb{R}\to (0,2)$ is a given continuous function. Thus the stability index near $t$ depends on the value of the process at $t$. In the case where $\alpha: \mathbb{R} \to (0,1)$, we first construct deterministic functions which satisfy a kind of autoregressive property involving sums over a plane point set $\Pi$. Taking $\Pi$ to be a Poisson point process then defines a random pure jump process, which we show has the desired localized distributions. When $\alpha$ may take values greater than 1, convergence of the considered sums may no longer be absolute. We generalize the construction in two stages, firstly by setting up a process based on a fixed point set but taking random signs of the summands, and then randomizing the point set to get a process with the desired local properties.
Exact spectral asymptotics of fractional processes and its applications
Prof. Marina KLEPTSYNA, University of Le Mans
Many results in the theory of Gaussian processes rely on the eigenstructure of the covariance operator. However, eigenproblems are notoriously hard to solve explicitly and closed form solutions are known only in a limited number of cases. In this talk we set up a framework for the spectral analysis of the fractional type covariance operators, corresponding to an important family of processes, which includes the fractional Brownian motion, its noise, the fractional Ornstein--Uhlenbeck process and the integrated fractional Brownian motion. We obtain accurate asymptotic approximations for the eigenvalues and the eigenfunctions. Our results provide a key to several problems, whose solution is long known in the standard Brownian case, but was missing in the more general fractional setting. This includes computation of the exact limits of $L^2$-small ball probabilities and asymptotic analysis of singularly perturbed integral equations, arising in mathematical physics and applied probability.
Time-Changed Fractional Ornstein-Uhlenbeck Process
Mr. Giacomo Ascione, Università degli studi di Napoli Federico II
In this talk I will focus on the description of some characteristics of a time-changed fractional Ornstein-Uhlenbeck process, i. e. a process obtained by considering a fractional Ornstein-Uhlenbeck process as introduced in [``Fractional Ornstein-Uhlenbeck Process" by Cheridito, Kawaguchi and Maejima] and we substitute the time by the inverse of an independent driftless subordinator. I will focus first on the existence and the asymptotics of the even-order moments. Moreover, I will discuss some generalized Fokker-Planck equations that arise from the Gaussian nature of the fractional Ornstein-Uhlenbeck process. Finally, I will focus on the stable case, for which more explicit results can be achieved. Such things are the result of a joint work with Yuliya Mishura from University of Kiev and Enrica Pirozzi from University of Naples and are all contained in [``Time-changed fractional Ornstein- Uhlenbeck process", to appear].
Financial markets with a memory
Prof. Yuliya Mishura, Taras Shevchenko National University of Kyiv
We present general conditions for the weak convergence of a discrete-time additive scheme to a stochastic process with memory in the space $D[0, T]$. Then we investigate the convergence of the related multiplicative scheme to a process that can be interpreted as an asset price with memory. As an example, we study an additive scheme that converges to fractional Brownian motion, which is based on the Cholesky decomposition of its covariance matrix. The second example is a scheme converging to the Riemann–Liouville fractional Brownian motion. The multiplicative counterparts for these two schemes are also considered. As an auxiliary result of independent interest, we obtain sufficient conditions for monotonicity along diagonals in the Cholesky decomposition of the covariance matrix of a stationary Gaussian process.
Integration-by-Parts Characterizations of Gaussian Processes
Prof. Tommi Sottinen, University of Vaasa
The Stein's lemma characterizes the one-dimensional Gaussian distribution via an integration-by-parts formula. We show that a similar integration-by-parts formula characterizes a wide class of Gaussian processes, the so-called Gaussian Fredholm processes. Examples include the Brownian motion and fractional Brownian motions. This talk is based on article Azmoodeh, E., Sottinen, T., Tudor, C.A. et al. Integration-by-parts characterizations of Gaussian processes. \textit{Collect. Math.} (2020). \\ https://doi.org/10.1007/s13348-019-00278-x
Decomposition formula for rough Volterra stochastic volatility models
Dr. Josep Vives, Universitat de Barcelona
The research presented in this paper provides an alternative option pricing approach for a class of rough fractional stochastic volatility models. These models are increasingly popular between academics and practitioners due to their surprising consistency with financial markets. However, they bring several challenges alongside. Most noticeably, even simple nonlinear financial derivatives as vanilla European options are typically priced by means of Monte–Carlo (MC) simulations which are more computationally demanding than similar MC schemes for standard stochastic volatility models. In this paper, we provide a proof of the prediction law for general Gaussian Volterra processes. The prediction law is then utilized to obtain an adapted projection of the future squared volatility — a cornerstone of the proposed pricing approximation. Firstly, a decomposition formula for European option prices under general Volterra volatility models is introduced. Then we focus on particular models with rough fractional volatility and we derive an explicit semiclosed approximation formula. Numerical properties of the approximation for a popular model — the rBergomi model — are studied and we propose a hybrid calibration scheme which combines the approximation formula alongside MC simulations. This scheme can significantly speed up the calibration to financial markets as illustrated on a set of AAPL options.
Modeling, approximation, and analysis of partial differential equations involving singular source terms (MS - ID 39)
Numerical Analysis and Scientific Computing
Analysis and approximation of fluids under singular forcing
Prof. Abner J. Salgado, University of Tennessee
Motivated by applications, like modeling of thin structures immersed in a fluid, we develop a well posedness theory for Newtonian and some non-Newtonian fluids under singular forcing in Lipschitz domains, and in convex polytopes. The main idea, that allows us to deal with such forces, is that we study the problem in suitably weighted Sobolev spaces. We develop an a priori approximation theory, which requires to develop the stability of the Stokes projector over weighted spaces. In the case that the forcing is a linear combination of Dirac deltas, we develop a posteriori error estimators for the stationary Stokes and Navier Stokes problems. We show that our estimators are reliable and locally efficient, and illustrate their performance within an adaptive method. We briefly comment on ongoing work regarding the Bousinessq system. Numerical experiments illustrate and complement our theory.
Regularity and finite element approximation for two-dimensional elliptic equations with line Dirac sources
Dr. Peimeng Yin, Wayne State University
We study the elliptic equation with a line Dirac delta function as the source term subject to the Dirichlet boundary condition in a two-dimensional domain. Such a line Dirac measure causes different types of solution singularities in the neighborhood of the line fracture. We establish new regularity results for the solution in a class of weighted Sobolev spaces and propose finite element algorithms that approximate the singular solution at the optimal convergence rate. Numerical tests are presented to justify the theoretical findings.
A priori error estimates of regularized elliptic problems
Dr. Wenyu Lei, SISSA-International School for Advanced Studies
Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. We show a-priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $H^1$ and $L^2$ error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method result in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories.
Discontinuous Galerkin Discretisations for Problems with Dirac Delta Source
Prof. Thomas Wihler, University of Bern
We investigate the symmetric interior penalty discontinuous Galerkin (SIPG) scheme for the numerical approximation of linear second-order elliptic PDE with Dirac delta right-hand side. We outline both an a priori and a (residual-type) a posteriori error analysis on the error measured in terms of the $L^2$--norm. Moreover, some computational results will be presented. Finally, a brief outlook on an inf-sup theory in weighted Sobolev spaces will be given.
Multicomponent diffusion in porous media (MS - ID 42)
Mathematical Physics
Gradient flow techniques for multicomponent diffusion-reaction
Prof. Martin Burger, FAU Erlangen-Nürnberg
In this talk we will discuss some gradient flow approaches to reaction-diffusion systems with nonlinear cross-diffusion, including the derivation from microscopic systems and the analysis of the macroscopic systems in terms of renormalized solutions. We will also discuss the case of coupled bulk-surface systems.
Nonisothermal Richards flow in porous media with cross diffusion
Dr. Nicola Zamponi, University of Mannheim
The existence of large-data weak solutions to a nonisothermal immiscible compressible two-phase unsaturated flow model in porous media is proved. The model is thermodynamically consistent and includes temperature gradients and cross-diffusion effects. So-called variational entropy solutions, which involve the integrated total energy balance, are considered in order to overcome the lack of integrability of some terms in the total energy flux. A priori estimates are derived from the entropy balance and the total energy balance. A sequence of approximated solutions is built via a time semi-discretization and several regularizations. The compactness of the sequence of approximated solutions is achieved by using the Div-Curl lemma.
Uniqueness for a cross-diffusion system issuing from seawater intrusion problems
Prof. Carole Rosier, Université du Littoral
ABSTRACT. This work is devoted to the mathematical analysis of the Cauchy problem for cross-diffusion systems without any assumption about its entropic structure. A global existence result of nonnegative solutions is obtained by applying a classical Schauder fixed point theorem. The proof is upgraded for enhancing the regularity of the solution, namely its gradient belongs to the space $ L ^ r ((0,T)\times \Omega )$ for some $ r>2$. To this aim, the Schauder's strategy is coupled with an extension of Meyers regularity result for linear parabolic equations. We show how this approach allows to prove the well-posedness of the problem using only assumptions prescribing and admissibility range for the {\it ratios} between the diffusion and cross-diffusion coefficients. Finally, the question of the maximal principle is also addressed, especially when source terms are incorporated in the equation in order to ensure the confinement of the solution.
Multiscale Modeling and Methods: Application in Engineering, Biology and Medicine (MS - ID 80)
Partial Differential Equations and Applications
FSI and reduced models for 3D hemodynamic simulations in time-dependent domains
Prof. Yuri Vassilevski, Marchuk Institute of Numerical Mathematics RAS
In the talk we briefly present our approach to fluid-structure interaction (FSI) simulations \cite{1,2} and consider three biomedical problems for flow in time-dependent domains which can be solved by simpler formulations than the FSI formulation: blood flow in the human ventricles \cite{3,4,5}, blood flow in the aortic bifurcation \cite{6}, coaptation characteristics of the aortic valve \cite{7}. The numerical schemes are summarized in the book \cite{8}. This is the joint work with M.Olshanskii, A.Lozovskiy, A.Danilov, T.Dobro\-serdova, G.Panassenko, V.Salamatova, A.Lyogkii. \begin{thebibliography}{99} \bibitem{1} Lozovskiy A., Olshanskii M., Salamatova V., Vassilevski Yu. An unconditionally stable semi-implicit FSI finite element method. Comput.Methods Appl.Mech.Engrg., 297 (2015), 437-454. \bibitem{2} Lozovskiy A., Olshanskii M., Vassilevski Yu. Analysis and assessment of a monolithic FSI finite element method. Comp. \& Fluids, 179 (2019), 277-288. \bibitem{3} Danilov A., Lozovskiy A., Olshanskii M., Vassilevski Yu. A finite element method for the Navier-Stokes equations in moving domain with application to hemodynamics of the left ventricle. Russian J. Numer. Anal. Math. Modelling, 32:4 (2017), 225-236. \bibitem{4} Lozovskiy A., Olshanskii M., Vassilevski Yu. A quasi-Lagrangian finite element method for the Navier-Stokes equations in a time-dependent domain. Comput.Methods Appl.Mech.Engrg., 333 (2018), 55-73. \bibitem{5} Vassilevski Yu., Danilov A. et al. A stable method for 4D CT-based CFD simulation in the right ventricle of a TGA patient. Russian J. Numer. Anal. Math. Modelling, 35:5 (2020), 315-324. \bibitem{6} Dobroserdova T., Liang F., Panasenko G., Vassilevski Yu. Multiscale models of blood flow in the compliant aortic bifurcation. Applied Mathematics Letters 93C (2019), 98-104. \bibitem{7} V.Salamatova, A.Liogky, et al. Numerical assessment of coaptation for auto-pericardium based aortic valve cusps. Russ. J. Numer. Anal. Math. Modelling 34(5):277-287, 2019 \bibitem{8} Vassilevski Yu., Olshanskii M., et al. Personalized Computational Hemodynamics. Models, Methods, and Applications for Vascular Surgery and Antitumor Therapy. Academic Press, 2020. \end{thebibliography}
Numerical solution of the viscous flows in a network of thin tubes: equations on the graph
Dr. Frédéric Chardard, Université Jean Monnet (Saint-Étienne)
A non-stationary flow in a network of thin tubes is considered. Its one-dimensional approximation was proposed in a paper by G.Panasenko and K.Pileckas~[1]. It consists of a set of equations with weakly singular kernels, on a graph, for the macroscopic pressure. A new difference scheme for this problem is proposed. Several variants are discussed. Stability and convergence are studied theoretically and numerically. Numerical results are compared to the direct numerical solution of the full dimension Navier-Stokes equations. More details about the kernels will be discussed in the talk by \'Eric Canon: On weakly singular kernels arising in equations set on a graph, modelling a flow in a networtk of thin tubes. \bigskip \noindent {\bf References} \begin{enumerate} \item {Panasenko G.\,P., Pileckas K.,} {Flows in a tube structure: equation on the graph,} \emph{Journal of Mathematical Physics}, \textbf{Vol} 55: 081505, 2014. \end{enumerate}
Noncommutative structures within order structures, semigroups and universal algebra (MS - ID 67)
Algebra
Quasibands and nonassociative, noncommutative lattices
Prof. Michael Kinyon, University of Denver
Nonassociative idempotent magmas arise naturally in various settings such as the faces of a building or chains in modular lattices. In this talk I will describe a variety of magmas we call \emph{quasibands}, which arise as (sub)reducts of bands (idempotent semigroups): in a band $(B,\cdot)$, define a new operation $\circ$ by $x\circ y = xyx$. This is analogous to how quandles arise as subreducts of groups under the conjugation operations. The quasiband operation $\circ$ is sometimes used as a notational shorthand (especially in the theory of noncommutative lattices) or to characterize band properties. For instance, $(B,\circ)$ is associative, hence a left regular band, if and only if $(B,\cdot)$ is a regular band. The variety of quasibands is defined by $4$ identities. A main result is that this is precisely the variety of $\circ$-subreducts of bands. In addition, I will talk about the natural preorder and natural partial order on a quasiband, the center of a quasiband, the relationship between free quasibands and free bands, and some enumeration of quasibands for low orders. In the case of a noncommutative lattice $(B,\land,\lor)$, the corresponding ``double quasiband'' $(B,\curlywedge,\curlyvee)$ can be viewed as a nonassociative, noncommutative lattice. I will discuss the relationship between the two structures for some of the more commonly studied classes of noncommutative lattices (quasilattices, paralattices, skew lattices, etc.) This is joint work with Toma\v{z} Pisanski (Ljubljana). My own work is partially supported by Simons Foundation Collaboration Grant 359872.
Duality for noncommutative frames
Dr. Jens Hemelaer, University of Antwerp
Noncommutative frames were introduced by Karin Cvetko-Vah as a noncommutative generalization of frames, similar to how skew lattices generalize lattices. The concept of a noncommutative frame was motivated by Lieven Le Bruyn's construction of a noncommutative topology on the points of the Arithmetic Site of Connes and Consani. In this talk, we will extend the duality between locales and frames to the noncommutative world. Our approach is inspired by an earlier paper of Bauer, Cvetko-Vah, Gehrke, van Gool and Kudryavtseva, that introduced a noncommutative Priestley duality. The talk is based on joint work with Karin Cvetko-Vah and Lieven Le Bruyn.
Noncommutativity in an algebraic theory of clones
Prof. Antonino Salibra, Università Ca'Foscari Venezia
We introduce the notion of clone algebra, intended to found a one-sorted, purely algebraic theory of clones. Clone algebras are defined by true identities and thus form a variety in the sense of universal algebra. The most natural clone algebras, the ones the axioms are intended to characterise, are algebras of functions, called functional clone algebras. The universe of a functional clone algebra, called omega-clone, is a set of infinitary operations containing the projections and closed under finitary compositions. We show that there exists a bijective correspondence between clones (of finitary operations) and a suitable subclass of functional clone algebras, called block algebras. Given a clone, the corresponding block algebra is obtained by extending the operations of the clone by countably many dummy arguments. One of the main results is the general representation theorem, where it is shown that every clone algebra is isomorphic to a functional clone algebra. In another result we prove that the variety of clone algebras is generated by the class of block algebras. This implies that every omega-clone is algebraically generated by a suitable family of clones by using direct products, subalgebras and homomorphic images. We present three applications. In the first one, we use clone algebras to answer a classical question about the lattices of equational theories. The second application is to the study of the category VAR of all varieties. We introduce the category CA of all clone algebras (of arbitrary similarity type) with pure homomorphisms as arrows. We show that the category VAR is categorically isomorphic to a full subcategory of CA. We use this result to provide a generalisation of a classical theorem on independent varieties. In the third application we show how skew Boolean algebras are related to clone algebras.
Covering skew lattices
Dr. Jurij Kovič, UP FAMNIT Koper and IMFM Ljubljana
We introduce the concept of covering skew lattices, using the analogy with the covering graphs, and we explain some new discovered phenomena in different forms.
Eilenberg-Moore, Kleisli and descent factorizations
Dr. Fernando Lucatelli Nunes, Utrecht University
\def\references#1{\vspace*{5mm}\noindent{\textbf{References:}}\list {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth \advance\leftmargin\labelsep \usecounter{enumi}} \def\newblock{\hskip .11em plus .33em minus .07em} \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax} \let\endthebibliog=\endlist \def\thefootnote{\fnsymbol{footnote}} Every functor that has a left adjoint has two well-known factorizations. The first one is through the category of Eilenberg-Moore algebras of the induced monad, while the second one is the factorization through the category of free coalgebras (co-Kelsili category) of the induced comonad. As usual in category theory, we also have the dual cases: a functor that has a right adjoint has a factorization through the category of Eilenberg-Moore coalgebras of the induced comonad, and other factorization through the Kleisli category of the induced monad. More generally, if the functor has a codensity monad (resp. a op-codensity comonad), the functor has a factorization through the category of Eilenberg-Moore algebras (resp. the free coalgebras) (see, for instance, \cite[Section~3]{3}). In \cite{3}, given a 2-category $\mathbb{A}$ satisfying suitable hypothesis, we show that every morphism inside a 2-category with opcomma objects (and pushouts) has a 2-dimensional cokernel diagram which, in the presence of the descent objects, induces a factorization of the morphism. We show that these factorizations generalize the usual Eilenberg-Moore and Kleisli factorizations. The result is new even in the case $ \mathbb{A} = \mathsf{Cat}$. In this case, we have that every functor has a factorization through the category of descent data of its 2-dimensional cokernel diagram. We show that, if a functor F has a left adjoint, this descent factorization coincides with the factorization through the category of algebras. Dually, if F has a right adjoint, this descent factorization coincides with the factorization through the category of coalgebras. This specializes in a new connection between monadicity and descent theory, which can be seen as a counterpart account to the celebrated B\'{e}nabou-Roubaud Theorem (see \cite{1} or, for instance, \cite[Theorem~1.4]{2} and \cite[Section~4]{4}). It also leads in particular to a (formal) monadicity theorem. In this talk, we shall give a sketch of the ideas and constructions involved in this particular case of $ \mathbb{A} = \mathsf{Cat} $. \begin{references}{99} \bibitem{1} J. B\'{e}nabou and J. Roubaud. Monades et descente. C. R. Acad. Sci. Paris Ser. A-B, 270:A96–A98, 1970. \bibitem{2} F. Lucatelli Nunes. Pseudo-Kan extensions and descent theory. Theory Appl. Categ., 33:No. 15, 390–444, 2018. \bibitem{3} F. Lucatelli Nunes. Semantic Factorization and Descent. DMUC preprints, 19-03, 2019. arXiv: 1902.01225 \bibitem{4} F. Lucatelli Nunes. Descent Data and Absolute Kan Extensions. DMUC preprints, 19-19, 2019. arXiv: 1606.04999 \end{references}
On modular skew lattices and their coset structure
Prof. Joao Pita Costa, Institute Jozef Stefan
Modular lattices are of great importance in many branches of mathematics, from Algebra to Topology. There are many well-known examples of lattices that are modular but not distributive: the lattice of subspaces of a vector space, the modules over a ring, the normal subgroups of a group, and many others. Modularity is a lattice property of somewhat topological flavour where, in the finite case, maximal chains have the same size, and decomposition theorems as the Krull-Schmidt-Remak for modules over a ring are derived from the modularity properties of its subspace lattice. In this talk, we will discuss several approaches to the generalisation of modularity to the non-commutative context of skew lattices, relate it to the known properties of skew distributivity and skew cancellation, and derive topological-like properties from that generalised concept. We will also discuss aspects of the coset structure of skew lattices of this nature, and derive some of their combinatorial properties.
Nonlinear analysis for continuum mechanics (MS - ID 33)
Analysis and its Applications
Dynamics of visco-elastic bodies with a cohesive interface
Prof. Matteo Negri, University of Pavia
We consider the dynamics of elastic materials with a common cohesive interface (or a domain with a prescribed cohesive fracture). In the bulk, the evolution is provided by linearized elasto-dynamics with Kelvin-Voigt visco-elastic dissipation, while on the interface the evolution is governed by a system of Karush-Kuhn-Tucker depending on the crack opening and on the internal variable. The weak formulation reads $$ \begin{cases} \rho \ddot u (t) + \partial_u \mathcal{E} ( u (t) ) - \langle F(t) , u \rangle + \partial_{[u]} \Psi ( \xi(t) , [u (t)] ) + \partial_{\dot{u}} \mathcal{R} ( \dot{u}) \ni0 , \\[5pt] \dot{\xi} (t) ( \xi(t) - | [ u (t) ] | ) = 0 \ \text{ and } \ | [u (t) ] | \le \xi (t) , \\[4pt] u (0) = u_0 , \ \dot{u} (0) = u_1 , \end{cases} $$ where $\mathcal{E}$ is the elastic energy, $F$ is the external force, $\mathcal{R}$ is Kelvin-Voigt visco-elastic dissipation, while $\Psi$ is the interface cohesive potential, concave under loading and quadratic under unloading. First we provide existence of a time-discrete evolution by means of incremental minimization problems (fully implicit in the displacement) and then its time-continuous limit, which satisfies the energy identity \begin{eqnarray*} \mathcal{E} ( u( t) ) + \Psi( [ u (t) ], \xi (t)) \!\!\! & + & \!\!\! \mathcal{K} ( \dot{u} (t) ) = \mathcal{E} ( u_0 ) + \Psi ( [ u_0 ] , \xi_0) + \mathcal{K} ( v_0 ) + \\ & & {\displaystyle + \int_0^{t} \langle F(s) , \dot{u} (s) \rangle \, ds - \int_0^{t} \partial_v \mathcal{R} ( \dot{u} (s) ) [\dot{u}(s)] \,ds} . \end{eqnarray*} Finally, we discuss the strong formulation of the system, with acceleration in $L^2$ and equilibrium of forces on the interface.
Critical semilinear fractional elliptic problems involving an inverse fractional operator
Prof. Eduardo Colorado, Universidad Carlos III de Madrid
\noindent In this talk we will study the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator, \begin{equation*} \left\{ \begin{tabular}{lcl} $(-\Delta)^{\alpha} u=\lambda u+ (-\Delta)^{\beta}|u|^{p-1}u$ & &in $\Omega$, \\ $\mkern+3mu(-\Delta)^{j}u=0$ & &on $\partial\Omega$, for $j\in\mathbb{Z}$, $0\leq j< [\alpha]$, \end{tabular} \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $0<\beta<1$, $\beta<\alpha<\beta+1$ and $\lambda>0$. In particular, we will show study the following fractional elliptic problem, \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{\alpha-\beta} u= \lambda(-\Delta)^{-\beta}u+ |u|^{p-1}u & \hbox{in} \quad \Omega, \\ \mkern+72.2mu u=0 & \hbox{on} \quad \partial\Omega, \end{array} \right. \end{equation*} proving existence or nonexistence of positive solutions depending on the parameter $\lambda>0$, up to the critical value of the exponent $p$, i.e., for $1<p\leq 2_{\mu}^*-1$ where $\mu:=\alpha-\beta$ and $2_{\mu}^*=\frac{2N}{N-2\mu}$ is the critical exponent of the Sobolev embedding. \ The results are mainly collected in the following paper, \ \'Alvarez-Caudevilla, P.; Colorado, E.; Ortega, Alejandro. {\it Positive solutions for semilinear fractional elliptic problems involving an inverse fractional operator.} Nonlinear Anal. Real World Appl. 2020.
Upscaling and spatial localization of non-local energies with applications to crystal plasticity
Dr. José Matias, Instituto Superior Técnico, U. of Lisbon
An integral representation result is obtained for the asymptotics of energies including both local and non-local terms, in the context of structured deformations. Starting from an initial energy featuring a local bulk and interfacial contribution and a non-local measure of the jump discontinuities, an iterated limiting procedure is performed. First, the initial energy is relaxed to structured deformation, and then the measure of non-locality is sent to zero, with the effect of obtaining an explicit local energy in which the non-linear contribution of submacroscopic slips and separations is accounted for. Two terms, different in nature, emerge in the bulk part of the final energy: one coming from the initial bulk energy and one arising from the non-local contribution to the initial energy. This structure turns out to be particularly useful for studying mechanical phenomena such as yielding and hysteresis. Moreover, in the class of invertible structured deformations, applications to crystal plasticity are presented.
Crystallization results in mechanics and collective behaviour
Prof. Marcello Ponsiglione, Sapienza, University of Rome
We will present variational approaches to crystallization in two dimensions. First, we will discuss classical pairwise interaction potentials leading to microscopic crystallization and hexagonal Wulff shapes. Then, we will show that potantials with non integrable tails may lead to round Wulff shapes minimizing fractional perimeters. Finally, we will discuss how this kind of analysis can be adapted to deal with problems arising in collective behaviour with possible applications to fish schooling .
Characterization of oscillation and mass concentrations occurring along sequences of $\mathcal A$-free functions
Dr. Adolfo Arroyo Rabasa, The University of Warwick
I will establish a functional dichotomy between $\mathcal A$-quasiconvex integrands, a notion of convexity associated to a constant coefficient differential operator $\mathcal A$, and the many ways a sequence $(u_j)$ of functions satisfying the differential constraint $\mathcal A u_j = 0$ may fail to converge strongly in $L^1$, due to oscillation and/or concentration effects. Concerning applications, I will discuss some interesting examples that showcase the failure of $L^1$-compensated compactness when concentration of mass is allowed. For instance, I will explain why a sequence of divergence-free vector-fields may develop any type of oscillation/concentration patterns.
Nonlocal operators and related topics (MS - ID 55)
Partial Differential Equations and Applications
The Bernstein technique for integro-differential equations
Prof. Xavier Cabré, ICREA and Universitat Politecnica de Catalunya
In this talk I will present a joint work with S. Dipierro and E. Valdinoci in which we extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully nonlinear equations of order smaller than two, for which we prove uniform estimates as their order approaches two. Our method is new even in the linear integro-differential case. We will also raise some intriguing open questions, one of them concerning the "pure" linear fractional Laplacian.
Nonlocal minimal graphs in the plane are generically sticky
Prof. Enrico Valdinoci, University of Western Australia
We discuss some recent boundary regularity results for nonlocal minimal surfaces in the plane. In particular, we show that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities. More precisely, if a nonlocal minimal graph in a slab is continuous up to the boundary, then arbitrarily small perturbations of the far-away data necessarily produce boundary discontinuities. Hence, either a nonlocal minimal graph is discontinuous at the boundary, or a small perturbation of the prescribed conditions produces boundary discontinuities. The proof relies on a sliding method combined with a fine boundary regularity analysis, based on a discontinuity/smoothness alternative. Namely, we establish that nonlocal minimal graphs are either discontinuous at the boundary or their derivative is Hölder continuous up to the boundary. In this spirit, we prove that the boundary regularity of nonlocal minimal graphs in the plane "jumps" from discontinuous to differentiable, with no intermediate possibilities allowed. In particular, we deduce that the nonlocal curvature equation is always satisfied up to the boundary. As a byproduct of our analysis, one describes the "switch" between the regime of continuous (and hence differentiable) nonlocal minimal graphs to that of discontinuous (and hence with differentiable inverse) ones. These results have been obtained in collaboration with Serena Dipierro and Ovidiu Savin.
On boundary decay of harmonic functions, Green kernels and heat kernels for some non-local operators
Prof. Zoran Vondraček, University of Zagreb
In this talk, I will discuss non-local operators in open subsets of Euclidean space with critical potentials and kernels admitting a decay at the boundary. The focus will be on the boundary decay of non-negative harmonic functions, Green kernels, and heat kernels. I will explain how decay depends on the critical potential and the possible decay of the kernel at the boundary
Local asymptotics and unique continuation from boundary points for fractional equations
Prof. Veronica Felli, Università di Milano - Bicocca
In this talk I will present some results in collaboration with A. De Luca and S. Vita on unique continuation and local asymptotics of solutions to fractional elliptic equations at boundary points, under some outer homogeneous Dirichlet boundary conditions. I will describe a blow-up procedure which involves an Almgren type monotonicity formula and provides a classification of all possible homogeneity degrees of limiting entire profiles. The Caffarelli-Silvestre extension provides an equivalent formulation of the fractional equation as a local degenerate or singular problem in one dimension more, with mixed Dirichlet and Neumann boundary conditions. In the development of a monotonicity argument, the mixed boundary condition raises delicate regularity issues, which turn out to be quite difficult in dimension $N\geq2$ due to the positive dimension of the junction set and some role played by the geometry of the domain. Such difficulties are overcome by a double approximation procedure: by approximating the potential with functions vanishing near the boundary and the Dirichlet $N$-dimensional region with smooth $(N+1)$-dimensional sets with straight vertical boundary, it is possible to construct a sequence of approximating solutions which enjoy enough regularity to derive Pohozaev type identities, needed to obtain Almgren type monotonicity formulas and consequently to perform blow-up analysis.
Fractional dissipations in fluid dynamics: the surface quasigeostrophic equation
Prof. Maria Colombo, EPFL Lausanne
The surface quasigeostrophic equation (SGQ) is a 2d physical model equation which emerges in meteorology and shares many of the essential difficulties of 3d fluid dynamics. In the supercritical regime for instance, where dissipation is modelled by a fractional Laplacian of order less than 1/2, it is not known whether or not smooth solutions blow-up in finite time. The goal of the talk is to show that every $L^2$ initial datum admits an a.e. smooth solution of the dissipative surface quasigeostrophic equation (SGQ); more precisely, we prove that those solutions are smooth outside a compact set (away from t=0) of quantifiable Hausdorff dimension. We draw analogies between SQG and other PDEs in fluid dynamics in several aspects, including the partial regularity results, and underline some extra structure that SQG enjoys. This is a joint work with Silja Haffter (EPFL).
Blow-up phenomena in nonlocal eigenvalue problems: when theories of $L^1$ and $L^2$ meet
Dr. Hardy Chan, ETH Zürich
We develop a linear theory of very weak solutions for nonlocal eigenvalue problems $\mathcal L u = \lambda u + f$ involving integro-differential operators posed in bounded domains with homogeneous Dirichlet exterior condition, with and without singular boundary data. We consider mild hypotheses on the Green's function and the standard eigenbasis of the operator. The main examples in mind are the fractional Laplacian operators. Without singular boundary datum and when $\lambda$ is not an eigenvalue of the operator, we construct an $L^2$-projected theory of solutions, which we extend to the optimal space of data for the operator $\mathcal L$. We present a Fredholm alternative as $\lambda$ tends to the eigenspace and characterise the possible blow-up limit. The main new ingredient is the transfer of orthogonality to the test function. We then extend the results to singular boundary data and study the so-called large solutions, which blow up at the boundary. For that problem we show that, for any regular value $\lambda$, there exist ``large eigenfunctions'' that are singular on the boundary and regular inside. We are also able to present a Fredholm alternative in this setting, as $\lambda$ approaches the values of the spectrum. We also obtain a maximum principle for weighted $L^1$ solutions when the operator is $L^2$-positive. It yields a global blow-up phenomenon as the first eigenvalue is approached from below. Finally, we recover the classical Dirichlet problem as the fractional exponent approaches one under mild assumptions on the Green's functions. Thus ``large eigenfunctions'' represent a purely nonlocal phenomenon.
Non-local ODEs in conformal geometry
Ms. María del Mar González Nogueras, Universidad Autónoma de Madrid
When one looks for radial solutions of an equation with fractional Laplacian, it is not generally possible to use usual ODE methods. If such equation has some conformal invariances, then one may rewrite it in Emden-Fowler (cylindrical) coordinates and to use the properties of the conformal fractional Laplacian on the cylinder. After giving the necessary background, we will briefly consider two particular applications of this technique: 1. Symmetry breaking, non-degeneracy and uniqueness for the fractional Caffarelli-Kohn-Nirenberg inequality (joint work with W. Ao and A. DelaTorre). 2. Existence and regularity for fractional Laplacian equations with drift and a critical Hardy potential (joint with H. Chan, M. Fontelos and J. Wei).
Should I stay or should I go? Zero-size jumps in random walks for L\'evy flights
Prof. Gianni Pagnini, BCAM - Basque Center for Applied Mathematics
Motivated by the fact that, in the literature dedicated to random walks for anomalous diffusion, it is disregarded if the walker does not move in the majority of the iterations because the most frequent jump-size is zero (i.e., the jump-size distribution is unimodal with mode located in zero) or, in opposition, if the walker always moves because the jumps with zero-size never occur (i.e., the jump-size distribution is bi-modal and equal to zero in zero), we provide an example in which indeed the shape of the jump-distribution plays a role. In particular, we show that the convergence of Markovian continuous-time random walk (CTRW) models for L\'evy flights to a density function that solves the fractional diffusion equation is not guaranteed when the jumps follow a bi-modal power-law distribution equal to zero in zero, but, as a matter of fact, the resulting diffusive process converges to a density function that solves a double-order fractional diffusion equation. Within this framework, self-similarity is lost. The consequence of this loss of self-similarity is the emergence of a time-scale for realizing the large-time limit. Such time-scale results to span from zero to infinity accordingly to the power-law displayed by the tails of the walker's density function. Hence, the large-time limit could not be reached in real systems. The significance of this result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to recurrence and the related concept of site fidelity in the framework of L\'evy-like motion for wild animals.
Nonsmooth Variational Methods for PDEs and Applications in Mechanics (MS - ID 8)
Partial Differential Equations and Applications
On hysteresis reaction-diffusion systems and application in population dynamics
Prof. Klemens Fellner, University of Graz
We consider a general class of PDE-ODE reaction-diffusion systems, which exhibits a singular fast-reaction limit towards a reaction-diffusion equation coupled to a scalar hysteresis operator. As applicational motivation, we present a PDE model for the growth of a population according to a given food supply coupled to an ODE for the turnover of a food stock. Under realistic conditions the stock turnover is much faster than the population growth yielding an intrinsic scaling parameter. We emphasise that the structural assumptions on the considered PDE-ODE models are quite general and that analogue systems might describe e.g. cell-biological buffer mechanisms, where proteins are stored and used at the same time. Finally, we present a new kind of hysteresis-diffusion driven instability caused by the nonlinear coupling between a reaction-diffusion equation and a scalar generalised play operator. We discuss in detail how this coupling with a generalised play operator can lead to spatially inhomogeneous large-time behaviour or equilibration to a homogeneous state.
Lagrange multipliers and nonconstant gradient constrained problem
Prof. Sofia Giuffre, Mediterranea University of Reggio Calabria
The talk is aimed at studying a gradient constrained problem associated to a linear operator. This classical problem was subject to an intense study a few decades ago (see \cite{Bre, Ci, Ev, Is,Wi}, but some very important issues were left open. In particular, we are able to prove two kinds of results (see \cite{Gi}): first, we prove the equivalence of a non-constant gradient constraint problem to a suitable obstacle problem, where the obstacles solve a Hamilton-Jacobi equation in the viscosity sense (see \cite{Li}) and, second, we obtain the existence of Lagrange multipliers associated to the problem. The Lagrange multipliers exist as a Radon measure in the case that the free term of the equation $f \in L^p$, $p > 1$, whereas, if $f$ is a positive constant, it is possible to regularize the result, namely to prove that they belong to $L^2$. These results have been obtained, using a new theory of infinite dimensional duality contained in \cite{Da}. The classical strong duality theory does not work in an infinite dimensional setting, when the interior of the ordering cone of the sign constraints is empty and this new theory overcomes this difficulty. \begin{thebibliography}{00} \bibitem{Bre} H. Brezis, G. Stampacchia, Sur la r\'egularit\'e de la solution d'in\'equations elliptiques, Bull. Soc. Math. France 96 (1968) 153--180. \bibitem{Ci}G. Cimatti, The plane stress problem of Ghizetti in elastoplasticity, Appl. Math. Optim.3 (1), 15-26 (1976). \bibitem{Da}P. Daniele, S. Giuffr\`e, A. Maugeri, F.Raciti, Duality theory and applications to unilateral problems, J. Optim. Theory Appl. (2014) 162:718-734. \bibitem{Ev} L. Evans, A second order elliptic equation with gradient constraint, Comm.Part. Diff. Eq., 4 (1979), 555-572. \bibitem{Gi} S. Giuffr\`e, Lagrange multipliers and non-constant gradient constrained problem, Journal of Differential Equations 269 (1), 542-562, 2020. \bibitem{Is} H. Ishii, S. Koike, Boundary regularity and uniqueness for an elliptic equation with gradient constraint, Comm. in Partial Diff. Eq., 8(4) (1983), 317- 346. \bibitem{Li} P.L. Lions, Generalized Solutions for Hamilton-Jacobi Equations, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1982. \bibitem{Wi} M. Wiegner, The $C^{1,1}$-character of solutions of second order elliptic equations with gradient constraint, Comm.Part. Diff. Eq., 6, 361-371 (1981). \end{thebibliography}
Number Theory (MS - ID 63)
Number Theory
Atkin-Lehner theory for Drinfeld Modular forms
Dr. Maria Valentino, University of Calabria
Let $S_k(N)$ be the space of cusp forms of level $\Gamma_0(N)$ with $N\in\mathbb{Z}$. Atkin-Lehner theory deals with the notion of oldforms, namely those coming from a lower level $M|N$, and newforms, i.e. the orthogonal complement of the space of oldforms with respect to the Petersson inner product. Moreover, it is also concerned with the construction of a basis for $S_k(N)$ made up by eigenfunctions for the Hecke operators $\mathbf{T}_n$ with $n$ prime to $N$. In this talk we shall present some recent advances on the analogous theory for Drinfeld modular forms, which are certain analogues over the function field $\mathbb{F}_q[T]$ of classical modular forms.
An epsilon constant conjecture for higher dimensional representations
Dr. Alessandro Cobbe, Universität der Bundeswehr München
The equivariant local epsilon constant conjecture was formulated in various forms by Fontaine and Perrin-Riou, Benois and Berger, Fukaya and Kato and others. If $N/K$ is a finite Galois extension of $p$-adic fields and $V$ a $p$-adic representation of $G_K$, then the above conjecture describes the epsilon constants attached to $V$ in terms of the Galois cohomology of $T$, where $T$ is a $G_K$-stable $\mathbb Z_p$-sublattice $T$ such that $V=T\otimes_{\mathbb Z_p}\mathbb Q_p$. Here we will discuss the case when $N/K$ is at most weakly ramified (this includes the case of tame ramification) and $T=\mathbb Z_p^r(\chi^\mathrm{cyc})(\rho^\mathrm{nr})$, i.e. the $\mathbb Z_p$-module $\mathbb Z_p^r$ with the trivial action of $G_K$ twisted by the cyclotomic character and by an unramified representation $\rho^\mathrm{nr}:G_K\to\mathrm{Gl}_r(\mathbb Z_p)$. The main results generalize previous work by Izychev, Venjakob, Bley and the author. This is a joint work with Werner Bley.
Composition series of a class of induced representations
Igor Ciganović, University of Zagreb, Faculty of Science
We determine compostion series of a class of parabolically induced representation $\delta([\nu^{-b}\rho,\nu^c\rho])\times \delta([\nu^{\frac{1}{2}}\rho,\nu^a\rho])\rtimes \sigma$ of $p$-adic symplectic group in terms of M\oe glin Tadi\'c clasification. Here $\frac{1}{2}\leq a<b<c\in \mathbb{Z}+\frac{1}{2}$ are half integers, $\nu=|\text{det} |_{F}$ where $F$ is a $p$-adic field, $\rho$ is a cuspidal representation of a general linear group, $\sigma$ is a cuspidal representation of a $p$-adic symplectic group such that $\nu^\frac{1}{2}\rho\rtimes \sigma$ reduces and $\delta([\nu^{x}\rho,\nu^y\rho])\hookrightarrow \nu^y\rho\times\cdots \times \nu^{x}\rho$ is a discrete series representation for $x\leq y\in \mathbb{Z}+\frac{1}{2}$.
$GCD$ results on semiabelian varieties and a conjecture of Silverman
Dr. Amos Turchet, Roma Tre University
A divisibility sequence of polynomials is a sequence $d_n$ such that whenever $m$ divides $n$ one has that $d_m$ divides $d_n$. Results of Ailon and Rudnick, among others, have shown that pairs of divisibility sequences corresponding to subgroups of the multiplicative group have only limited common factors. Silverman conjectured that a similar behaviour should appear in (a large class of) all algebraic groups. We show, extending work of Ghioca-Hsia-Tucker and Silverman on elliptic curves, how to prove Silverman's conjecture over function fields for abelian and split semi-abelian varieties and some generalizations.
Operator Algebras (MS - ID 14)
Analysis and its Applications
Cartan subalgebras in classifiable C*-algebras
Prof. Xin Li, University of Glasgow
This talk is about Cartan subalgebras in classifiable C*-algebras. We will give an overview of some recent results, including a general construction of Cartan subalgebras in all stably finite classifiable C*-algebras and a detailed analysis of these Cartan subalgebras in example classes.
Quantum groups in the heat
Prof. Amaury Freslon, Université Paris-Saclay
In this talk, I will consider the diffusion of the heat semi-group on free orthogonal quantum groups. In the case of classical orthogonal groups, this is the Markov semi-group associated to the Brownian motion, and it is known to spread very abruptly in the sense that it exhibits a \emph{cut-off phenomenon} (this is a result of P.-L. Meliot). I will explain what this means and show that this phenomenon also occurs in the quantum setting. I will further detail how one can get a more precise description of the behaviour of the semi-group around the mixing time by computing the so-called \emph{limiting profile}. This is based on a joint work with L. Teyssier and S. Wang.
Noncommutative ergodic theory of higher-rank lattices
Prof. Cyril Houdayer, Université Paris-Saclay
I will survey recent results in the study of dynamical properties of the space of positive definite functions and characters of higher-rank lattices. I will explain that for a large class of higher-rank lattices, all their Uniformly Recurrent Subgroups (URS) are finite and all their weakly mixing unitary representations weakly contain the left regular representation. These results strengthen celebrated results by Margulis (1978), Stuck-Zimmer (1992) and Nevo-Zimmer (2000). The key novelty in our work is a structure theorem for equivariant normal unital completely positive maps between von Neumann algebras and function spaces associated with Poisson boundaries. Based on joint works with R. Boutonnet and U. Bader, R. Boutonnet, J. Peterson.
Boundary theory and amenability: from Furstenberg's Poisson formula to boundaries of Drinfeld doubles of quantum groups
Prof. Sergey Neshveyev, University of Oslo
In his work on the Poisson formula for semisimple Lie groups Furstenberg attached two boundaries to every locally compact group $G$, which are now called the Poisson and Furstenberg boundaries of $G$. As has been observed over the years, both constructions can be approached from an operator algebraic point of view and extended to the noncommutative setting, leading to the theories of noncommutative Poisson boundaries by Izumi and of injective envelopes by Hamana. The noncommutative Poisson boundaries have been computed in a number of cases. An important aspect of the computations, both in the classical and noncommutative settings, is that the boundaries can be interpreted as universal objects measuring nonamenability of a (quantum) group $G$. I'll explain how such universal properties can be used to identify noncommutative Poisson and Furstenberg-Hamana boundaries in some cases, leading to quantum analogues of classical results of Furstenberg and Moore. (Joint work with Erik Habbestad and Lucas Hataishi.)
Random quantum graphs are asymmetric
Dr. Mateusz Wasilewski, KU Leuven
The study of quantum graphs emerged from quantum information theory. One way to define them is to replace the space of functions on a vertex set of a classical graph with a noncommutative algebra and find a satisfactory counterpart of an adjacency matrix in this context. Another approach is to view undirected graphs as symmetric, reflexive relations and ``quantize'' the notion of a relation on a set. In this case quantum graphs are operator systems and the definitions are equivalent. Doing this has some consequences already for classical graphs; viewing them as operator systems of a special type has already led to the introduction of a few new ``quantum'' invariants. Motivated by developing the general theory of quantum graphs, I will take a look at random quantum graphs, having in mind that the study of random classical graphs is very fruitful. I will show how having multiple perspectives on the notion of a quantum graph is useful in determining the symmetries of these objects – as expected, a generic quantum graph is asymmetric.
Higher Kazhdan projections and the Baum-Connes conjecture
Dr. Sanaz Pooya, Institute of Mathematics Polish Academy of Sciences (IM PAN)
The Baum-Connes conjecture, if it holds for a certain group, provides topological tools to compute the K-theory of its reduced group C*-algebra. This conjecture has been confirmed for large classes of groups, such as amenable groups, but also for some Kazhdan's property (T) groups. Property (T) and its strengthening are driving forces in the search for potential counterexamples to the conjecture. Having property (T) for a group is characterised by the existence of a certain projection in the universal group C*-algebra of the group, known as the Kazhdan projection. It is this projection and its analogues in other completions of the group ring, which obstruct known methods of proof for the Baum-Connes conjecture. In this talk, I will introduce a generalisation of Kazhdan projections. Employing these projections we provide a link between surjectivity of the Baum-Connes map and the $\ell^2$-Betti numbers of the group. A similar relation can be obtained in the context of the coarse Baum-Connes conjecture. This is based on joint work with Kang Li and Piotr Nowak.
Rigidity of Roe algebras
Dr. Kang Li, KU Leuven
(Uniform) Roe algebras are $C^*$-algebras associated to metric spaces, which reflect coarse properties of the underlying metric spaces. It is well-known that if $X$ and $Y$ are coarsely equivalent metric spaces with bounded geometry, then their (uniform) Roe algebras are (stably) $*$-isomorphic. The rigidity problem refers to the converse implication. The first result in this direction was provided by Ján Špakula and Rufus Willett, who showed that the rigidity problem has a positive answer if the underlying metric spaces have Yu's property A. I will in this talk review all previously existing results in literature, and then report on the latest development in the rigidity problem. This is joint work with Ján Špakula and Jiawen Zhang.
Operator Semigroups and Evolution Equations (MS - ID 29)
Analysis and its Applications
Bounded functional calculi for unbounded operators
Prof. Charles Batty, University of Oxford
Several new functional calculi for unbounded operators have been discovered recently. In particular, many semigroup generators have a bounded functional calculus for a Banach algebra of so-called analytic Besov functions. I will describe properties and significance of this calculus, and then briefly mention further extensions of that calculus for generators of bounded semigroups on Hilbert spaces and for generators of bounded holomorphic semigroups on Banach spaces. This work has been in collaboration with Alexander Gomilko and Yuri Tomilov.
Chernoff approximation of operator semigroups and applications
Dr. Yana Kinderknecht (Butko), Technische Universität Braunschweig, Germany
We present a method to approximate operator semigroups with the help of the Chernoff theorem. We discuss different approaches to construct Chernoff approximations for semigroups, generated by Markov processes, and for Schr\"{o}dinger groups. This method provides simultaneousely some numerical schemes for PDEs and pseudo-differential equations (in particular, the operator splitting method), Euler--Maruyama schemes for the corresponding SDEs and other Markov chain approximations to the corresponding Markov processes, can be understood as a numerical path integration method. In some cases, Chernoff approximations have the form of limits of $n$ iterated integrals of elementary functions as $n\to\infty$ (in this case, they are called \emph{Feynman formulae}) and can be used for direct computations and simulations of stochastic processes. The limits in Feynman formulae sometimes coincide with (or give rise to) path integrals with respect to probability measures (such path integrals are usually called \emph{Feynman-Kac formulae}) or with respect to Feynman type pseudomeasures. Therefore, Feynman formulae can be used to approximate the corresponding path integrals and to establish relations between different path integrals. In this talk, we discuss Chernoff approximations for semigroups generated by Feller processes in $\mathbb{R}^d$. We are also interested in constructing Chernoff approximations for semigroups, generated by Markov processes which are obtained by different operations from some original Markov processes. In this talk, we discuss Chernoff approximations for such operations as: a random time change via an additive functional of a process, a subordination (i.e., a random time change via an independent a.s. nondecreasing $1$-dim. L\'{e}vy process), killing of a process upon leaving a given domain, reflecting of a process. These results allow, in particular, to obtain Chernoff approximations for subordinate diffusions on strar graphs and compact Riemannian manifolds. Moreover, the constructed Chernoff approximations for evolution semigroups can be used further to approximate solutions of some time-fractional evolution equations describing anomalous diffusion (solutions of such equations do not posess the semigroup property and are related to some non-Markov processes). \begin{thebibliography}{00} \item Ya. A. Butko. The method of Chernoff approximation, to appear in ```Semigroups of Operators: Theory and Applications SOTA 2018'', Springer Proceedings in Mathematics, 2020; \texttt{https://arxiv.org/pdf/1905.07309.pdf}. \item Ya. A. Butko. Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker--Planck--Kolmogorov equations. \emph{Fract. Calc. Appl. Anal.} \textbf{21} N 5 (2018), 35 pp. %\texttt{URL: https://arxiv.org/pdf/1708.02503.pdf}. \item Ya. A. Butko. Chernoff approximation of subordinate semigroups. \emph{Stoch. Dyn.} . Stoch. Dyn., \textbf{18} N 3 (2018), 1850021, 19 pp., DOI: 10.1142/S0219493718500211. \item Ya. A. Butko, M.~Grothaus and O.G. Smolyanov. Feynman formulae and phase space Feynman path integrals for tau-quantization of some L\'evy-Khintchine type Hamilton functions. \emph{J. Math. Phys.} \textbf{57} 023508 (2016), 22 p. \item Ya. A. Butko. Description of quantum and classical dynamics via Feynman formulae. \emph{Mathematical Results in Quantum Mechanics: Proceedings of the QMath12 Conference}, p.227-234. World Scientific, 2014. ISBN: 978-981-4618-13-7 (hardcover), ISBN: 978-981-4618-15-1 (ebook). \item Ya. A. Butko, R.L. Schilling and O.G. Smolyanov. Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbations, \emph{Inf. Dim. Anal. Quant. Probab. Rel. Top.}, \textbf{15} N 3 (2012), 26 p. \item Ya. A. Butko, M.~Grothaus and O.G. Smolyanov. Lagrangian Feynman formulae for second order parabolic equations in bounded and unbounded domains, \emph{Inf. Dim. Anal. Quant. Probab. Rel. Top.} \textbf{13} N3 (2010), 377-392. \item Ya. A. Butko. Feynman formulas and functional integrals for diffusion with drift in a domain on a manifold, \emph{Math. Notes} \textbf{83} N3 (2008), 301--316. %%\item Ya.A. Butko, R.L. Schilling and O.G. Smolyanov. Feynman formulae for Feller semigroups, \emph{Dokl. Math.} \textbf{82} N2 (2010), 697--683. %%\item Ya.A. Butko, M.~Grothaus and O.G. Smolyanov. Feynman formula for a class of second order parabolic equations in a %% bounded domain, \emph{Dokl. Math.} \textbf{78} N1 (2008), 590--595. %%\item Ya.A. Butko. Function integrals corresponding to a solution of the Cauchy-Dirichlet problem for %%the heat equation in a domain of a Riemannian manifold, \emph{ J. Math. Sci.}, {\bf 151} N1 (2008), 2629--2638. %%\item Ya.A. Butko. Feynman formulae for evolution semigroups (in Russian). \emph{Electronic scientific and technical periodical "Science and education"}, DOI: 10.7463/0314.0701581 , N 3 (2014), 95-132. %%\item Ya.A. Butko. Functional integrals over Smolyanov surface measures for evolutionary equations on a Riemannian manifold, %%\emph{Quant. Probab. White Noise Anal. } \textbf{20} (2007), 145--155. %%\item B. B\"{o}ttcher, Ya.A. Butko, R.L. Schilling and O.G. Smolyanov. Feynman formulae and path integrals for some evolutionary semigroups related to $\tau$-quantization, \emph{Rus. J. Math. Phys.} \textbf{18} N4 (2011), 387--399. %%\item Ya.A. Butko, R.L. Schilling and O.G. Smolyanov. Hamiltonian Feynman--Kac and Feynman formulae for dynamics of particles with position-dependent mass, %% \emph{Int. J. Theor. Phys.} \textbf{50} (2011), 2009--2018. \end{thebibliography}
Fractional semidiscrete evolution equations in Lebesgue sequence spaces
Prof. Pedro J. Miana, Universidad de Zaragoza
In this talk, we give representations for solutions of time-fractional differential equations that involve operators on Lebesgue spaces of sequences defined by discrete convolutions involving kernels through the discrete Fourier transform. We consider finite difference operators of first and second orders, which are generators of uniformly continuous semigroups and cosine functions. We present the linear and algebraic structures (in particular, factorization properties) and their norms and spectra in the Lebesgue space of summable sequences. We identify fractional powers of these generators and apply to them the subordination principle. We also give some applications and consequences of our results. These results have been published in a joint paper with Carlos Lizama and Jorge González-Camus from the Universidad de Santiago de Chile.
Subclasses of Partial Contraction Mapping and generating evolution system on Semigroup of Linear Operators
Prof. Kamilu Rauf , University of Ilorin, Ilorin, Nigeria
Some properties of $C_0$-Semigroup are investigated and used to derive properties of $\omega$-Order Preserving and Reversing Partial Contraction Mapping where homogeneous, inhomogeneous and regularity of mild solution for analytic semigroups are engaged. Furthermore, the subclasses performed like semigroup of linear operators. Moreover, semigroup of linear operator generated by $\omega$-order reversing partial contraction mapping ($\omega$-$ORCP_{n}$) as the infinitesimal generator of a $C_{0}$-semigroup is discussed. It is an attempt to obtain results on evolution systems and stable families of generators considering the homogeneous and inhomogeneous initial value problem.
Spectral aspects of eventually positive $C_0$-semigroups
Ms. Sahiba Arora, Technical University Dresden
The theory of positive one-parameter semigroups is rich and has applications in various mathematical fields. A closely related notion that appeared recently is that of \emph{eventually positive} semigroups, i.e., semigroups that become (and stay) positive for sufficiently large times. A systematic study of this concept revealed that such semigroups exhibit spectral properties similar to those already known for positive semigroups. We start with a few examples of eventually positive semigroups and then highlight several spectral theoretic behaviours of this notion. We follow this up with criteria for \emph{strong convergence} of an eventually positive semigroup. Further, we look at a characterization of convergence of eventually positive semigroups in the operator norm which generalizes a result that was previously only known for positive semigroups. Towards the end, we look at some spectral and convergence implications of \emph{locally eventual positive} semigroups. In a loose sense, this means that the solution of the corresponding Cauchy problem becomes positive in a part of the domain for large times. While examples of this concept have been known for quite some time, a systematic study of it has only been recently initiated. %\medskip \begin{center}\textbf{References}\end{center} [1] Rainer Nagel, editor. One-parameter semigroups of positive operators. Vol. 1184. Springer, Cham, 1986. [2] Sahiba Arora. Locally eventually positive operator semigroups. To appear in J. Oper. Theory. \\ Preprint available online at https://arxiv.org/abs/2101.11386. (2021). [3] Sahiba Arora and Jochen Glück. Spectrum and convergence of eventually positive operator semigroups. Preprint.\\ Available online at https://arxiv.org/abs/2011.04296v2. (2021). [4] Daniel Daners, Jochen Glück, and James B. Kennedy. Eventually positive semigroups of linear operators. J. Math. Anal. Appl. 433. 2(2016): 1561–1593. [5] Filippo Gazzola and Hans-Christoph Grunau. Eventual local positivity for a biharmonic heat equation in $\mathbb R^n$. Discrete Contin. Dyn. Syst., Ser. S 1. 1(2008): 83–87.
Observability for Non-Autonomous Systems
Mr. Fabian Gabel, Hamburg University of Technology
We study non-autonomous abstract Cauchy problems \begin{align*} \dot{x}(t) = A(t) x(t)\,,\; y(t) = C(t) x(t)\,,\; t>0\,,\quad x(0) = x_0\in X\,, \end{align*} where $A(t): D(A) \to X$ is a strongly measurable family of operators on a Banach space $X$ and $C(t) \in \mathcal{L}(X,Y)$ is a family of bounded observation operators from $X$ to a Banach space $Y$. For measurable subsets $E \subseteq (0,T)$, $T > 0$, we provide sufficient conditions such that the Cauchy problem satisfies a \textit{final state observability estimate} \begin{align*} \|x(T)\|_{X} \lesssim \left(\;\int_{E} \; \|y(t)\|_Y^r \; \mathrm{d}t\;\right)^{1/r} , \quad r \in [1,\infty)\,, \end{align*} where an analogous estimate holds for the case $r = \infty$. An application of the above result to families of strongly elliptic differential operators $A(t)$ and observation operators \begin{align*} C(t)u := \mathbf{1}_{\Omega(t)} u\,,\quad \Omega(t) \subseteq \mathbb{R}^d\,, u \in \mathrm{L}^p(\mathbb{R}^d)\,, \end{align*} is presented. In this setting, we give sufficient and necessary geometric conditions on the family of sets $(\Omega(t))$ such that the corresponding Cauchy problem satisfies a final state observability estimate.
On decay rates of the solutions of parabolic Cauchy problems
Dr. Jari Taskinen, University of Helsinki
We consider the Cauchy problem in the Euclidean space $\mathbb{R}^N \ni x$ for the parabolic equation $\partial_t u (x,t) = Au(x,t)$, where the operator $A$ (e.g. the Laplacian) is assumed, among other things, to be a generator of a $C_0$ semigroup in a weighted $L^p$-space $L_w^p(\mathbb{R}^N)$ with $1 \leq p < \infty$ and a fast growing weight $w$. We show that there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_w^p(\mathbb{R}^N)$ with the following property: given an arbitrary positive integer $m $ there exists $n_m > 0$ such that, if the initial data $f$ belongs to the closed linear span of $e_n$ with $n \geq n_m$, then the decay rate of the solution of the problem is at least $t^{-m}$ for large times $t$. In other words, the Banach space of the initial data can be split into two components, where the data in the infinite-dimensional component leads to decay with any pre-determined speed $t^{-m}$, and the exceptional component is finite dimensional. We discuss in detail the needed assumptions of the integral kernel of the semigroup $e^{tA}$. We present variants of the result having different methods of proofs and also consider finite polynomial decay rates instead of unlimited $m$. The results are contained in the following of papers published together with Jos\'e Bonet (Valencia) and Wolfgang Lusky (Paderborn). \smallskip \noindent [1] J.Bonet, W.Lusky, J.Taskinen, Schauder basis and the decay rate of the heat equation. J. Evol. Equations 19 (2019), 717--728. \noindent [2] J.Bonet, W.Lusky, J.Taskinen: On decay rates of the solutions of parabolic Cauchy problems, Proc. Royal Soc. Edinburgh, to appear. \hfill \break DOI:10.1017/prm.2020.48
Orthogonal Polynomials and Special Functions (MS - ID 10)
Analysis and its Applications
Periodic random tiling models and non-Hermitian orthogonality
Prof. Arno Kuijlaars, Katholieke Universiteit Leuven
Certain random tiling models show characteristic features that are similar to the eigenvalues of large random matrices. They can be recast as determinantal point process from which limiting laws as the Tracy-Widom distribution at the edge and the Gaussian free field in the bulk can be deduced. I will discuss a new technique, developed in collaboration with Maurice Duits, to study tiling models with periodic weights. The technique relies on a formulation of a correlation kernel as a double contour integral containing non-Hermitian orthogonal polynomials. In the case of periodic weightings, the orthogonal polynomials are matrix valued. We use the Riemann-Hilbert problem for matrix valued orthogonal polynomials to obtain asymptotics for the two-periodic Aztec diamond. This model is remarkable since it gives rise to a gaseous phase, in addition to the more familiar solid and liquid phases.
Dual bases and orthogonal polynomials
Paweł Woźny, University of Wrocław
%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Dual bases and orthogonal polynomials % Paweł Woźny, University of Wrocław % %%%%%%%%%%%%%%%%%%%%%%%%%%% Let $b_0,b_1,\ldots,b_n$ be linearly independent functions. Let us consider the linear space ${\cal B}_n$ generated by these functions with an inner product $\left<\cdot,\cdot\right>\,:\,{\cal B}_n\times {\cal B}_n\rightarrow\mathbb C$. We say that the functions $D_n:=\{\displaystyle d^{(n)}_0,d^{(n)}_1,\ldots,d^{(n)}_n\}$ form \textit{a dual basis} of the space ${\cal B}_n$ with respect to the inner product $\left<\cdot,\cdot\right>$, if the following conditions hold: \begin{equation*} \left\{ \begin{array}{l} \mbox{span}\left\{d^{(n)}_0,d^{(n)}_1,\ldots,d^{(n)}_n\right\}= {\cal B}_n,\\[2ex] \left<b_i,d^{(n)}_j\right>=\delta_{ij}\qquad (0\leq i,j\leq n), \end{array} \right. \end{equation*} where $\delta_{ii}=1$, and $\delta_{ij}=0$ for $i\neq j$. In general, the dual basis $D_n$ can be found with $O(n^2)$ computational complexity. However, if the dual basis $D_n$ is known, it is possible to construct the dual basis $D_{n+1}$ faster, i.e., with $O(n)$ computational complexity. Dual bases have many applications in numerical analysis, approximation theory or in computer aided geometric design. For example, skillful use of these bases often results in less costly algorithms which solve some computational problems. It is also important that dual bases are very closely related to orthogonal bases. In the first part of the talk, we present general results on dual bases. Next, we focus on some important families of polynomial dual bases and their connections with classical, discrete and $q$-orthogonal polynomials. For example, the so-called dual Bernstein polynomials are related to orthogonal Hahn, dual Hahn and Jacobi polynomials. Using some of these connections, one can find differential-recurrence formulas, differential equation or recurrence relation satisfied by dual Bernstein polynomials. There also exists a first-order non-homogeneous recurrence relation linking dual Bernstein and orthogonal Jacobi polynomials. When used properly, it allows to propose fast and numerically efficient algorithms for evaluating all $n+1$ dual Bernstein polynomials of degree $n$ with $O(n)$ computational complexity.
Khrushchev formulas for orthogonal polynomials
Prof. Luis Velázquez, University of Zaragoza & IUMA
Since their origin in the early 20th century, the theory of orthogonal polynomials on the unit circle (OPUC) has proved to be intimately connected to harmonic analysis via the so called Schur functions. The beginning of this century has witnessed a renewed interest in this connection due to a revolutionary approach to OPUC by the hand of Sergei Khrushchev, which emphasizes the role of continued fractions and Schur functions. The name ``Khrushchev theory”, coined by Barry Simon, refers to a body of methods and results on OPUC originated by this new approach, whose cornerstone is the so called ``Khrushchev formula”. The interest of Schur functions and Khrushchev formulas has been fueled even more by a recently uncovered link between OPUC theory and the study of quantum walks, the quantum version of random walks, where Schur functions are central to develop the quantum version of P\'olya’s renewal theory. This has led to a very general understanding of Khrushchev formula, susceptible to be applied to a wide range of situations covering not only OPUC, but also orthogonal polynomials on the real line (OPRL), matrix valued measures built out of scalar OPUC and OPRL, as well as matrix valued OPUC and OPRL. This talk will give an overview of these different versions of Khrushchev formula, pointing also to their implications in other areas such as harmonic analysis or quantum renewal theory.
A proof of a conjecture of Elbert and Laforgia on the zeros of cylinder functions
Dr. Gergő Nemes, Alfréd Rényi Institute of Mathematics
We prove the enveloping property of the known divergent asymptotic expansion of the large real zeros of the cylinder functions, and thereby answering in the affirmative a conjecture posed by Elbert and Laforgia in 2001 (\emph{J. Comput. Appl. Math.} {\bf 133} (2001), no. 1--2, p. 683). The essence of the proof is the construction of an analytic function that returns the zeros when evaluated along certain discrete sets of real numbers. By manipulating contour integrals of this function, we derive the asymptotic expansion of the large zeros truncated after a finite number of terms plus a remainder that can be estimated efficiently. The conjecture is then deduced as a corollary of this estimate.
Local asymptotics for some $q$-hypergeometric polynomials
Dr. Juan F. Mañas-Mañas, Universidad de Almería
The basic $q$-hypergeometric function $ _r\phi_s$ is defined by the series \begin{equation}\label{q-basic} _r\phi_{s}\left(\begin{array}{l} a_{1}, \ldots, a_{r} \\ b_{1}, \ldots, b_{s} \end{array} ; q, z\right) =\sum_{k=0}^{\infty} \frac{\left(a_{1}; q\right)_{k}\cdots\left(a_{r} ; q\right)_{k}}{\left(b_{1}; q\right)_{k}\cdots\left(b_{s} ; q\right)_{k}}\left((-1)^kq^{\binom{k}{2}}\right)^{1+s-r}\frac{z^{k}}{(q ; q)_{k}}, \end{equation} where $0<q<1$ and $ \left(a_{j}; q\right)_{k}$ and $\left(b_{j}; q\right)_{k}$ denote the $q$-analogues of the Pochhammer symbol. When one of the parameters $a_j $ in (\ref{q-basic}) is equal to $q^{-n}$ the basic $q$-hypergeo-metric function is a polynomial of degree at most $n$ in the variable $z$. Our objective is to obtain a type of local asymptotics, known as Mehler--Heine asymptotics, for $q$-hypergeometric polynomials when $r=s$. Concretely, by scaling adequately these polynomials we intend to get a limit relation between them and a $q$-analogue of the Bessel function of the first kind. Originally, this type of local asymptotics was introduced for Legendre orthogonal polynomials (OP) by the German mathematicians H. E. Heine and G. F. Mehler in the 19th century. Later, it was extended to the families of classical OP (Jacobi, Laguerre, Hermite), and more recently, these formulae were obtained for other families as discrete OP, generalized Freud OP, multiple OP or Sobolev OP, among others. These formulae have a nice consequence about the scaled zeros of the polynomials, i.e. using the well--known Hurwitz's theorem we can establish a limit relation between these scaled zeros and the ones of a Bessel function of the first kind. In this way, we are looking for a similar result in the context of the $q$-analysis and we will illustrate the results with numerical examples.
PDE models in life and social sciences (MS - ID 71)
Partial Differential Equations and Applications
Classification and stability analysis of polarising and depolarising travelling wave solutions for a model of collective cell migration.
Dr. Dietmar Oelz, University of Queensland
We study travelling wave solutions of a 1D continuum model for collective cell migration in which cells are characterised by position and polarity. Four different types of travelling wave solutions are identified which represent polarisation and depolarisation waves resulting from either colliding or departing cell sheets as observed in model wound experiments. We study the linear stability of the travelling wave solutions numerically and using spectral theory. This involves the computation of the Evans function most of which we are able to carry out explicitly, with one final step left to numerical simulation.
Asymptotic behaviour of the run and tumble equation for bacterial chemotaxis
Ms. Havva Yoldaş, Claude Bernard University of Lyon 1
In this talk, I will present a recent work (arXiv:2103.16524) on the long-time behaviour of the run and tumble equation which is a kinetic-transport equation modelling the bacteria movement under the effect of chemical stimulus. The movement of bacteria is a combination of a transport with a constant velocity, run, and a random change in the direction of the movement, tumble. We show the exponential convergence to unique stationary state for the linear run and tumble equation. This result is an improvement of a recent work by Mischler and Weng (Kinet. Relat. Models, 10 (3), 799-822 (2017)). We also consider a weakly nonlinear equation with a nonlocal coupling on the chemoattractant concentration. We construct a unique stationary solution for the weakly non-linear equation and show the exponential convergence towards it. I will also mention how this result give insights of tackling the higher nonlinearities with more physically relevant couplings. The last part is a subject of an ongoing work. This talk is based on joint works with Josephine Evans (University of Warwick).
Asymptotic consensus in the Hegselmann-Krause model with finite speed of information propagation
Dr. Jan Haskovec, King Abdullah University of Science and Technology
We introduce a variant of the Hegselmann-Krause model of consensus formation where information between agents propagates with a finite speed ${\mathfrak{c}}>0$. This leads to a system of ordinary differential equations (ODE) with state-dependent delay. Observing that the classical well-posedness theory for ODE systems does not apply, we provide a proof of global existence and uniqueness of solutions of the model. We prove that asymptotic consensus is always reached in the spatially one-dimensional setting of the model, as long as agents travel slower than ${\mathfrak{c}}$. We also provide sufficient conditions for asymptotic consensus in the spatially multi-dimensional setting. Finally, we discuss the mean-field limit of the model, showing that it does not facilitate a description in terms of a Fokker-Planck equation.
Partial differential equations describing far-from-equilibrium open systems (MS - ID 51)
Partial Differential Equations and Applications
Thermodynamics of viscoelastic rate-type fluids and its implications for stability analysis
Vít Průša, Charles University
We introduce a thermodynamic basis for some non-Newtonian fluids, namely we explicitly characterise energy storage and entropy production mechanisms that lead to the frequently used viscoelastic rate-type models such as the Oldroyd-B model, the Giesekus model, the Phan-Thien--Tanner model, the Johnson--Segalman model, the Bautista--Manero--Puig model and their diffusive variants. Knowing the thermodynamical basis of the models, we show how this knowledge can be used in nonlinear (finite amplitude) stability analysis of steady flows of these fluids.
Ergodic theory for energetically open fluid systems
Eduard Feireisl, Czech Academy of Sciences
We consider global in time (weak) solutions for the complete Navier-Stokes-Fourier system describing the motion of a compressible, viscous and heat conducting fluid driven by inhomogeneous boundary conditions. We show that any globally bounded trajectory generates a stationary statistical solution. Then the Birkhoff-Khinchin theorem can be applied to show the validity of the (weak) ergodic hypothesis on the associated omega-limit set.
A simple thermodynamic framework for heat-conducting flows of mixtures of two mechanically interacting fluids
Prof. Josef Malek, Charles University, Faculty of Mathematics and Physics
Within the theory of interacting continua, we develop a model for a heat conducting mixture of two mechanically interacting fluids described in the terms of the densities and the velocities for each fluid and the temperature field for the whole mixture. We use a general thermodynamic framework that determines the response of the material from the knowledge of two pieces of information, namely how the material stores the energy and how the energy of material is dissipated. This information is expressed in the form of the constitutive equations for two scalars: the Helmholtz free energy and the entropy production. Additionally, we follow the goal to determine the response of a mixture from a small (minimal) set of material parameters (bulk and shear viscosity, heat conductivity, the drag coefficient) that can be associated with the mixture as the whole. The same thermodynamic approach is used to obtain the model when the whole mixture responses as an incompressible material. For both the compressible and incompressible variants, we investigate two variants stemming from different definitions of the velocity associated with the mixture as a whole.
Non-isothermal viscoelastic flows with conservation laws and relaxation
Mr. Mark Dostalík, Charles University
We propose a system of balance laws for non-isothermal viscoelastic flows of Maxwell fluids. The system is an extension of the polyconvex elastodynamics of hyperelastic bodies using additional structure variables. We establish a strictly convex mathematical entropy to show that the proposed system is symmetrizable and, as a consequence, we obtain the short-time existence and uniqueness of smooth solutions which define genuinely causal viscoelastic flows with waves propagating at finite speed. To model heat-conductors the system is complemented by the hyperbolic Maxwell–Cattaneo heat conduction law.
Polynomial decay of solutions to integrodifferential equations
Mr. Tomas Barta, Charles University, Prague
We study convergence to equilibrium for second order integrodifferential equations. The speed of convergence depends on the decay of the convolution kernel, the right-hand side and the Lojasiewicz exponent. The result is valid provided one a-priori knows that the solution has a precompact trajectory.
Hydrodynamic stability for the dynamic slip
Mr. Michael Zelina, Charles University, Faculty of Mathematics and Physics
We consider the incompressible Navier-Stokes equation with the dynamic slip boundary condition. Our first goal is to prove the so-called linearization principle in the class of weak solutions satisfying the energy inequality. By this we mean that if the spectrum of certain operator has only positive real parts, then the stationary solution $u^*$ of the Navier-Stokes equation is stable with respect to sufficiently small initial perturbations. We deal further with two explicit geometries, namely with either two infinite parallel planes or two concentric cylinders, where the solution $u^*$ corresponds either to Couette/Poiseuille or Taylor-Couette flow. We eventually compare our results with well-known analogue results in the case of Dirchlet boundary condition.
Rational approximation for data-driven modeling and complexity reduction of linear and nonlinear dynamical systems (MS - ID 69)
Dynamical Systems and Ordinary Differential Equations and Applications
Spurious poles
Prof. Nick Trefethen, Oxford University
Spurious poles of rational approximations are fascinating in theory and troublesome in practice. This talk reviews the mathematical and computational sides of this subject and explores the use of the two-step method of AAA-least squares approximation for avoiding the problem.
Mixed interpolatory and inference for non-intrusive reduced order nonlinear modelling
Dr. Charles Poussot-Vassal, Onera
Based on input-output time-domain raw data collected from a complex simulator, the Mixed Interpolatory Inference (\textbf{MII}) process approach allows to infer a reduced-order linear or nonlinear (e.g. bilinear or quadratic) time invariant dynamical model of the form \begin{equation} \left \lbrace \begin{array}{rcl} \mathbf{\dot{\hat x}} &=& \hat A \mathbf{\hat x} + \hat B \mathbf{u} + \mathbf{\hat f}(\mathbf{\hat x},\mathbf{u})\\ \mathbf{\hat y} &=& \hat C \mathbf{\hat x} + \hat D \mathbf{u}+ \mathbf{\hat g}(\mathbf{\hat x},\mathbf{u}) \end{array} \right. , \label{eq:fODE} \end{equation} that accurately reproduces the underlying phenomena dictated by the raw data. In \eqref{eq:fODE}, $\mathbf{\hat x}(\cdot)\in\mathbb R^r$, $\mathbf{u}(\cdot)\in\mathbb R^{n_u}$ and $\mathbf{\hat y}(\cdot)\in\mathbb R^{n_y}$, denote the reduced internal, input and approximated output variables respectively. Moreover, $\mathbf{\hat f}$ and $\mathbf{\hat g}$ denote either quadratic or bilinear functions. The approach is essentially based on the sequential combination of \textbf{rational interpolation} (e.g. Pencil, Loewner, AAA) with a \textbf{linear least square} resolution. With respect to intrusive methods, no prior knowledge on the operator is needed. In addition, compared to the traditional non-intrusive operator inference approaches, the proposed rationale alleviates the need of measuring and storing the original full-order model internal variables. It is thus applicable to a wider range of applications than the standard intrusive and non-intrusive methods. It is therefore very close to the identification field. The \textbf{MII} is successfully applied on different numerically challenging application related to pollutant dispersion. First \emph{(i)} a large eddy simulation of a pollutants dispersion case over an airport area, and second \emph{(ii)} a flow simulation over a building, both involving multi-scale and multi-physics dynamical phenomena. Despite the simplicity of the resulting low complexity model, the proposed approach shows satisfactory results to predict the pollutants plume pattern while being significantly faster to simulate.
Numerical aspects of the Koopman and the dynamic mode decomposition for model reduction
Prof. Zlatko Drmac, University of Zagreb, Faculty of Science
The Dynamic Mode Decomposition (DMD) is a tool of trade in computational data driven analysis of complex dynamical systems, e.g. fluid flows, where it can be used to decompose the flow field into component fluid structures, called DMD modes, that describe the evolution of the flow. The DMD is deeply connected with the analysis of nonlinear dynamical systems in terms of the spectral data of the associated Koopman operator. The main tasks of the analysis are approximations of eigenfunctions and eigenvalues of the operator, and computation of a spatio-temporal representation of the data snapshots, based on the computed eigenvalues and eigenfunctions. The latter includes possibly ill-conditioned least squares problem. Exceptional performances motivated developments of several modifications that make the DMD an attractive method for identification, analysis and model reduction of nonlinear systems in data driven settings. In this talk, we will present our recent results on the numerical aspects of the DMD/Koopman analysis. We show how the state of the art numerical linear algebra can be deployed to improve the numerical performances in cases that are usually considered notoriously ill-conditioned. The numerical framework is based on algorithms that also apply e.g. to matrix rational approximations in modeling by Vector Fitting.
The use of rational approximation for linearization of models that are nonlinear in the frequency
Prof. Karl Meerbergen, KU Leuven
Finite element models for the analysis of vibrations typically have a quadratic dependency on the frequency. This makes the finite element method suitable for eigenvalue computations and time integration, by a formulation as a first order system, which we call a linearization. The study of new damping materials often leads to nonlinear frequency dependencies, sometimes represented by a rational functions but, often, by truly nonlinear functions. In classical analyses, vibrations are studied in the frequency domain. The nonlinear frequency dependency is an issue for algorithms for fast frequency sweeping. In the context of numerical algorithms for digital twins, time integration of mathematical models is required, which is not straightforward for models that are not linear or polynomial in the frequency. We will discuss rational approximation and linearization of nonlinear frequency dependencies and their use for fast frequency sweeping and time integration. In particular, we use the AAA rational approximation and the associated linearization based on the barycentric Lagrange formulation of rational functions, which is successfully used for solving nonlinear eigenvalue problems. We show how real valued matrices can be obtained. We also show how the parameters can be tuned to obtain a stable linear model.
Structured Realization Based on Time-Domain Data
Philipp Schulze, Technische Universität Berlin
In this talk we present a method for constructing a continuous-time linear time-invariant system based on discrete samples of an input/output trajectory of the system. Especially, our approach allows to impose different structures on the constructed system including structures like second-order systems, systems with time-delay in the state, and fractional systems. The proposed method consists of first using the measured time-domain data to estimate the transfer function at selected points in the frequency domain by using a modified version of the empirical transfer function estimation presented in [Peherstorfer, Gugercin, Willcox, SIAM J. Sci. Comput., 39(5):2152–2178, 2017]. Afterward, we construct a structured realization based on the estimated frequency data such that the transfer function of the obtained realization interpolates the frequency data. The effectiveness of this new approach is illustrated by means of a numerical example.
Structure-Preserving Interpolation for Bilinear Systems
Mr. Steffen W. R. Werner, Max Planck Institute for Dynamics of Complex Technical Systems
The modeling of natural processes as population growth, mechanical structures and fluid dynamics, or stochastic modeling often results in bilinear time-invariant dynamical systems \begin{align} \label{eqn:bsys} \begin{aligned} E\dot{x}(t) & = Ax(t) + \sum\limits_{j = 1}^{k}N_{j}x(t)u_{j}(t) + Bu(t),\\ y(t) & = Cx(t), \end{aligned} \end{align} with $E, A, N_{j} \in \mathbb{R}^{n \times n}$, for $j = 1, \ldots, m$, $B \in \mathbb{R}^{n \times m}$ and $C \in \mathbb{R}^{p \times n}$. The aim of model reduction for~\eqref{eqn:bsys} is the reduction of related computational resources, like time and memory for the simulation of~\eqref{eqn:bsys}, by the reduction of internal states $n$, while approximating the input-to-output behavior of the system. Often related to the underlying applications, bilinear systems~\eqref{eqn:bsys} can have special structures that one wants to preserve in the reduced-order model as, e.g., in case of bilinear mechanical systems \begin{align} \label{eqn:bsosys} \begin{aligned} M \ddot{q}(t) + D \dot{q}(t) + K q(t) & = \sum\limits_{j = 1}^{m} N_{\mathrm{p},j} q(t) u_{j}(t) + \sum\limits_{j = 1}^{m} N_{\mathrm{v},j} \dot{q}(t) u_{j}(t) + B_{\mathrm{u}}u(t),\\ y(t) & = C_{\mathrm{p}} q(t) + C_{\mathrm{v}} \dot{q}(t), \end{aligned} \end{align} with $M, D, K, N_{\mathrm{p}, j}, N_{\mathrm{v}, j} \in \mathbb{R}^{n \times n}$, for $j = 1, \ldots, m$, $B_{\mathrm{u}} \in \mathbb{R}^{n \times m}$ and $C_{\mathrm{p}}, C_{\mathrm{v}} \in \mathrm{R}^{p \times n}$. In case of linear systems, structured-preserving interpolation of the underlying transfer function in the frequency domain can be used to efficiently construct reduced-order models with the same structure as the original system~\cite{morBeaG09}. We present an extension of the structure-preserving interpolation framework to the bilinear system case, which we describe in the frequency domain by general multivariate transfer functions \begin{align} \label{eqn:btf} \begin{aligned} G_{k}(s_{1}, \ldots, s_{k}) & = \mathcal{C}(s_{k})\mathcal{K}(s_{k})^{-1} \left(\prod\limits_{j = 1}^{k-1} (I_{m^{j-1}} \otimes \mathcal{N}(s_{k-j}))(I_{m^{j}} \otimes \mathcal{K}(s_{k-j})^{-1}) \right)\\ & \quad{}\times{} (I_{m^{k-1}} \otimes \mathcal{B}(s_{1})), \end{aligned} \end{align} for $k \geq 1$ and where $\mathcal{N}(s) = \begin{bmatrix} \mathcal{N}_{1}(s) & \ldots & \mathcal{N}_{m}(s) \end{bmatrix}$ with the matrix functions $\mathcal{C}: \mathbb{C} \rightarrow \mathbb{C}^{p \times n}$, $\mathcal{K}: \mathbb{C} \rightarrow \mathbb{C}^{n \times n}$, $\mathcal{B}: \mathbb{C} \rightarrow \mathbb{C}^{n \times m}$, $\mathcal{N}_{j}: \mathbb{C} \rightarrow \mathbb{C}^{n \times n}$ for $j = 1, \ldots, m$. We develop a projection-based, structure-preserving interpolation framework for bilinear systems associated with~\eqref{eqn:btf} that allows the efficient construction of reduced-order structured bilinear systems. % *** REFERENCES *** \bibliographystyle{plainurl} \begin{thebibliography}{1} \bibitem{morBeaG09} C.~A. Beattie and S.~Gugercin. \newblock Interpolatory projection methods for structure-preserving model reduction. \newblock {\em Syst. Control Lett.}, 58(3):225--232, 2009. \newblock doi:10.1016/j.sysconle.2008.10.016. \end{thebibliography}
Rational interpolation and model order reduction for data-driven controller design
Dr. Pauline Kergus, LTH
In many control applications, a mathematical description of the system, derived from physical laws, is not available. In this case, the controller has to be designed on the basis of experimental measurements. This work presents a data-driven control strategy based on the Loewner framework, where a reduced-order controller is directly obtained from the available experimental data. Rational interpolation is also used to build achievable specifications and to ensure closed-loop stability for the controlled system. No parametric model of the system is used, allowing to handle applications in which the model of the system might be too complicated or too difficult to obtain for traditional model-based strategies. This technique is particularly appealing to control infinite dimensional systems, such as the ones described by linear partial differential equations. Such an example, a crystallizer (common in the chemical industry), is tackled in this work.
Comparison of greedy-type approaches involving the Loewner matrix for rational modeling
Dr. Sanda Lefteriu, IMT Lille Douai
The Loewner framework has established itself as a popular choice for building rational approximations in barycentric form. The Loewner together with the shifted Loewner matrices are built from measurements and, together with the data matrices, yield a high-order rational model for a potentially non-rational function. To eliminate the redundancy in the data, an SVD step of the Loewner matrix pencil is involved. However, as the size of these matrices is equal to half of the number of measurements, this step is rather costly for large data sets when the traditional SVD is employed. This talk compares several greedy-type approaches which use the same principle: starting from an order 1 approximant, points from the available data set are added in a greedy fashion by minimizing the error measure of choice. These approaches considered in the comparison are: AAA \cite{AAA}, the CUR decomposition \cite{CUR}, DEIM-CUR decomposition \cite{DEIM}, as well as the adaptive and recursive approaches \cite{TCAD}. \begin{thebibliography}{1} \bibitem{AAA} Y. Nakatsukasa, O. Sète, L. N. Trefethen. \textit{The AAA algorithm for rational approximation}. SIAM Journal on Scientific Computing, 40(3):A1494–A1522, Jan. 2018. \bibitem{CUR} D. S. Karachalios, I. V. Gosea, A. C. Antoulas. \textit{Data-driven approximation methods applied to non-rational functions}. Proc. Appl. Math. Mech., 18(1), 2018. \bibitem{DEIM} D. C. Sorensen, M. Embree. \textit{A DEIM induced CUR factorization}, SIAM Journal on Scientific Computing, 38(3): A1454–A1482, 2016. \bibitem{TCAD} S. Lefteriu, A. C. Antoulas. \textit{A New Approach to Modeling Multiport Systems From Frequency-Domain Data}, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 29(1):14-27, Jan. 2010. \end{thebibliography}
Recent Developments on Preservers (MS - ID 38)
Analysis and its Applications
Linear maps which are (triple) derivable or anti-derivable at a point
Prof. Antonio M. Peralta, Universidad de Granada
A typical challenge in the setting of preservers asks whether a linear map $T$ from a C$^*$-algebra $A$ into a Banach $A$-bimodule $X$ behaving like a derivation (i.e. $D(a b) = D(a) b + a D(b)$) or like an anti-derivation ($D(a b) = D(b) a + b D(a)$) only on those pairs of elements $(a,b)$ in a proper subset $\mathfrak{D}\subset A^2$ is in fact a derivation or an anti-derivation. A protagonist role is played by sets of the form $\mathfrak{D}_z :=\{ (a,b)\in A^2 : a b =z\},$ where $z$ is a fixed point in $A$. A linear map $T: A\to X$ is said to be a \emph{derivation} or an \emph{anti-derivation at a point} $z\in A$ if it behaves like a derivation or like an anti-derivation on pairs $(a,b)\in\mathfrak{D}_z$. These maps are usually called \emph{derivable or anti-derivable at $z$}. Let us simply observe that applying a similar method to define linear maps which are homomorphisms at zero, we find a natural link with the fruitful line of results on zero products preservers.\smallskip %It is known that every continuous linear map $\delta$ on a von Neumann algebra is a generalized derivation whenever it is derivable at zero, and if we further asume $\delta(1)=0$ the mapping $\delta$ is in fact a derivation (see \cite[Theorem 4]{JingLuLi2002}). For each infinite dimensional Hilbert space $H$, a linear map $\delta : B(H) \to B(H)$ which is a generalized Jordan derivation at zero, or at 1, is a generalized derivation, even if $\delta$ is not assumed to be a priori continuous (cf. \cite{Jing2009}). Concerning $B(H)$, a linear map $\delta :B(H)\to B(H)$ is a derivation if and only if it is a derivation at a non-zero point in $B(H)$.\smallskip A recent study developed by B. Fadaee and H. Ghahramani in \cite{FaaGhahra2019} characterizes continuous linear maps from a C$^*$-algebra $A$ into its bidual which are derivable at zero. A similar problem was considered by H. Ghahramani and Z. Pan for linear maps on a complex Banach algebra which is zero product determined \cite{GhaPan2018}. These authors also find necessary conditions to guarantee that a continuous linear map $T : A \to A^{**}$ is anti-derivable at zero, where $A$ is a C$^*$-algebra, and also for linear maps on a zero product determined unital $^*$-algebra to be anti-derivable at zero. \smallskip We have been involved in the study of those linear maps on C$^*$-algebras which are derivations or triple derivations at zero or at the unit \cite{EssPe2018}. We shall revisit some of the main conclusions on these kind of maps from the perspective of preservers. We have further explored in \cite{AbulJamPe19} whether a full characterization of those (continuous) linear maps on a C$^*$-algebra which are ($^*$-)anti-derivable at zero can be given in pure algebraic terms. In this talk we shall present the latest advances in \cite{AbulJamPe19}, which provide a complete solution to this problem. \begin{thebibliography}{99} \bibitem{AbulJamPe19} D.A. Abulhamail, F.B. Jamjoom, A.M. Peralta, Linear maps which are anti-derivable at zero, to appear in \emph{Bull. Malays. Math. Sci. Soc.}. arXiv: 1911.04134v2 \bibitem{EssPe2018} A.B.A. Essaleh, A.M. Peralta, Linear maps on C$^*$-algebras which are derivations or triple derivations at a point, \emph{Linear Algebra and its Applications} \textbf{538}, 1-21 (2018). \bibitem{FaaGhahra2019} B. Fadaee, H. Ghahramani, Linear maps behaving like derivations or anti-derivations at orthogonal elements on C$^*$-algebras, to appear in \emph{Bull. Malays. Math. Sci. Soc.} https://doi.org/10.1007/s40840-019-00841-6. arXiv:1907.03594v1 \bibitem{GhaPan2018} H. Ghahramani, Z. Pan, Linear maps on $^*$-algebras acting on orthogonal elements like derivations or anti-derivations, \emph{Filomat} \textbf{32}, no. 13, 4543-4554 (2018). \end{thebibliography}
On the linearity of order-isomorphisms
Dr. Bas Lemmens, University of Kent
A basic problem in the theory of partially ordered vector spaces is to understand when order-isomorphisms are affine. This depends in a subtle way on the geometry of the cones involved. In this talk I will discuss some recent progress on this problem, mainly based on joint with van Gaans and van Imhoff. We will introduce a new condition on the extreme rays of the cone, which ensures that all order-isomorphisms are affine. The condition is milder than existing ones and is satisfied by, for example, the cone of positive operators in the space of bounded self-adjoint operators on a Hilbert space.
Meet preservers between lattices of real-valued continuous functions
Prof. Kristopher Lee, Iowa State University
It is well-know that the set $C(X)$ of real-valued continuous functions defined on a compact Hausdorff space $X$ becomes a lattice when equipped with the usual point-wise ordering; in particular, the join and meet of $f,g \in C(X)$ are given by \begin{equation*} (f \vee g)(x) = \max \{f(x),g(x)\} \quad \textnormal{and} \quad (f \wedge g)(x) = \min \{f(x),g(x)\}, \end{equation*} respectively. We will demonstrate that any surjective $T \colon C(X) \to C(Y)$ satisfying \begin{equation*} \operatorname{Ran}_\pi(f \wedge g) = \operatorname{Ran}_\pi(T(f) \wedge T(g)) \end{equation*} for all $f,g \in C(X)$, where $\operatorname{Ran}_\pi(\cdot)$ denotes the set of range values of maximum absolute value, induces a homeomorphism $\psi \colon Y \to X$ such that \begin{equation*} T(f) = f \circ \psi \end{equation*} holds for all $f \in C(X)$ with $0 \leq f$.
Maps on positive definite cones of $C^*$-algebras preserving the Wasserstein mean
Prof. Lajos Molnar, University of Szeged
In the past years, we have obtained some structural results for bijective maps between positive definite cones of operator algebras which preserve some specific Kubo-Ando means (power means, geometric mean). Recently, Bhatia, Jain, and Lim have introduced a new mean for positive definite matrices called Wasserstein mean. Its importance lies in its close connection to the Bures-Wasserstein metric. In this talk, we give the complete description of all (continuous) isomorphisms between positive definite cones of $C^*$-algebras with respect to the operation of the Wasserstein mean and present a result concerning the nonexistence of nonconstant such morphisms into the positive reals. Comments on the algebraic properties of the Wasserstein mean relating associativity are also made.
Isometries of Wasserstein spaces
Dr. Daniel Virosztek, Institute of Science and Technology Austria
I will report on our systematic study of isometries of classical Wasserstein spaces with various underlying spaces. The starting point was the case of the discrete underlying space, where we found a rich family of non-surjective isometries. However, we proved isometric rigidity on the contrary, which means that every surjective isometry is governed by a permutation of the underlying space [G.-T.-V., J. Math. Anal. Appl. 480 (2019), 123435]. \par The next step was the description of the isometries of $\mathcal{W}_p(\mathbb{R})$ for $p \neq 2.$ Here, we proved isometric rigidity and classified the non-surjective isometries for $p>1,$ as well. The study of $\mathcal{W}_p([0,1])$ led to the discovery of a mass-splitting isometry for $p=1,$ which turned out to be also a key step in giving an affirmative answer to Kloeckner's questions from 2010 concerning the existence of exotic and mass-splitting isometries on quadratic Wasserstein spaces [G.-T.-V., Trans. Amer. Math. Soc. 373 (2020), 5855-5883]. \par The most recent work of ours concerns Wasserstein-Hilbert spaces. We extended Kloeckner's result on the quadratic case to the infinite-dimensional setting and proved isometric rigidity for the non-quadratic cases. The main tool we introduced to study the non-quadratic cases is the Wasserstein potential of measures. As a byproduct of our results, we showed that $\mathcal{W}_p(X)$ is isometrically rigid for every Polish space $X$ and parameter $0<p<1$ [G.-T.-V., arXiv:2102.02037 (2021)].
Wigner's theorem in normed spaces
Prof. Dijana Ilišević, University of Zagreb
Let $(H,(\cdot,\cdot))$ and $(K,(\cdot,\cdot))$ be inner product spaces over $\mathbb F\in\{\mathbb R,\mathbb C\}$ and suppose that $f \colon H\to K$ is a mapping satisfying \begin{equation}| (f(x),f(y))|=|(x,y)|,\quad x,y\in H.\end{equation} Then the famous Wigner's unitary--antiunitary theorem says that $f$ is a solution of (1) if and only if it is phase equivalent to a linear or an anti-linear isometry, say $U$, that is, $$f(x)=\sigma(x) Ux,\quad x\in H,$$ where $\sigma \colon H\to\mathbb F$, $|\sigma(x)|=1$, $x\in H$, is a so called phase function. In this talk several generalizations of this theorem to the setting of normed spaces will be presented.
On maps preserving products equal to fixed elements
Prof. Louisa Catalano, Union College
In this talk, we will characterize the form of bijective linear maps $f: M_n(\mathbb{C})\to M_n(\mathbb{C})$ satisfying $f(A)f(B) = M$ whenever $AB=N$, where $M$ and $N$ are fixed $n\times n$ complex matrices.
Linear functions preserving Green's relations over fields
Mr. Artem Maksaev, Lomonosov Moscow State University
The talk is based on the joined work with A. Guterman, M. Johnson, and M. Kambites \cite{GJKM}. Green's relations are a number of equivalence relations and pre-orders which are defined upon any semigroup. Introduced by J. Green \cite{Green} in 1951, they encapsulate the ideal structure of the semigroup, and play a central role in almost every aspect of semigroup theory. The following are the definitions of Green's relations ${\mathcal R}, {\mathcal L}, {\mathcal J}, {\mathcal H}, {\mathcal D}$. \vspace{4pt} \textbf{Definition.} Let ${\mathcal M}$ be a monoid. For $a, b \in {\mathcal M}$, we say that: \begin{itemize} \item[(i)] $a \, {\mathcal R} \, b$ if $a{\mathcal M} = b{\mathcal M}$; \item[(ii)] $a \, {\mathcal L}\, b$ if ${\mathcal M} a = {\mathcal M} b$; \item[(iii)] $a \, {\mathcal J}\, b$ if ${\mathcal M} a {\mathcal M} = {\mathcal M} b {\mathcal M}$; \item[(iv)] $a \, {\mathcal H}\, b$ if $a \, {\mathcal R}\, b$ and $a \, {\mathcal L}\, b$; \item[(v)] $a \, {\mathcal D}\, b$ if there exists $ c \in {\mathcal M}\colon a \, {\mathcal R}\, c$ and $c \, {\mathcal L}\, b$. \end{itemize} In particular, these are natural relations to define upon the set of $n \times n$ matrices over any semiring, when viewed as a semigroup under matrix multiplication. The investigation of linear transformations preserving natural functions, invariants and relations on matrices has a long history, dating back to a result of Frobenius \cite{Frobenius} describing maps which preserve the determinant. During the past century a lot of effort has been devoted to the development of this theory, particularly, for matrices over semirings. In 2018, motivated by recent interest in the structure of the tropical semifield, A. Guterman, M. Johnson, and M. Kambites \cite{GutermanJohnsonKambites} characterized bijective linear maps which preserve (or strongly preserve) each of Green's relations on the space of $n \times n$ matrices over an anti-negative semifield. The results of \cite{GutermanJohnsonKambites} are rather unusual, since they provided a classification for all semifields except fields. Thus, the question arises, what the corresponding maps over fields are. The aim of this talk is to answer it. Let $\mathbb{F}$ be a field and $M_n(\mathbb{F})$ be the monoid of $n \times n$ matrices over $\mathbb{F}$. Based on a convenient description of Green's relations for $M_n(\mathbb{F})$, we present a complete classification of linear maps $T\colon M_n(\mathbb{F}) \to M_n(\mathbb{F})$ preserving each of these relations. However, some additional assumptions are needed, namely: \begin{itemize} \item for all relations (${\mathcal R}, {\mathcal L}, {\mathcal J}, {\mathcal H}, {\mathcal D}$), the classification was carried out under the assumption that $T$ is bijective; \item for the relations ${\mathcal R}, {\mathcal L}, {\mathcal H}$, it was done under the assumption that the field $\mathbb{F}$ contains roots of all polynomials from $\mathbb{F}[x]$ of degree $n$ (particularly, for algebraically closed field); \item for the relations ${\mathcal J}, {\mathcal D}$, no additional assumptions are needed. \end{itemize} Also, for the relation ${\mathcal H}$, it holds that non-zero ${\mathcal H}$-preservers coincide exactly with invertibility preservers, which classification is a well-known preserver problem. Over an arbitrary field, linear maps preserving invertibility have been fully described by de Seguins Pazzis \cite{dpazzis}. Furthermore, we show some examples of non-bijective linear ${\mathcal L}$-, ${\mathcal R}$-, and ${\mathcal H}$-preservers that are not of the form mentioned in the classification. This shows the importance of the restrictions on the field or map stated above. \begin{thebibliography}{99} \bibitem{Green} J. A. Green. On the structure of semigroups. Annals of Math. {\bf 54} (1951), 163-172. \bibitem{Frobenius} G. Frobenius. Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsber. Deutsch. Akad. Wiss. (1897), 994-1015. \bibitem{GutermanJohnsonKambites} A. Guterman, M. Johnson, M. Kambites. Linear isomorphisms preserving Green's relations for matrices over anti-negative semifields, Linear Algebra Appl. {\bf 545} (2018), 1-14. \bibitem{dpazzis} C. de Seguins Pazzis. The singular linear preservers of non-singular matrices, Linear Algebra Appl. {\bf 433} (2010), 483--490. \bibitem{GJKM} A. Guterman, M. Johnson, M. Kambites, A. Maksaev. Linear functions preserving Green's relations over fields, Linear Algebra Appl. {\bf 611} (2021), 310-333. \end{thebibliography}
Singularity preserving maps on matrix algebras
Mr. Valentin Promyslov, Lomonosov Moscow State University
The talk is based on the joined work with Alexander Guterman and Artem Maksaev. The first result on linear preservers was obtained by Ferdinand Georg Frobenius, who characterized linear maps on complex matrix algebra preserving the determinant. Let $M_n(\mathbb{F})$ be the $n \times n$ matrix algebra over a field $\mathbb{F}$ and $\mathcal{Y}$ be a subset of $M_n(\mathbb{F})$. We say that a transformation $T\colon \mathcal{Y} \to~M_n(\mathbb{F})$ is of a {\it standard form} if there exist non-singular matrices $P, Q$ such that \begin{equation}\label{standart_form} T(A)=PAQ \quad \text{or} \quad T(A)=PA^TQ \quad \text{ for all } A \in \mathcal{Y}. \end{equation} Frobenius \cite{Frobenius} proved that if $T\colon M_n(\mathbb{C}) \to~M_n(\mathbb{C})$ is linear and preserves the determinant, i.\,e., $\operatorname{det}(T(A))=\operatorname{det}(A) \text{ for all } A \in M_n(\mathbb{C}),$ then $T$ is of the standard form (\ref{standart_form}) with $\operatorname{det}(PQ)=1$. In 1949 Jean Dieudonn\'e \cite{Dieudonne} generalized this result for an arbitrary field $\mathbb{F}$. He replaced the determinant preserving condition by the singularity preserving condition and proved the corresponding result for a bijective map $T$. In 2002 Gregor Dolinar and Peter \v{S}emrl \cite{DS} modified the classical result of Frobenius by removing the linearity and replacing the determinant preserving condition by \begin{equation} \label{bundle_conditions_on_determinant} \operatorname{det}(A +\lambda B)=\operatorname{det}(T(A)+\lambda T(B)) \quad \text{for all } A,B \in M_n(\mathbb{F}) \text{ and all } \lambda \in \mathbb{F} \end{equation} for $\mathbb{F}=\mathbb{C}.$ They proved that if $T\colon M_n(\mathbb{C}) \to~M_n(\mathbb{C})$ is surjective and satisfies (\ref{bundle_conditions_on_determinant}), then $T$ is linear and hence is of the standard form (\ref{standart_form}) with $\operatorname{det}(PQ)=1$. Soon after that, Victor Tan and Fei Wang \cite{Tan-Wang} generalized this proof for a field $\mathbb{F}$ with $|\mathbb{F}|>n$ and showed that under the condition (\ref{bundle_conditions_on_determinant}) the map $T$ is linear even without the surjectivity condition. Moreover, they revealed that if $T$ is surjective, then only two different values of $\lambda$ are required in (\ref{bundle_conditions_on_determinant}). To be more precise, if $|\mathbb{F}|>n$ and $T\colon M_n(\mathbb{F}) \to~M_n(\mathbb{F})$ is a surjective map satisfying \begin{equation*} \operatorname{det}(A +\lambda_i B)=\operatorname{det}(T(A)+\lambda_i T(B)) \quad \text{for all } A,B \in M_n(\mathbb{F}) \text{ and } i=1,2, \end{equation*} where $\lambda_i\neq 0$ and $(\lambda_1/\lambda_2)^k \neq 1$ for $1\leqslant k \leqslant n-2$, then $T$ is of the standard form (\ref{standart_form}). Nevertheless, this result was also further generalized by Constantin Costara~\cite{Costara}. Suppose $|\mathbb{F}|>n^2$ and $\lambda_0 \in \mathbb{F}$. Let $T\colon M_n(\mathbb{F}) \to~M_n(\mathbb{F})$ be a surjective map satisfying (\ref{bundle_conditions_on_determinant}) only for one fixed value of $\lambda=\lambda_0:\ \operatorname{det}(A +\lambda_0 B)=\operatorname{det}(T(A)+\lambda_0 T(B)) \quad \text{for all } A,B \in M_n(\mathbb{F}).$ Costara obtained that if $\lambda_0 \neq -1$, then such $T$ is of the standard form (\ref{standart_form}) with $\operatorname{det}(PQ)=1$. For $\lambda_0 = -1$, he showed that there exist $P,Q \in GL_n(\mathbb{F}),\ \operatorname{det}(PQ)=1$, and $A_0 \in M_n(\mathbb{F})$ such that $$T(A)=P(A+A_0)Q \quad \text{or} \quad T(A)=P(A+A_0)^TQ \quad \text{ for all } A \in \mathcal{Y}.$$ The aim of this work is to relax the condition (\ref{bundle_conditions_on_determinant}) for $T$. It has been revealed that if $\mathbb{F}$ is an algebraically closed field, then the conditions on determinant in the above results of Dieudonn\'e or Tan and Wang can be replaced by less restrictive. The following result has been obtained. {\bf Theorem.} Suppose $\mathcal{Y}=GL_n(\mathbb{F})$ or $\mathcal{Y}=M_n(\mathbb{F}),$ $T\colon \mathcal{Y} \to~M_n(\mathbb{F})$ is a map satisfying the following conditions: \begin{itemize} \item for all $A, B \in \mathcal{Y}$ and $\lambda \in \mathbb{F}$, the singularity of $A+\lambda B$ implies the singularity of $T(A)+\lambda T(B)$; \item the image of $T$ contains at least one non-singular matrix. %there exists a matrix $A \in GL_n(\mathbb{F})$ such that $T(A) \in GL_n(\mathbb{F}).$ \end{itemize} Then $T$ is of the standard form (\ref{standart_form}). (Note that in the theorem above $\operatorname{det}(PQ)$ possibly differs from $1$.) \begin{thebibliography}{99} \bibitem{Frobenius} G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsber. Deutsch. Akad. Wiss. (1897), pp. 994–1015. \bibitem{Dieudonne} D. J. Dieudonn\'e, Sur une g\'en\'eralisation du groupe orthogonal \'a quatre variables, Arch. Math. 1 (1949), pp. 282–287. \bibitem{DS} G. Dolinar, P. \v{S}emrl, Determinant preserving maps on matrix algebras, Linear Algebra Appl. 348 (2002), pp. 189–192. \bibitem{Tan-Wang} V. Tan, F. Wang, On determinant preserver problems, Linear Algebra Appl., 369 (2003), pp. 311-317. \bibitem{Costara} C. Costara, Nonlinear determinant preserving maps on matrix algebras, Linear Algebra Appl., 583 (2019), pp. 165–170. \end{thebibliography}
Spectral Graph Theory (MS - ID 46)
Combinatorics and Discrete Mathematics
Almost mixed Moore graphs and their spectra
Dr. NACHO (IGNACIO) LOPEZ LORENZO, Universitat de Lleida
{\em Almost mixed Moore graphs\/} appear in the context of the {\em Degree$/$Diameter problem} as a class of extremal mixed graphs, in the sense that their order is one less than the Moore bound for mixed graphs. In this talk we will give some necessary conditions for the existence of almost mixed Moore graphs derived from the factorization in $\mathbb{Q}[x]$ of their characteristic polynomial. In this context, we deal with the irreducibility of certain polynomials $\Phi_i(x) \circ f(x)$, where $\Phi_i(x)$ denotes the $i$-th cyclotomic polynomial.
Optimal Grid Drawings of Complete Multipartite Graphs and an Integer Variant of the Algebraic Connectivity
Mr. Clemens Huemer, Universitat Politècnica de Catalunya
We use spectral graph theory to show how to draw the vertices of a complete multipartite graph G on different points of a bounded d-dimensional integer grid, such that the sum of squared distances between vertices of G is (i) minimized or (ii) maximized. For both problems we provide a characterization of the solutions. For the particular case d = 1, our solution for (i) also settles the minimum-2-sum problem for complete bipartite graphs; the minimum-2-sum problem was defined by Juvan and Mohar in 1992. Weighted centroidal Voronoi tessellations are the solution for (ii). Such drawings are related with Laplacian eigenvalues of graphs. This motivates us to study which properties of the algebraic connectivity of graphs carry over to the restricted setting of drawings of graphs with integer coordinates.
Characterizing identifying codes from the spectrum of a graph or digraph
Ms. Berenice Martínez Barona, Universitat Politècnica de Catalunya
A $(1,\le \ell)$-identifying code in digraph $D$ is a dominating subset $C$ of vertices of $D$, such that all distinct subsets of vertices of $D$ with cardinality at most $\ell$ have distinct closed in-neighborhoods within $C$. In this talk we give a new method to obtain an upper bound on $\ell$ for digraphs. The results obtained here can also be applied to graphs. As far as we know, it is the first time that the spectral graph theory has been applied to the identifying codes.
On the rank of pseudo walk matrices
Dr. Alexander Farrugia, G.F. Abela Junior College, University of Malta
\newcommand{\bb}[1]{\textup{\textbf{#1}}} \newcommand{\vertset}[1]{\mathcal{V}(#1)} \begin{abstract} In the literature, the walk matrix $\bb{W}_\bb{b}$ associated with a graph $G$ having vertex set $\vertset{G}$ is the matrix with columns $\bb{b},\bb{A}\bb{b},\bb{A}^2\bb{b},\ldots,\allowbreak\bb{A}^{r-1}\bb{b}$ that enumerates the number of all possible walks on $G$ of length $0,1,2,\ldots,\allowbreak r-1$ starting from each vertex of $G$ and ending at any of the vertices indicated by $\bb{b}$. We generalize walk matrices further to obtain pseudo walk matrices $\bb{W}_\bb{v}$ having any walk vector $\bb{v}$. For any subset $S$ of $\vertset{G}\times\vertset{G}$, the total number of walks $N_0(S),N_1(S),N_2(S),\ldots$ of length $0,1,2,\ldots$ in $G$ that start from vertex $i$ and end at vertex $j$ for all $(i,j)\in S$ is considered. Various results on such pseudo walk matrices are presented, particularly related to their rank. The matrix rank of pseudo walk matrices allows the consideration of controllable and recalcitrant pairs. \end{abstract}
Eigenvalues and $[a,b]$-factors in regular graphs
Prof. Suil O, SUNY Korea
For positive integers, $r \ge 3, h \ge 1$, and $k \ge 1$, Bollob{\' a}s, Saito, and Wormald proved some sufficient conditions for an $h$-edge-connected $r$-regular graph to have a $k$-factor in 1985. Lu gave an upper bound for the third-largest eigenvalue in a connected $r$-regular graph to have a $k$-factor in 2010. Gu found an upper bound for certain eigenvalues in an $h$-edge-connected $r$-regular graph to have a $k$-factor in 2014. For positive integers $a \le b$, an even (or odd) $[a, b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for each vertex $v \in V (G)$, $d_H(v)$ is even (or odd) and $a \le d_H(v) \le b$. In this talk, we provide best upper bounds (in terms of $a, b,$ and $r$) for certain eigenvalues (in terms of $a, b, r,$ and $h$) in an $h$-edge-connected $r$-regular graph $G$ to guarantee the existence of an even $[a, b]$-factor or an odd $[a, b]$-factor. This result extends the one of Bollb{\' a}s, Saito, and Wormald, the one of Lu, and the one of Gu.
Isospectral magnetic graphs
Prof. Fernando Lledó, University Carlos III, Madrid and Insitute for Mathematical Sciencies (ICMAT), Madrid
We present a new geometrical construction leading to an infinite collection of families of graphs, where all the elements in each family are (finite) isospectral non-isomorphic graphs for the discrete magnetic Laplacian with normalised weights (in particular for standard weights). The construction is based on the notion of isospectral frames which, together with the $s$-partition of a natural number $r$, define the isospectral families of graphs by contraction of distinguished vertices. The isospectral frames have high symmetry and we use a spectral preorder of graphs studied in [2,3] to control the spectral spreading of the eigenvalues under elementary perturbations of the graph like vertex contraction and vertex virtualisation.\\[2mm] \emph{References:}\\[1mm] [1] J.S. Fabila-Carrasco, F. Lledó and O. Post, {\em A geometric construction of isospectral magnetic graphs}, 2021 (in preparation).\\[1mm] [2] J.S. Fabila-Carrasco, F. Lledó and O. Post, {\em Spectral preorder and perturbations of discrete weighted graphs}, Math. Ann. 2020, \\[0mm] DOI: https://doi.org/10.1007/s00208-020-02091-5\\[1mm] [3] J.S. Fabila-Carrasco, F. Lledó and O. Post, {\em Spectral gaps and discrete magnetic Laplacians}, Lin. Alg. Appl. {\bf 547} (2018) 183-216
Maximal cliques in strongly regular graphs
Dr. Gary Greaves, Nanyang Technological University
In this talk, I will introduce a cubic polynomial that can be associated to a strongly regular graph $\Gamma$. The roots of this polynomial give rise to upper and lower bounds for the size of a maximal clique in $\Gamma$. I will explain how we can use this cubic polynomial to rule out the existence of strongly regular graphs that correspond to an infinite family of otherwise feasible parameters. This talk is based on joint work with Jack Koolen and Jongyook Park.
Spectral Theory and Integrable Systems (MS - ID 57)
Dynamical Systems and Ordinary Differential Equations and Applications
Control of eigenfunctions on negatively curved surfaces
Prof. Semyon Dyatlov, MIT
Given an $L^2$-normalized eigenfunction with eigenvalue $\lambda^2$ on a compact Riemannian manifold $(M,g)$ and a nonempty open set subset $\Omega$ of $M$, what lower bound can we prove on the $L^2$-mass of the eigenfunction on $\Omega$? The unique continuation principle gives a bound for any $\Omega$ which is exponentially small as $\lambda$ goes to infinity. On the other hand, microlocal analysis gives a $\lambda$-independent lower bound if $\Omega$ is large enough, i.e. it satisfies the geometric control condition. This talk presents a $\lambda$-independent lower bound for any set $\Omega$ in the case when $M$ is a negatively curved surface, or more generally a surface with Anosov geodesic flow. The proof uses microlocal analysis, the chaotic behavior of the geodesic flow, and a new ingredient from harmonic analysis called the Fractal Uncertainty Principle. Applications include control for Schrodinger equation and exponential decay of damped waves. Joint work with Jean Bourgain, Long Jin, and St\'ephane Nonnenmacher.
On a weighted inequality for fractional integrals
Mr. Giorgi Imerlishvili, Georgian Technical University
\noindent The weighted inequality \begin{equation}\label{1} \| I_{\alpha}f \|_{L^q_V} \leq C \|f\|_{L^p} \end{equation} for the Riesz potential $I_{\alpha}$, $0<\alpha<n$, plays an important role in the theory of PDEs. It is worth mentioning its applications to the theory of Sobolev embeddings (see, e.g., [Maz]), its connection with eigenvalue estimates for the Schr\"odinger operator $H= -\Delta- V$ with a potential $V$ (see, e.g., [FJW], PP. 91-94), etc. In 1972 D. Adams proved that the above mentioned weighted inequality holds for $1<p<q<\infty$ if and only if there is a positive constant $C$ such that for all balls $B$ in ${\Bbb{R}}^n$, $$ V(B) \leq C |B|^{(1/p- \alpha/n)q}. $$ In the diagonal case, i.e., when $p=q$, necessity of this condition remains valid, however, it is not sufficient for \eqref{1} (see, e.g., [Le]). We proved that the condition $$ V(B) \leq C |B|^{1-p \alpha/n}$$ is simultaneously necessary and sufficient for the boundedness of $I_{\alpha}$ from the Lorentz space $L^{p,1}$ to the weighted Lebesgue space $L^{p}_V$. Some other related results are also derived. \vskip+0.2cm {\bf Acknowledgment.} The work was supported by the Shota Rustaveli National Foundation grant of Georgia (Project No. DI-18-118). \vskip+0.5cm \begin{center} {\bf References} \end{center} \vskip+0.4cm [FJW] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of function spaces, {\em Regional Conference Series in Mathematics}, Vol. 79, {\em Amer. Math. Soc. Providence. Rl,} 1991. [Le] P.-G. Lemari\'e, Multipliers and Morrey spaces, {\em Potential Analysis}, {\bf 38}(2013), 741-752. [Maz] V. Maz'ya, Sobolev spaces, {\em Springer, Berlin}, 1984.
A note on the multiple fractional integrals defined on the product of quasi-metric measure spaces
Dr. Tsira Tsanava, Georgian Technical University
A complete characterization of a vector-measure $\overrightarrow{\mu}= (\mu_1, \cdots, \mu_n)$ governing the boundedness of the multiple fractional integral operator $$I^{\overrightarrow{\gamma}}f(x_1, \ldots, x_n)=\int\limits_{X_1}\cdots \int\limits_{X_n} \frac{f(y_1,\ldots,y_{n})d\mu_1(y_1) \cdots d\mu_n(y_n)}{\prod\limits_{j=1}^{n}\left(d_j(x_j,y_j)\right)^{1-\gamma j}}, \quad \overrightarrow{\gamma}=(\gamma_{1},\ldots,\gamma_{n})$$ from one mixed norm Lebesgue space $L_{\overrightarrow{\mu}}^{\overrightarrow{p}}$ to another one $L_{\overrightarrow{\mu}}^{\overrightarrow{q}}$ is obtained, where $(X_i, d_i, \mu_i)$, $i=1, \ldots, n$, are quasi-metric measure spaces (spaces of nonhomogeneous type). \vskip+0.2cm {\bf Acknowledgement.} The work was supported by the Shota Rustaveli National Foundation grant of Georgia (Project No. DI-18-118).
A scalar Riemann--Hilbert problem on the torus
Mr. Mateusz Piorkowski, University of Vienna
In this talk we will present a case study of a scalar Riemann--Hilbert problem on the torus. This work has been motivated by the analysis of the KdV equation with steplike initial data via the nonlinear steepest descent method. It turns out that the model problem for the transition region can be naturally formulated as a scalar Riemann--Hilbert problem on the torus. This approach does not only lead to the explicit Riemann--Hilbert solutions given in terms of Jacobi theta functions, but also illuminates the uniqueness and ill-posedness issues raised in an earlier paper.
Computing Eigenvalues of the Laplacian on Rough Domains
Dr. Frank Rösler, Cardiff University
We discuss a recent work in which we prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincar\'e-type inequality, which is of independent interest. These results are applied construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. This immediately leads to new classifications in the so-called ``Solvability Complexity Index Hierarchy'' recently inroduced by Hansen et al.
Stochastic Evolution Equations (MS - ID 68)
Probability
Local characteristics and tangent martingales in Banach spaces
Dr. Ivan Yaroslavtsev, Max Planck Institute for Mathematics in the Sciences
Let $L$ be a real-valued L\'evy martingale. Then we know that by the L\'evy-Khinchin formula \[ \mathbb E e^{i\theta L_t} = \exp\Bigl\{ t\Bigl(-\frac 12 \sigma^2 \theta^2 + \int_{\mathbb R} e^{i\theta x} - 1 -i\theta x \nu(d x)\Bigr)\Bigr\},\;\;\; t\geq 0,\;\;\theta\in \mathbb R, \] where $\sigma \geq 0$ is responsible for the {\em continuous} part of $L$ and the measure $\nu$ on $\mathbb R$ in responsible for the {\em discontinuous} part of $L$, i.e.\ if we decompose $L = W + N$ into a sum of a Brownian motion $W$ and a purely discontinuous Poisson martingale $N$, then the distributions of $W$ and $N$ are uniquely determined by $\sigma$ and $\nu$ respectively. Any real-valued martingale $M$ has a analogue of $(\sigma, \nu)$ which is called the {\em local characteristics} of $M$ and which is defined as the pair $([M^c], \nu^M)$ of a quadratic variation $[M^c]$ of the continuous part $M^c$ of $M$ and the compensator $\nu^M$ of the jump measure of $M$. The local characteristic have been defined and intensively studied in a number of works by Jacod, Kallenberg, Kwapie\'{n}, Shiryaev, and Woyczy\'{n}ski, and in particular it turned out that the local characteristics uniquely determine the distribution of the corresponding martingale if and only if they are constant. Therefore the notion of {\em tangent} martingales (i.e.\ martingales with the same local characteristics) was introduced. In 2017 Kallenberg have shown sharp $L^p$ bounds for tangent real-valued martingales. \medskip In the present talk we will discuss the generalisation of local characteristics to Banach space-valued martingales. In particular, we will prove sharp $L^p$ estimates for tangent martingales with values in infinite dimensions. This will help us to provide new sharp bounds for vector-valued stochastic integrals with respect to a general martingale.
Weighted Energy--Dissipation principle for nonlinear stochastic evolution equations
Dr. Luca Scarpa, Politecnico di Milano
We present the Weighted Energy--Dissipation (WED) principle for nonlinear stochastic evolution equations in variational form. The approach consists in minimizing suitable convex WED functionals, defined on spaces of entire trajectories, and depending on an approximation parameter. The corresponding Euler--Lagrange equation is characterized as an elliptic-in-time regularization of the original problem, which can be equivalently seen as a forward--backward nonlinear stochastic evolution system. Finally, WED minimizers are shown to converge to the solution of the original nonlinear evolution equation as the approximation parameter vanishes. This study is based on a joint work with Ulisse Stefanelli (University of Vienna, Austria).
Invariant measures for a stochastic nonlinear and damped 2D Schr\"odinger equation
Dr. Margherita Zanella, Politecnico di Milano
We consider a two-dimensional stochastic nonlinear defocusing Schr\"odinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in Stratonovich form and a nonlinear noise in It\^o form. We work at the same time on compact Riemannian manifolds without boundary and on compact smooth domains with either Dirichlet or Neumann boundary conditions. We construct a martingale solution using a modified Faedo-Galerkin's method, then by means of suitable Strichartz estimates we show the pathwise uniqueness of solutions. Finally, we prove the existence of invariant measures by means of a version of the Krylov-Bogoliubov method, which involves the weak topology, as proposed by Maslowski and Seidler . Some remarks on the uniqueness in a particular case are provided as well. The talk is based on a joint work with B. Ferrario and Z. Brze\'zniak.
Symmetry of Graphs, Maps and Polytopes (MS - ID 9)
Combinatorics and Discrete Mathematics
Having fun with designs
Prof. Cheryl Praeger, University of Western Australia
The kind of design we explore is a finite $2$-design: a point-block incidence structure where each block is a $k$-subset of points, and each pair of points lies in a constant number $\lambda$ of blocks. We ask that the design admits a block-transitive group $G$ of automorphisms which preserves also a nontrivial partition of the point set. One famous study of these designs, by Delandtsheer and Doyen in 1989 introduced two parameters, now called Delandtsheer--Doyen parameters, that linked the design structure with the point-partition. Another even earlier study, by Davies in 1987, showed that, if $G$ is transitive on flags (incident point-block pairs) then the number of points is bounded above by some function of $\lambda$ (but no function was specified). Recently, with Alice Devillers, we have been exploring these two results. For flag-transitive designs: we showed that $4\lambda^6$ could be taken for the Davies function – though this is not a tight upper bound. Moreover, while investigating possible examples with small parameters, we found a rather beautiful flag-transitive design where the point set is a $6\times 6$ grid, the block size is $8$, $\lambda=4$, and the full automorphism group is the symmetric group $S_6$. Then, with Carmen Amarra and Alice Devillers, while exploring bounds and several extreme cases of the Delandtsheer--Doyen parameters, we constructed a (probably infinite) new family of designs. Whether the family is infinite depends on the Bunyakovsky Conjecture about the prime numbers, and the number of examples in a certain range seems to be related to another number theoretic conjecture, the Bateman—Horn Conjecture. We are grateful to Sasha Zvonkin and Gareth Jones for their interest in the number theoretic puzzles our construction spawned: their computer enumerations related to these conjectures showed that there are $12,357,532$ designs in the family where the point-partition has classes of prime cardinality less than $10^8$.
Some observations about regular maps
Prof. Marston Conder, University of Auckland
A map on an orientable surface is called `orientably-regular' if its automorphism group has a single orbit on arcs (incident vertex-edge pairs), and is then called `reflexible' or `chiral' depending on whether or not it admits reflections (for example, fixing an arc but swapping the two faces incident with it). In every such map with a high degree of symmetry, all vertices have the same valency, say $k$, and all faces have the same size, say $m$, and then the (ordered) pair $\{m,k\}$ is called the `type' of the map. Writing a book with Gareth Jones, Jozef {\v S}ir{\' a}{\v n} and Tom Tucker) on regular maps has prompted us to review and extend what is known about them. Various questions have arisen naturally, including what I believe is the most important unanswered one, namely whether chirality is more prevalent than reflexibility. Other questions include these: What types $\{m,k\}$ occur the most frequently among orientably-regular maps on hyperbolic surfaces? What kinds of groups are the most prevalent as the group of orientation-preserving automorphisms? (Simple groups? insoluble groups? soluble groups? 2-groups?) Is chirality is more prevalent than reflexibility for a given type? I will give some partial answers to these questions, with reference to computational evidence. These answers may be surprising.
Realisation of groups as automorphism groups of maps and hypermaps
Prof. Gareth Jones, University of Southampton
I will show that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, or of coverings of a suitable topological space, every countable group $A$ is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely many of which are finite if $A$ is finite. In particular, the latter applies to dessins d'enfants, regarded as finite oriented hypermaps. The objects realising $A$ are obtained as regular coverings by $A$ of certain basic objects with primitive monodromy groups, corresponding to maximal subgroups of triangle groups. The constructions of these generalise results of Bernhard Neumann on maximal subgroups of infinite index in the modular group, and of Marston Conder on maximal subgroups of finite index in various cocompact triangle groups.
On the Asymmetrizing Cost and Density of Graphs
Dr. Wilfried Imrich, Montanuniversität Leoben
A set $S$ of vertices in a graph $G$ with nontrivial automorphism group is \emph{asymmetrizing} if the identity mapping is the only automorphism of $G$ that preserves $S$ as a set. If such sets exist, then their minimum cardinality is the \emph{asymmetrizing cost} $\rho(G)$ of $G$. For finite graphs the \emph{asymmetrizing density} $\delta(G)$ of $G$ is the quotient of the size of $S$ by the order of $G$. For infinite graphs $\delta(G)$ is defined by a limit process. The talk discusses bounds on the asymmetrizing cost, classes of graphs with asymmetrizing density zero, and infinite graphs with finite asymmetrizing cost. It is easy to construct graphs with positive asymmetrizing density, unless they are vertex transitive. Hitherto no infinite vertex transitive graphs with $\delta(G) > 0$ seem to have been known. Here we construct connected, infinite vertex transitive cubic graphs of asymmetrizing density $\delta(G)= n^{-1}2^{-n-1}$ for each $n\geq 1$. We also construct finite vertex transitive cubic graphs of arbitrarily large asymmetrizing cost. The examples are Split Praeger--Xu graphs, for which we provide another characterization.
On compact Riemann surfaces and hypermaps of genus $p+1$ where $p$ is prime
Dr. Sebastián Reyes-Carocca, Universidad de La Frontera
In this talk we shall discuss a classification of compact Riemann surfaces of genus $g$, where $g-1$ is a prime $p$, which have a group of automorphisms of order $\rho(g - 1)$ for some integer $\rho \geqslant 1$. We also provide isogeny decompositions of their Jacobian varieties. As a consequence, a classification of orientably regular hypermaps of genus $p + 1$ with automorphism group of order divisible by the prime $p$ is obtained.
Base sizes for the symmetric and alternating groups
Prof. Joy Morris, University of Lethbridge
A base of a permutation group is a set of the elements on which it is acting, that is only fixed by the identity element of that permutation group. The base size of a permutation group is the minimum possible size for a base. These concepts have been rediscovered and studied in the context of automorphism groups of combinatorial objects: if a permutation group is the automorphism group of an object, then a distinguishing set is a collection of "points" of the object that are only fixed by the identity automorphism. The distinguishing number is the smallest size of such a set. I will present new results about the base sizes for two non-standard actions of the symmetric group Sym(n): its action on partitions, and its action on subsets of a fixed cardinality. For the first of these, I will also present results about the corresponding base sizes for the alternating group.
On automorphisms of direct products of abelian Cayley graphs
Dave Morris, University of Lethbridge, Canada
The direct product of two graphs $X$ and $Y$ is denoted $X \times Y$. (It is also known as the ``tensor product'' or ``categorical product'' or ``Kronecker product'' or ``conjunction'' of $X$ and~$Y$.) This is a natural construction, so any isomorphism from $X$ to $X'$ can be combined with any isomorphism from $Y$ to $Y'$ to obtain an isomorphism from $X \times Y$ to $X' \times Y'$. Therefore, the automorphism group $\mathrm{Aut}(X \times Y)$ contains a copy of $(\mathrm{Aut}\, X) \times (\mathrm{Aut}\, Y)$. It is not known when this inclusion is an equality, even for the special case where $Y = K_2$ is the complete graph with only $2$ vertices. (The direct product $X \times K_2$ is also known as the ``canonical bipartite double cover'' of~$X$. The graph $X$ is said to be ``stable'' if equality holds in this special case.) When $X$ is a circulant graph with an odd number of vertices (and $Y = K_2$), recent work of B.\,Fernandez and A.\,Hujdurovi\'c shows that equality holds if and only if $X$ is connected and no two vertices of~$X$ have exactly the same neighbours. We will present a short, elementary proof that generalizes this theorem to the case where $X$ is a Cayley graph on an abelian group of odd order.
A family of non-Cayley cores that are constructed from vertex-transitive or strongly regular self-complementary graphs
Dr. Marko Orel, University of Primorska, IMFM
Let $\Gamma$ be a finite simple graph on $n$ vertices. In the talk I will consider the graph $\Gamma\equiv \bar{\Gamma}$ on $2n$ vertices, which is obtained as the disjoint union of $\Gamma$ and its complement $\bar{\Gamma}$, where we add a perfect matching such that each its edge joins two copies of the same vertex in $\Gamma$ and $\bar{\Gamma}$. The graph $\Gamma\equiv \bar{\Gamma}$ generalizes the Petersen graph, which is obtained if $\Gamma$ is the pentagon. It is a non-Cayley graph if $n>1$, and is vertex-transitive if and only if $\Gamma$ is vertex-transitive and self-complementary. In this case $\Gamma\equiv \bar{\Gamma}$ is Hamiltonian-connected whenever $n>5$. It is shown that the fraction between the cardinalities of the automorphism groups of $\Gamma\equiv \bar{\Gamma}$ and $\Gamma$ can attain only values 1, 2, 4, or 12, and the corresponding four classes of graphs are described. The spectrum of the adjacency matrix of $\Gamma\equiv \bar{\Gamma}$ is computed whenever $\Gamma$ is regular. The main results involve the endomorphisms of $\Gamma\equiv \bar{\Gamma}$. It is shown that the graph $\Gamma\equiv \bar{\Gamma}$ is a core, i.e. all its endomorphisms are automorphisms, whenever $\Gamma$ is strongly regular and self-complementary. The same type of a result is obtained for many cases, where $\Gamma$ is vertex-transitive and self-complementary.
Geometry and Combinatorics of Semiregular Polytopes
Mr. Egon Schulte, Northeastern University
Traditionally, a polyhedron or polytope is semiregular if its facets are regular and its symmetry group is transitive on vertices. We briefly review the semiregular convex polytopes, and then discuss semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices. Our focus is on alternating semiregular polytopes, with two kinds of regular facets occurring in an alternating fashion. The cuboctahedron is a familiar example in rank 3. We then describe recent progress on the assembly problem for alternating semiregular polytopes: which pairs of regular n-polytopes can occur as facets of a semiregular (n+1)-polytope? If time permits, we brief discuss semiregularity in the context of skeletal polyhedra in 3-space. Most work is joint with Barry Monson.
Topics in complex and quaternionic geometry (MS - ID 74)
Algebraic and Complex Geometry
Balanced Hermitian metrics
Prof. Anna Fino, Università di Torino
A Hermitian metric on a complex manifold is balanced if its fundamental form is co-closed. In the talk I will review some general results about balanced metrics and present new smooth solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds.
On cohomogeneity one Hermitian non-K\"ahler manifolds
Daniele Angella, Università di Firenze
We study Hermitian manifolds with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following B\'erard-Bergery, which includes, among the others, the holomorphic line bundles on $\mathbb{C} P ^{m-1}$, the linear Hopf manifolds and the Hirzebruch surfaces. We characterize their invariant special Hermitian metrics, such as balanced, K\"ahler-like, pluriclosed, locally conformally K\"ahler, Vaisman, Gauduchon. Furthermore, we construct new examples of cohomogeneity one Hermitian metrics solving the second-Chern-Einstein equation and the constant Chern-scalar curvature equation.
Geometry of Kato manifolds
Ms. Alexandra-Iulia Otiman, Universita' di Firenze
Kato manifolds are compact complex manifolds containing a global spherical shell. Their modern study has been widely carried out in complex dimension 2 and originates in the seminal work of Inoue, Kato, Nakamura and Hirzebruch. In this talk I plan to describe a special class of Kato manifolds in arbitrary complex dimension, whose construction arises from toric geometry. I will present several of their analytic and geometric properties, including existence of special complex submanifolds and partial results on their Dolbeault cohomology. Moreover, since they are compact complex manifolds of non-Kähler type, I will investigate what special Hermitian metrics they support.
Special geometries with torus symmetry
Prof. Andrew Swann, Aarhus University
A survey of recent results on torus symmetry for metrics with special holonomy, particularly $G_2$ and Spin(7), and related geometries such as nearly Kähler six-manifolds. The known explicit examples of these geometries all have a large compact symmetry group and in particular an action of some torus. Studying the orbit structure and using ideas such as multi-moment maps, one gets pictures related to the Delzant description for toric manifolds, and/or certain geometric flows.
On a construction of quaternionic and octonionic Riemann surfaces
Prof. Fabio Vlacci, University of Trieste
We present an original way to introduce quaternionic and octonionic analogs of the classical Riemann surfaces. The construction of these manifolds has nice peculiarities and the scrutiny of Bernhard Riemann approach to Riemann surfaces, mainly based on conformality, leads to the definition of slice conformal or slice isothermal parameterization of Riemann $4$-manifold and $8$-manifold. These new classes of manifolds include slice regular quaternionic and octonionic curves, graphs of slice regular functions, the $4$ and $8$ dimensional sphere and helicoidal and catenoidal manifolds. This is a joint work with Graziano Gentili and Jasna Prezelj.
Topics in sub-elliptic and elliptic PDEs (MS - ID 31)
Partial Differential Equations and Applications
Time-dependent focusing Mean Field Games with strong aggregation
Ms. Daria Ghilli, LUISS University of Rome
Mean Field Games (MFG) theory models the behavior of an in finite number of indistinguishable rational agents aiming at minimising a common cost. A large part of MFG literature is devoted to the study of MFG systems with increasing coupling ("non focusing"case). Heuristically, this assumption means that agents prefer sparsely populated areas (indeed concentration costs), and it is well-suited to model competitive cases. Moreover, the increasing monotonicity of the coupling ensures existence and regularity of solutions in many circumstances. We are interested in the "focusing"case, that is, where the coupling is monotone decreasing and it is a local function of the distribution, so that no regularising effect can be expected. These systems describe Nash equilibria of games with a large number of agents aiming at aggregation. In this talk, we will introduce the model in the focusing case and we will show that there is a threshold for the growth of the coupling, after which the solutions to the MFG system may not exist. This is coherent with the focusing character of the MFG, which induces solutions to concentrate and develop singularities.
Exponentially Subelliptic Harmonic Maps
Prof. Yuan-Jen Chiang, University of Mary Washington
Exponentially harmonic maps were first introduced by Eells and Lemaire [5] in 1990. Exponential wave maps are exponentially harmonic maps on Minkowski spaces, which were first studied by Chiang and Yang [1, 4] since 2007. Firstly, we deal with the critical points of maps $\phi: \mathbb H_n \to S^m$ from the Heiserberg group into a sphere with energy $E_1(\phi)=\int_\Omega \exp (\frac 12 ||\nabla^H \phi||^2_\theta )\theta \wedge (d\theta)^n$ for domains $\Omega \subset \subset \mathbb H_n$ and a contact structure $\theta$ on $\mathbb H_n.$ They are solutions to the 2nd order quasi-linear subelliptic PDE system \begin{eqnarray} -\triangle_b \phi^j + 2 e_b(\phi) \phi^j +G_\theta(\nabla^H e_b(\phi), \, \nabla^H \phi^j)=0, 1\le j\le m+1, \nonumber\end{eqnarray} and arise through Fefferman's construction, i.e. as base maps $\phi: \mathbb H_n \to S^m$ associated to $S^1$ invariant exponential wave maps $\Phi: C(\mathbb H^n) \to S^m$ from the total space of the canonical circle bundle $S^1\to C(\mathbb H^n)\to S^m$ endowed with the Fefferman's metric $F_\theta.$ We establish Caccioppoli type estimates \begin{eqnarray}\int_{B_{r(x)}} \exp \Big ( \frac Q 2 ||\nabla^H\phi||^2_\theta \Big ){|| \nabla^H u ||}_\theta^Q \theta \wedge (d\theta)^n \le C r^\beta \, \,(0<\beta <1)\nonumber \end{eqnarray} with $Q=2n+2$ (the homogeneous dimension of $\mathbb H^n$), and show that any weak solution $\phi \in \bigcap_{p\ge Q} W_H^{1,p} (\Omega, S^n)$ of finite p-energy $E_p (\phi) < \infty$ for some $p\ge 2Q$ is locally H$\ddot {o}$lder continuous, i.e. $\phi^j \in S_{loc}^{0,\alpha} (\Omega)$ (H$\ddot {o}$lder like spaces) for $0<\alpha \le 1,$ built in terms of the Carnot-Carath$\acute e$odory metric $\rho_{\theta}.$ The main theorems and results are based on [2]. Secondly, we study exponentially subelliptic harmonic (e.s.h.) maps from a compact pseudo hermitian manifold $(M, \,\theta)$ into a Riemannian manifold $(N,\,h)$, i.e. $C^2 $ solutions of $\phi: M\to N$ to nonlinear PDE system $\tau_b(\phi)+\phi_* \nabla^H e_b(\phi)=0$ which are the Euler-Legrange equation of $\delta E_b (\phi)=0$ with $ E_b (\phi)=\int_M \exp (e_b(\phi)) \theta\wedge d\theta^n , $ where e.s.h. maps arise in a similar way as the first setting. We study the second variation formula and stability of exponentially subelliptic harmonic maps based on [3]. References: \begin{enumerate}[{[1]}] \item\label{Ch13} Y. J. Chiang, \emph{Developments of harmonic maps, wave maps and Yang-Mills fields into biharmonic maps, biwave maps and bi-Yang-Mills fields}, Frontiers in Math, Birkh$\rm{\ddot a}$user, Springer, Basel, xxi+399, 2013, Chapters 7 \& 8 : Exponentially harmonic maps \& Exponential wave maps. \item\label{ChDrEs19} Y. J. Chiang, S. Dragomir and F. Esposito, \emph{Exponentially subelliptic harmonic maps from the Heisengberg group into a sphere}, Cal. of Variations and PDEs, 1--45, online July 2019. \item\label{ChDrEs20} Y. J. Chiang, S. Dragomir and F. Esposito, \emph{Second variation formula and stability of exponentially subelliptic harmonic maps}, preprint. \item\label{ChYa07} Y. J. Chiang and Y. H. Yang, \emph{Exponential wave maps}, J. of Geom. Phys. $\bold{57}$ (12) (2007), 2521--2532. \item\label{EeLe90} J. Eells and L. Lemaire, \emph{Some properties of exponentially harmonic maps}, in: Partial Differential Equations, Part 1, 2 (Warsaw, 1990), 129--136, Banach Center Publ. 27, Polish Acad. Sci., Warsaw, 1992. \end{enumerate}
Topological Methods in Differential Equations (MS - ID 13)
Analysis and its Applications
Existence results of fourth order equations with perturbed two-point boundary conditions
Prof. Alberto Cabada, University of Santiago de Compostela
In this talk we establish the existence and multiplicity of positive solutions for a fourth-order boundary value problem. Integral perturbations of some kind of two-point boundary conditions are considered. After the construction of a Green’s function and the study of its constant sign, it is defined a positive cone, where to apply the Krasnoselskii compression/expansion and Leggett-Williams fixed point theorems in cones. A generalization for a higher order case is also considered. Some particular examples are given.
Quasilinear conservation laws with discontinuous flux as singular limit of semilinear parabolic equations
Prof. Graziano Guerra, Università degli Studi di Milano-Bicocca
Scalar conservation laws with a discontinuous flux with respect to the space variable arise in many applications where the conservation laws describe physical models in rough non homogeneous media. For example, traffic flows with rough road conditions and polymer flooding in porous media. We are interested in solutions to this type of equations obtained by approximating them with solutions to semilinear parabolic equations. We show that the Crandall Liggett theory~\cite{CL} of nonlinear semigroups provides a very elegant framework for proving existence and uniqueness of solutions to the Cauchy problem for the semilinear parabolic equation \begin{equation} u_{t}+f\left(x,u\right)_{x}=\varepsilon u_{xx}\label{eq:1} \end{equation} where the function $f$ is only $\mathbf{L}^{\infty}$ regular with respect to $x$. Then, when the flux has a single discontinuity at $x=0$, we show that the Brezis \& Pazy convergence theorem~\cite{BP} can be used to show existence and uniqueness of the singular limit, as $\varepsilon\to 0$, of solutions to~\eqref{eq:1}~\cite{2019-SIMA}. The limits obtained in this way are the unique vanishing viscosity solutions to the quasilinear conservation law~\eqref{eq:1} with $\varepsilon=0$. The convergence result can be generalised to fluxes less regular than BV~\cite{2018-JDE} but counterexamples show that it is not true for the simple $\mathbf{L}^{\infty}$ regularity. \begin{thebibliography}{00} \bibitem{2018-JDE} A.~Bressan, G.~Guerra, and W.~Shen. \newblock Vanishing viscosity solutions for conservation laws with regulated flux. \newblock {\em Journal of Differential Equations}, 266(1):312 -- 351, 2019. \bibitem{BP} H.~Br\'ezis and A.~Pazy. \newblock Convergence and approximation of semigroups of nonlinear operators in {B}anach spaces. \newblock {\em J. Functional Analysis}, 9:63--74, 1972. \bibitem{CL} M.~G. Crandall and T.~M. Liggett. \newblock Generation of semi-groups of nonlinear transformations on general {B}anach spaces. \newblock {\em Amer. J. Math.}, 93:265--298, 1971. \bibitem{2019-SIMA} G.~Guerra and W.~Shen. \newblock Vanishing {V}iscosity and {B}ackward {E}uler {A}pproximations for {C}onservation {L}aws with {D}iscontinuous {F}lux. \newblock {\em SIAM J. Math. Anal.}, 51(4):3112--3144, 2019. \end{thebibliography}
Periodic solutions to a forced Kepler problem in the plane
Dr. Duccio Papini, Università degli Studi di Udine
We investigate the following forced Kepler problem in the plane: \[ \ddot x = - \dfrac{x}{|x|^3} + \nabla_x U(t,x),\qquad x\in \mathbb{R}^2\setminus\{(0,0)\}, \] where $U(t,x)$ is $T$-periodic in the first variable and satisfies $U(t,x) = \mathcal{O}(\vert x \vert^{\alpha})$ for some $\alpha \in (0,2)$ as $\vert x \vert \to \infty$. We look for a $T$-periodic solution which minimimizes the corresponding action functional on a space of loops which are not null-homotopic in the punctured plane. On one hand, we do not impose further symmetry conditions on the perturbation's potential $U$. On the other, the solution we find is generalised, according to the definition given in the paper [Boscaggin, Ortega, Zhao, \emph{Periodic solutions and regularization of a Kepler problem with time-dependent perturbation}, Trans. Amer. Math. Soc. \textbf{372} (2018), 677--703]. In particular, such solution may have a finite number of collisions with the origin in each period, while its energy and bouncing directions behave in a regular way at each collision time.
Multiple oscillating BV-solutions for a mean-curvature Neumann problem
Dr. Francesca Colasuonno, Università di Bologna
In this talk, I will focus on a one-dimensional mean-curvature problem under Neumann boundary conditions. I will describe how we can find multiple positive oscillating BV-solutions, using an approximation of the mean curvature operator and the shooting method. This is joint work with Alberto Boscaggin and Colette De Coster.
A fixed point approach for decaying solutions of difference equations
Prof. Serena Matucci, University of Florence
A boundary value problem associated to the difference equation with advanced argument \begin{equation} \label{*}\Delta\bigl (a_{n}\Phi(\Delta x_{n})\bigr)+b_{n}\Phi(x_{n+p}% )=0,\ \ n\geq1 \tag{$*$} \end{equation} is presented, where $\Phi(u)=|u|^{\alpha}$sgn $u,$ $\alpha>0,p$ is a positive integer and the sequences $a,b,$ are positive. We deal with a particular type of decaying solutions of (\ref{*}), the so-called intermediate solutions, that is solutions $x$ of (\ref{*}) such that $x_{n}>0,\Delta x_{n}<0$ for large $n$ and \begin{equation*} \lim_{n}x_{n}=0, \ \ \lim_{n}x_{n}^{[1]}=a_{n}\Phi(\Delta x_{n})=-\infty, \end{equation*} where $x^{[1]}$ is called the quasidifference of $x.$ In particular, we prove the existence of these type of solutions for (\ref{*}) by reducing it to a suitable boundary value problem associated to a difference equation without deviating argument. Our approach is based on a fixed point result for difference equations, which originates from existing ones stated in the continuous case, but take into account some peculiarities of the discrete case.
About coupled gradient-type quasilinear elliptic systems with supercritical growth
Prof. Anna Maria Candela, Università degli Studi di Bari Aldo Moro
The aim of this talk is pointing out some recent results on the coupled gradient--type quasilinear elliptic system \[(P)\; \left\{ \begin{array}{ll} - {\rm div} (A(x,u)\vert\nabla u\vert^{p_1 -2} \nabla u) + \frac{1}{p_1}A_u (x,u)\vert\nabla u\vert^{p_1} = G_u(x,u,v) &\hbox{ in $\Omega$,}\\ [10pt] - {\rm div} (B(x,v)\vert\nabla v\vert^{p_2 -2} \nabla v) +\frac{1}{p_2}B_v(x,v)\vert\nabla v\vert^{p_2} = G_v(x,u,v) &\hbox{ in $\Omega$,}\\[10pt] u = v = 0 &\hbox{ on $\partial\Omega$,} \end{array} \right.\] where $\Omega \subset \mathbb{R}^N$ is an open bounded domain, $p_1$, $p_2 > 1$ and $A(x,u)$, $B(x,v)$ are $\mathcal{C}^1$--Carath\'eo\-do\-ry functions on $\Omega \times \mathbb{R}$ with partial derivatives $A_u(x,u)$, respectively $B_v(x,v)$. Here, $(G_u(x,u,v),G_v(x,u,v)) = \nabla G(x,u,v)$ where $G(x,u,v)$ is a given function on $\Omega \times \mathbb{R}^2$. Even if the coefficients $A(x,u)$ and $B(x,v)$ make the variational approach more difficult, suitable hypotheses allow us to prove that the weak bounded solutions of problem $(P)$ are critical points of the functional \[ {\cal J}(u,v) = \frac{1}{p_1}\int_{\Omega} A(x,u) \vert\nabla u\vert^{p_1} dx + \frac{1}{p_2}\int_{\Omega} B(x,v) \vert\nabla v\vert^{p_2} dx - \int_{\Omega} G(x,u,v) dx \] in the Banach space $X= X_1\times X_2$, where $X_i = W^{1,p_i}_0(\Omega) \cap L^\infty(\Omega)$ for $i=1, 2$. Unluckily, classical variational theorems cannot apply to ${\cal J}$ in $X$ but, following an approach which exploits the interaction between $\|\cdot\|_X$ and the standard norm on $W^{1,p_1}_0(\Omega) \times W^{1,p_2}_0(\Omega)$, the existence of critical points of ${\cal J}$ can be proved by means of a generalized Mountain Pass Theorem. In particular, if the coefficients $A(x,u)$ and $B(x,v)$ grow in the ``right'' way then $G(x,u,v)$ can have a suitable supercritical growth and if ${\cal J}$ is even then $(P)$ has infinitely many weak bounded solutions. \smallskip These results are part of joint works with Caterina Sportelli and Addolorata Salvatore.
Existence of radial bounded solutions for some quasilinear elliptic equation in ${\mathbb R}^N$
Prof. Addolorata Salvatore, Università degli studi di Bari Aldo Moro
We study the quasilinear equation \[(P)\quad - {\rm div} (A(x,u) |\nabla u|^{p-2} \nabla u) + \frac1p\ A_t(x,u) |\nabla u|^p + |u|^{p-2}u\ =\ g(x,u) \quad \hbox{in ${\mathbb R}^N$,} \] with $N\ge 3$, $p > 1$, where $A(x,t)$, $A_t(x,t) = \frac{\partial A}{\partial t}(x,t)$ and $g(x,t)$ are Cara\-th\'eo\-do\-ry functions on ${\mathbb R}^N \times {\mathbb R}$. Under suitable assumptions on $A(x,t)$ and $g(x,t)$ the problem has a good variational structure, i.e. the weak bounded solutions of problem $(P)$ are critical points of the $C^1$ functional \[ {\cal J}(u)\ =\ \frac1p\ \int_{{\mathbb R}^N} A(x,u)|\nabla u|^p dx + \frac1p\ \int_{{\mathbb R}^N} |u|^p dx - \int_{{\mathbb R}^N} G(x,u) dx, \] on the Banach space $X = W^{1,p}({\mathbb R}^N) \cap L^\infty({\mathbb R}^N)$, with $G(x,t) = \int_0^tg(x,s)ds$. In order to overcome the lack of compactness, we assume that the problem has radial symmetry, then we look for critical points of ${\cal J}$ restricted to $X_r$, subspace of the radial functions in $X$. Following an approach which exploits the interaction between $\|\cdot\|_X$ and the norm on $W^{1,p}({\mathbb R}^N)$, we prove the existence of at least one weak bounded radial solution of $(P)$ by applying a generalized version of the Ambrosetti--Rabinowitz Mountain Pass Theorem. \smallskip The result is contained in a joint work with Anna Maria Candela.
Compactness properties of operator of translation along trajectories in evolution equations
Mr. Władysław Klinikowski, Nicolaus Copernicus University in Toruń
\begin{center} \textbf{Abstract} \end{center} Our aim is to present some results about compactness properties of operator of translation along trajectories (which is also known as Poincaré operator) which is associated with some evolution equation. Fixed points of this operator are periodic solutions of connected evolution equation. In order to apply some kind of topological degree we study just compactness properties of operator of translation. We consider two types of evolution equations: first is linked with parabolic problems and second with hyperbolic problems. In case of parabolic equations we discuss results which come from A. Ćwiszewski and R. Łukasiak ([1], [2]). Next we present approach (but not yet with specific results) to hyperbolic problems which is a part of collaboration with A. Ćwiszewski. Our method uses so-called ,,tail estimates" (which were firstly introduced by B. Wang in [4]) and is based on work of D. Fall and Y. You (see [3], but note that they looked for global attractors).\\ \\ \textbf{References}\\ \\ $[1]$ A. Ćwiszewski, R. Łukasiak, \textit{Forced periodic solutions for nonresonant parabolic equations on }$\mathbb{R}^N$, https://arxiv.org/pdf/1404.0256v2.pdf\\ $[2]$ A. Ćwiszewski, R. Łukasiak, \textit{A Landesman-Lazer type for periodic parabolic problems on }$\mathbb{R}^N$\textit{ at resonance}, Nonlinear An. TMA 125 (2015), 608-625\\ $[3]$ D. Fall, Y. You, \textit{Global attractors for the damped nonlinear wave equation in unbounded domains}, https://arxiv.org/pdf/1801.00104.pdf\\ $[4]$ B. Wang, \textit{Attractors for reaction-diffusion equations in unbounded domains}, Phys. D 128 (1999), 41-52
A Neumann $p$-Laplacian problem on metric spaces
Dr. Antonella Nastasi, University of Palermo
We use a variational approach to study existence and regularity of solutions for a Neumann p-Laplacian problem with a reaction term on metric spaces equipped with a doubling measure and supporting a Poincare' inequality. Trace theorems for functions with bounded variation are applied in the de finition of the variational functional and minimizers are shown to satisfy De Giorgi type conditions.
Topological methods in dynamical systems (MS - ID 65)
Dynamical Systems and Ordinary Differential Equations and Applications
On the boundary of the basins of attraction for the secant method applied to polynomials
Dr. Xavier jarque , Universitat de Barcelona
We investigate the discrete dynamical system $S$ defined on $\mathbb R^2$ given by the secant method applied to a real polynomial $p$. Every simple root $\alpha$ of $p$ has associated its basin of attraction $\mathcal A(\alpha)$ formed by the set of points converging under $S$ towards $\alpha$ and $\mathcal A^*(\alpha)$ its immediate basin of attraction. We focus on the structure and dynamical behaviour of the boundary of the immediate basin of attraction of a root of $p$. We call {\it external} roots of $p$ the smallest and largest value and {\it internal} all the rest. If $\alpha$ is an internal root of $p$ then $\partial \mathcal A^*(\alpha)$ is given by the stable manifold of a 4-cycle. Moreover we show that, under some hypothesis, those internal basins are simply connected.
Topological estimates of the number of vertices of minimal triangulation
Prof. Waclaw Marzantowicz, Adam Mickiewicz University in Poznan
From the beginning of the algebraic topology, then also called the combinatorial topology, i.e. from the beginning of the 20th century, its basic object is the simplicial complex. The representation of a given topological space $X$ as a simplicial complex (i.e. a homeomorphism with it) is called triangulation. The name comes from the fact that when space $X$ is a two-dimensional surface, triangulation means representing it as the union of adjacent triangles with edges meeting at the vertices. In the case of higher dimensions, the basic cells are the $"i"$ - dimensional simplices. One of the natural questions is to find a triangulation with the minimum number of vertices, respectively of all simplexes (or estimate these numbers). This lecture will be devoted to this problem. We will present a new method based on the notion of covering type estimating from below the number of vertices by the weighted length of the elements in the cohomology ring $ H^*(X) $, or the weighted Lusternik-Schirelmann category theory. As a consequence, we got not only a unified method of proof of estimates of the number of vertices of the minimal triangulations derived originally by ad hock combinatorial methods, but also sharper estimates, or estimates for the families of manifolds not studied earlier. \centerline{References} \begin{itemize} \item[[DMZ]] {Duan, W. Marzantowicz, and X. Zhao, {\emph{On the number of simplices required to triangulate a Lie group}}, Topology \& Applications, Volume 293, 15 April 2021, https://doi.org/10.1016/j.topol.2020.107559} \item[[GMP1]] {D.~Govc, W.~Marzantowicz and P.~Pave\v si\'c, \emph{Estimates of covering type and the number of vertices of minimal triangulations}, Discr. Comp. Geom. (2019), 63 (2020), no. 1, 31–48.} \item[[GMP2]]{ D.~Govc, W.~Marzantowicz and P.~Pave\v si\'c, \emph{How many simplices are needed to triangulate a Grassmannian?}, TMNA, Volume 56, No. 2, 2020, 501- 518.} \item[[KW]] {M.~Karoubi, C.~Weibel, \emph{On the covering type of a space}, L'Enseignement Math. {\bf 62} (2016), 457--474.} \end{itemize}
\v Cech cohomology and the region of influence of non-saddle sets
Dr. Héctor Barge, Universidad Politécnica de Madrid
This talk is devoted to show some properties of a class of isolated invariant compacta known as non-saddle sets. We shall see that while the local dynamics near these sets is fairly easy, flows having this kind of invariant sets may exhibit a great deal of global complexity. As we will see, this global complexity is closely related to the way in which the non-saddle set sits in the phase space at the cohomological level.
The topological conjugacy criterion for surface Morse-Smale flows with a finite number of moduli
Mr. Vladislav Kruglov, National Research University Higher School of Economics
The Morse-Smale flows have been classified in sense of topological equivalence for several times during the last century. The most known invariants for such flows are invariants by E. Leontovich and A. Mayer \cite{LeMa1}, \cite{LeMa2}, M. Peixoto \cite{Peix}, A. Oshemkov and V. Sharko \cite{OS}. Besides, the $\Omega$-stable flows on surfaces have been classified in such sense too by D. Neumann and T. O'Brien \cite{Neu} and V. Kruglov, D.Malyshev and O. Pochinka \cite{KruMaPo}. Attempts were also made to classify Morse-Smale flows in sense of topological conjugacy: in particular, V. Kruglov \cite{Kru} proved that the classes of topological equivalence and topological conjugacy for gradient-like flows on surfaces coincide. J. Palis \cite{Palis} considered a flow in a neighbourhood of a separatrix which connects two saddle points. He showed that in each topological equivalence class there is continuum of topological conjugacy classes, that is a flow with a separatrix-connection has analitical conjugacy invariants called {\it moduli of stability} or {\it moduli of topological conjugacy}. Each limit cycle likewise generates at least one modulus associated with its period. V. Kruglov, O. Pochinka and G. Talanova \cite{KruPoTa} proved that non-singular flows on an annulus with only two limit cycles on the annulus's boundary components have infinite number of moduli, expressed by a function. The first result of this report is the following. \begin{theorem} A Morse-Smale surface flow has finite number of moduli iff it has no a trajectory going from one limit cycle to another. \end{theorem} To construct the topological classification in sense of conjugacy we use the complete topological classifications in sense of equivalence from \cite{OS}, \cite{KruMaPo}. Namely, there is one-to-one correspondence between equivalent classes of a Morse-Smale flow $\phi^t$ on a surface and isomorphic classes of the {\it equipped graph} $\Upsilon^*_{\phi^t}$, which contains an information about partition the ambient manifold into cells with similar trajectories behaviour and the limit cycles types. To distinguish topological conjugacy classes we add to the equipped graph an information on the periods of the limit cycles. It gives a new equipped graph $\Upsilon^{**}_{\phi^t}$, and here is the second result. \begin{theorem} Morse-Smale surface flows $\phi^t, \phi'^t$ without trajectories going from one limit cycle to another one are topologically conjugate iff their equipped graphs $\Upsilon^{**}_{\phi^t}$ and $\Upsilon^{**}_{\phi'^t}$ are isomorphic. \end{theorem} Acknowledgements: The authors are partially supported by Laboratory of Dynamical Systems and Applications NRU HSE, grant No 075-15-2019-1931 of the Ministry of Science and Higher Education of Russian Federation. \begin{thebibliography}{99} \bibitem{LeMa1} Leontovich E. A., Mayer A. G. (1937). On the trajectories determining qualitative structure of sphere partition into trajectories (in Russian), Doklady Akademii nauk SSSR 14, No.5, 251-257. \bibitem{LeMa2} Leontovich E. A., Mayer A. G. (1955). On the scheme determining topological structure of partition into trajectories (in Russian), Doklady Akademii nauk SSSR 103, No.4, 557-560. \bibitem{Peix} Peixoto M. (1971). On the classification of flows on two-manifolds. Dynamical systems Proc. Symp. held at the Univ.of Bahia, Salvador, Brasil. \bibitem{OS} Oshemkov A. A., Sharko V. V. (1998). On the classification of Morse-Smale flows on 2-manifolds (in Russian), Matematicheskiy sbornik 189, No.8, 93-140. \bibitem{Neu} Neumann D., O'Brien T.(1976). Global structure of continuous flows on 2-manifolds, J. DifF. Eq. 22, 89-110. \bibitem{KruMaPo} Kruglov V. E., Malyshev D. S., Pochinka O. V. (2018). Topological classification of $\Omega$-stable flows by means of effectively distinguishable multigraphs, Discrete and Continuous Dynamical Systems 38, No.9, 4305-4327. \bibitem{Kru} Kruglov V. E. (2018). Topological conjugacy of gradient-like flows on surfaces, Dinamicheskie sistemy 8(36), No.1, 15-21. \bibitem{Palis} Palis J. (1978). A differentiable invariant of topological conjugacies and moduli of stability, Ast{\'e}risque 51, 335-346. \bibitem{KruPoTa} Kruglov V. E., Pochinka O. V., Talanova G. N. (2020) On functional moduli of surface flows, Proceedings of the International Geometry Center 13, No.1, 49-60. \end{thebibliography}
Attractors of dissipative homeomorphisms of the infinite surface homeomorphic to a punctured sphere
Prof. Grzegorz Graff, Gdansk University of Technology
A class of dissipative orientation preserving homeomorphisms of the infinite annulus, pairs of pants, and generally infinite surface homeomorphic to a punctured sphere is considered. We prove that in some isotopy classes the local behavior of such homeomorphisms at a fixed point, namely the existence of so-called inverse saddle, impacts the topology of the attractor - it cannot be arcwise connected. \vskip 3mm This work was supported by the National Science Centre, Poland within the grant Sheng 1 UMO-2018/30/Q/ST1/00228 (G. Graff).
Variational and evolutionary models involving local/nonlocal interactions (MS - ID 58)
Analysis and its Applications
Phase Separation in Nonlocal Multispecies Models
Prof. Martin Burger, FAU Erlangen-Nürnberg
In this talk we discuss some nonlocal variational problems arising as macroscopic steady states of some many particle systems with different species. We will demonstrate the the emergence of phase separation effects, explained from relations to Cahn-Hilliard Systems, and network formation, whose theoretical understanding is still rather open. We will also comment on corresponding evolution equations formulated in terms gradient flows, e.g. in Wasserstein spaces.
Minimal planar N-partitions for large N
Prof. Giovanni Alberti, University of Pisa
By minimal $N$-partition of a given planar domain $E$ we mean a partition which consists of $N$ sets (cells) with equal area, that minimize the total perimeter (that is, the length of the union of the boundaries of all cells). \\ T. C. Hales proved in 2001 that if $E$ is a flat 2-dimensional torus, then a regular hexagonal $N$-partition (if there is any) is minimal. It is then interesting to understand what happens for a planar domain $E$ that does not admit a regular hexagonal N-partition; in particular the following questions naturally arise: Are the cells asymptotically hexagonal as N tends to infinity, and to which extent the partition looks locally hexagonal? Is the partition rigid, in the sense that the orientation of the cells is (essentially) the same through the domain? \\ In this talk I will describe some results obtained in these directions together with Marco Caroccia (Politecnico di Milano) and Giacomo Del Nin (University of Warwick).
Minimisers of a fractional seminorm and nonlocal minimal surfaces
Prof. Serena Dipierro, University of Western Australia
The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev-Gagliardo spaces, and the ones related to the minimisation of fractional perimeters that act on measurable sets of the Euclidean space. In this talk, we relate these two types of variational problems. In particular, we investigate the connection between the nonlocal minimal surfaces and the minimisers of a Gagliardo seminorm, showing that a function is a minimiser for the fractional seminorm if and only if its level sets are minimisers for the fractional perimeter, and that the characteristic function of a nonlocal minimal surface is a minimiser for the fractional seminorm. We also discuss an existence result for minimisers of the fractional seminorm, an explicit non-uniqueness example for nonlocal minimal surfaces, and a Yin-Yang result describing the full and void patterns of nonlocal minimal surfaces. This is a joint work with Claudia Bucur, Luca Lombardini and Enrico Valdinoci.
Stability of nonlocal geometric evolutions
Prof. Matteo Novaga, University of Pisa
We introduce a notion of uniform convergence for local and nonlocal curvatures, and we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general method to characterize the limits of fractional mean curvature flows and Riesz curvature flows.
Convexity properties of the isoperimetric profile
Dr. Giorgio Saracco, Scuola Internazionale di Studi Superiori Avanzati
Given an open, bounded set $\Omega$ we consider the isoperimetric profile $\mathcal{J}$ that to each volume $V \in [0, |\Omega|]$ associates the least perimeter $P(E)$ among Borel subsets $E$ of $\Omega$ needed to enclose the given volume. We shall prove that for a wide class of planar sets, which encompasses convex sets, there exists a threshold $\bar V$ such that $\mathcal{J}$ is concave below it and convex above it. Moreover, $\mathcal{J}^2$ is globally convex. In order to prove these properties, a full characterization of the isoperimetric sets will be provided. Some comments on the $n$-dimensional case will be given.
Striped patterns for generalized antiferromagnetic functionals with power law kernels of exponent smaller than $d+2$
Ms. Alicja Kerschbaum, FAU Erlangen-Nürnberg
In this talk I will consider a class of continuous sharp interface generalized antiferromagnetic models previously studied by Daneri, Goldman and Runa. The functional consists of a perimeter term (retaining discrete symmetry) and a repulsive nonlocal term with a power law kernel. In a suitable regime the two terms enter in competition and symmetry breaking with formation of periodic striped patterns is expected to occur. We will show that the results obtained by Daneri and Runa showing striped pattern formation for power law kernels with exponents $p\geq d+2$ can be extended to power law kernels within a range of exponents strictly smaller than $d+2$ and strictly larger than $d+1$, being $d$ the dimension of the underlying space. Notice that the exponent $p=d+1$ corresponds to an anisotropic version (retaining discrete symmetry) of the model for pattern formation in thin magnetic films.
Torus-like solutions for the Landau-de Gennes model
Prof. Adriano Pisante, University of Roma "La Sapienza"
We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals in three-dimensional domains, possibly in a restricted class of axisymmetric configurations. Assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a corresponding physically relevant norm constraint (Lyuksyutov constraint) in the interior, we discuss partial regularity of the minimizers away from a possible finite set of interior singularities lying on the symmetry axis or even full regularity when no symmetry is imposed. As a consequence, we discuss boundary data which yield as minimizers smooth configuration with maximally biaxial set carrying nontrivial topology. In the axially symmetric case we show how singular (split) solutions or smooth (torus) solutions (or even both) for the Euler-Lagrange equations do appear for boundary data and/or domains which are smooth deformation of the radial hedgehog in a nematic droplet.
Abel Lecture
Graph limits and Markov spaces
László Lovász, Eötvös Loránd University
Limit objects for sequences of finite structures, larger and largerin size but more and more similar in some sense, have been constructedsporadically, perhaps since von Neumann constructed continuous geometries, butthis research has become quite extensive in the last decade and a half. Graphsare perhaps the simplest structures, and accordingly, the limit theory ofgraphs has made the most progress. The theory of graph limits is only understood, to a somewhat satisfactorydegree, in the cases of bounded degree graphs (initiated by Benjamini andSchramm) and of dense graphs (initiated by Borgs, Chayes, Lov\'asz, Szegedy, S\'osand Vesztergombi). More recently there is a lot of interest in the intermediatecases. It appears that the most important constituents of graph limits in the general case will be Markov spaces (Markov chains on measurable spaces with astationary distribution). Several important theorems can be extended from finite graphs to Markov spacesor, more generally, to measurable spaces: flow theory, expanders and spectra,mixing of random walks, etc. In this talk we will give a glimpse into thisemerging theory, based on the work of \'A. Backhaus, D. Kunszenti-Kov\'acs, B.Szegedy and the speaker.
Algebra
Real Cayley-Dickson algebras: doubly alternative elements and zero divisor graphs
Ms. Svetlana Zhilina, Lomonosov Moscow State University
Due to the non-alternativity of Cayley-Dickson algebras with dimension at least $16$, there appear zero divisors which are hard to study and to classify, except for some particular cases. At present most of the authors restrict their attention to the algebras of the main sequence where all parameters which determine the Cayley-Dickson construction are assumed to be equal to $-1$. We mention in particular the works by Moreno and Biss, Dugger, and Isaksen. Moreno's key idea was to study doubly alternative zero divisors, that is, such elements that their components are both alternative elements of the previous algebra. This notion was then extended to the Cayley-Dickson split-algebras and led to similar results. The talk is devoted to zero divisors in arbitrary real Cayley-Dickson algebras whose components satisfy some additional conditions on the norm and alternativity. We are interested in the patterns which they form in orthogonality and zero divisor graphs, and these patterns appear to be hexagonal. In the case of the algebras of the main sequence, these hexagons can be extended to the so-called double hexagons. Moreover, the vertices of a double hexagon have a convenient multiplication table which has a block structure.
Analysis and its Applications
On rapidly varying sequences
Dr. Valentina Timotić, University of East Sarajevo
In this paper we investigate the connection between the class $R_{\infty,s}$, of rapidly varying sequences (in the sense of de Haan) and the rapid equivalence, selection principles and game theory.
On system of split generalised mixed equilibrium problem and fixed point problems for multivalued mappings with no prior knowledge of operator norm
Prof. OLUWATOSIN MEWOMO, UNIVERSITY OF KWAZULU-NATAL
Abstract: \noindent In this paper, we introduce system of split generalised mixed equilibrium problem, which is more general than the existing well known split equilibrium problems and their generalisations, split variational inequality problems and several other related problems. We propose a new iterative scheme with inertial term, which is independent on the operator norm and obtain strong convergence result for approximating a common solution of the problem and fixed point of finite family of multivalued demicontractive mappings. We obtain consequent results which complement several existing results in this direction in the current literature. We also apply our results to approximate the solution of split convex minimisation problems and present numerical examples to demonstrate the efficiency of our algorithm in comparison with some existing algorithms in the literature.\\ \noindent {\bf {Keywords: Inertial algorithm, system of split generalised mixed equilibrium problems, fixed point problems, multivalued demicontractive mappings, strong convergence.}}\\ \noindent {\bf {2010 Mathematics Subject Classification: 65K15, 47J25, 65J15, 90C33}}.\\ \bibliography{xampl} \begin{thebibliography}{1111} \bibitem{alakoya} T.O. Alakoya,, L.O. O.T. Mewomo, Modified inertia subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, {\it Optimization} (2020), DOI:10.1080/02331934.2020.1723586. \bibitem{CAAM} C. Izuchukwu, K.O. Aremu, A.A. Mebawondu and O.T. Mewomo, A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space, {\it Appl. Gen. Topol.,} {\bf 20} (1) (2019), 193-210. \bibitem{izu} C. Izuchukwu, A. A. Mebawondu, K. O. Aremu, H. A. Abass and O. T. Mewomo, Viscosity iterative techniques for approximating a common zero of monotone operators in an Hadamard space, {\it Rend. Circ. Mat. Palermo II}, (2019), doi.org/10.1007/s12215-019-00415-2. \bibitem{iumka} C. Izuchukwu, G.C. Ugwunnadi, O.T. Mewomo, A.R. Khan and M. Abbas, Proximal-type algorithms for split minimization problem in p-uniformly convex metric space, {\it Numer. Algorithms}, {\bf 82} (3), (2019), 909 – 935. \bibitem{jolaoso} L.O. Jolaoso, T.O. Alakoya, A. Taiwo and O.T. Mewomo, A parallel combination extragradient method with Armijo line searching for finding common solution of finite families of equilibrium and fixed point problems, {\it Rend. Circ. Mat. Palermo II}, (2019), DOI:10.1007/s12215-019-00431-2 \bibitem{jola} L.O. Jolaoso, T.O. Alakoya, A. Taiwo, O.T. Mewomo, Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space, {\it Optimization}, (2020), DOI:10.1080/02331934.2020.1716752 \bibitem{jolaoso3} L.O. Jolaoso, A. Taiwo, T.O. Alakoya, O.T. Mewomo, A strong convergence theorem for solving variational inequalities using an inertial viscosity subgradient extragradient algorithm with self adaptive stepsize, {\it Demonstr. Math.}, (2019), {\bf 52} (1), 183-203. \bibitem{jolaoso4} L.O. Jolaoso, A. Taiwo, T.O. Alakoya, O.T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, {\it Comput. Appl. Math.}, (2019), DOI: 10.1007/s40314-019-1014-2. \bibitem{aaa1} O.T. Mewomo, F.U. Ogbuisi, Convergence analysis of an iterative method for solving multiple-set split feasibility problems in certain Banach spaces, {\it Quest. Math.}, {\bf 41} (1), (2018), 129-148. \bibitem{om1} F.U. Ogbuisi, O.T. Mewomo, On split generalized mixed equilibrium problems and fixed point problems with no prior knowledge of operator norm, {\it J. Fixed Point Theory Appl.}, {\bf 19} (3) (2017), 2109-2128. \bibitem{dd} F.U. Ogbuisi, O.T. Mewomo, Iterative solution of split variational inclusion problem in real Banach space, {\it Afr. Mat.}, {\bf 28} (1-2) (2017), 295-309. \bibitem{om} F.U. Ogbuisi, O.T. Mewomo, Convergence analysis of common solution of certain nonlinear problems, {\it Fixed Point Theory}, {\bf 19} (1) (2018), 335-358. \bibitem{aa3} F.U. Ogbuisi, O.T. Mewomo, Iterative solution of split variational inclusion problem in a real Banach space, {\it Afr. Mat.}, {\bf 28} (1-2), (2017), 295-309. \bibitem{ogwo} G. N. Ogwo, C. Izuchukwu, K.O. Aremu, O.T. Mewomo, A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space, {\it Bull. Belg. Math. Soc. Simon Stevin}, (2019), (accepted, to appear). \bibitem{okeke} C.C. Okeke, O.T. Mewomo, On split equilibrim problem, variational inequality problem and fixed point problem for multi-valued mappings, {\it Ann. Acad. Rom. Sci. Ser. Math. Appl.}, {\bf 9} (2), (2017), 255-280. \bibitem{AJM1} {\rm A. Taiwo, L. O. Jolaoso, O. T. Mewomo}, A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces, {\it Comput. Appl. Math}, {\bf 38} (2), (2019), Article 77. \bibitem{taiwo1} {\rm A. Taiwo, L.O. Jolaoso and O.T. Mewomo}, Parallel hybrid algorithm for solving pseudomonotone equilibrium and Split Common Fixed point problems, {\it Bull. Malays. Math. Sci. Soc.}, (2019), DOI: 10.1007/s40840-019-00781-1. \bibitem{taiwo2} A. Taiwo, L.O. Jolaoso and O.T. Mewomo, General alternative regularization method for solving Split Equality Common Fixed Point Problem for quasi-pseudocontractive mappings in Hilbert spaces, {\it Ricerche Mat.}, (2019), DOI: 10.1007/s11587-019-00460-0. \end{thebibliography}{1111}
Numerical solution of Optimal Transport Problem on graphs
Mr. enrico facca, Scuola Normale Superiore Pisa
The Dynamical Monge-Kantorovich (DMK) model is dynamical system of equations whose steady state has been related to the solution of the Optimal Transport Problem with cost equal to the Euclidean distance. In this talk we present a re-adaption into a graph setting of the DMK model, that can be rewritten in the form of a Gradient Flow. Using this formulation, we can solve the OTP on graphs with cost given by the shortest path distance looking at the long-time solution of the Gradient Flow equations. To this aim, we discretized the Gradient Flow equation via backward Euler time-stepping, in order to use larger time steps, getting faster convergence toward the optimal solution. The non-linear equations arising from such implicit time-stepping scheme are solved via Newton-Raphson Method. Thus, the optimization problem is reduced to the solution of a sequence of large and sparse saddle point linear systems, for which efficient preconditioners have to be build. In this talk we present different preconditioning approaches to tackle this problem.
Combinatorics and Discrete Mathematics
1-skeleton of the polytope of pyramidal tours with step-backs
Dr. Andrei Nikolaev, P. G. Demidov Yaroslavl State University
\newtheorem{Theorem}{Theorem} \newtheorem{lemma}{Lemma} The 1-skeleton of the polytope $P$ is the graph whose vertex set is the vertex set of $P$ and edge set is the set of $1$-dimensional faces of $P$. We consider the following 3 characteristics of a 1-skeleton: the vertex adjacency, the diameter, and the clique number. The adjacency relation is the key for the study and analysis of 1-skeleton, as well as the basis for the edge-following algorithms. While the diameter and the clique number serve as lower bounds on complexity in some computation models and classes of algorithms (see [1,3], for example). We consider a classic asymmetric traveling salesperson problem: for a given complete weighted digraph $K_n$ it is required to find a Hamiltonian cycle of minimum weight. For a Hamiltonian cycle $\tau$ we denote by $\tau(i)$ the successor of a vertex $i$ and by $\tau^{-1}(i)$ the predecessor of $i$. A vertex $i$, satisfying $\tau^{-1}(i) < i$ and $\tau(i) < i$, is called a \textit{peak}. A \textit{step-back peak} is the vertex $i$, such that \[\tau^{-1}(i) < i, \ \tau(i) = i - 1, \ \tau^{2}(i) > i, \text{ or } \tau^{-2}(i) > i, \ \tau^{-1}(i) = i - 1, \ \tau(i) < i.\] A \textit{proper peak} is a peak $i$ which is not a step-back peak. A \textit{pyramidal tour with step-backs}, introduced by Enomoto, Oda and Ota [2], is a Hamiltonian cycle with exactly one proper peak $n$. These tours are of interest, since, on the one hand, the minimum cost tour can be determined in $O(n^2)$ time by dynamic programming, and, on the other hand, there are known restrictions on the distance matrix that guarantee the existence of an optimal solution that is a pyramidal tour with step-backs. We define the traveling salesperson polytope $\mathrm{TSP}(n)$ and the polytope of pyramidal tours with step-backs $\mathrm{PSB}(n)$ as the convex hulls of characteristic vectors of all corresponding tours in the complete digraph $K_n$. It is known that for the polytope $\mathrm{TSP}(n)$ the question of whether two vertices are nonadjacent is NP-complete [4], the diameter equals 2 for asymmetric polytope [3] and is at most 4 for symmetric polytope [5], and the clique number of 1-skeleton is superpolynomial in $n$ [1]. Let $x$ and $y$ be two pyramidal tours with step-backs. We denote by $x^v$ and $y^v$ the corresponding vertices of the $\mathrm{PSB}(n)$ polytope and by $x \cup y$ a regular directed multigraph that contains all edges of both tours $x$ and $y$. \begin{lemma}[Sufficient condition for nonadjacency] Given two tours $x$ and $y$, if the multigraph $x \cup y$ includes a pair of edge-disjoint pyramidal tours with step-backs, different from $x$ and $y$, then the corresponding vertices $x^v$ and $y^v$ of the polytope $\mathrm{PSB}(n)$ are not adjacent. \end{lemma} \begin{lemma}[Necessary condition for nonadjacency] If the vertices $x^v$ and $y^v$ of the polytope $\mathrm{PSB}(n)$ are not adjacent, then the multigraph $x \cup y$ includes at least two pyramidal tours with step-backs, different from $x$ and $y$. \end{lemma} Our main result is that both necessary and sufficient conditions for nonadjacency can be verified in linear time. \begin{Theorem} The question of whether two vertices of the polytope $\mathrm{PSB}(n)$ are adjacent or nonadjacent can be verified in linear time $O(n)$. \end{Theorem} Based on the vertex adjacency relation, we can investigate other characteristics of 1-skeleton of the polytope $\mathrm{PSB}(n)$. \begin{Theorem} The diameter of 1-skeleton of the polytope $\mathrm{PSB}(n)$ is bounded above by 4. The clique number of 1-skeleton of the polytope $\mathrm{PSB}(n)$ is quadratic in n: \[\omega(\mathrm{PSB}(n)) = \Theta(n^2).\] \end{Theorem} Thus, the polyhedral characteristics of the traveling salesperson problem on Hamiltonian cycles and pyramidal tours with step-backs are fundamentally different and correspond to the computational complexity of the problem. \vspace{15pt} \setlength{\parindent}{0cm}{\textbf{References:}} [1] Bondarenko V. A., Nonpolynomial lower bounds for the complexity of the traveling salesman problem in a class of algorithms, \textit{Autom. Remote Control}, \textbf{44} (1983), 1137--1142. [2] Enomoto H., Oda Y., Ota K., Pyramidal tours with step-backs and the asymmetric traveling salesman problem, \textit{Discrete Appl. Math.}, \textbf{87} (1998) 57--65. [3] Padberg M. W., Rao M. R., The travelling salesman problem and a class of polyhedra of diameter two, \textit{Math. Program.}, \textbf{7} (1974), 32–45. [4] Papadimitriou C. H., The adjacency relation on the traveling salesman polytope is NP-Complete, \textit{Math. Program.}, \textbf{14} (1978), 312--324. [5] Rispoli F. J., Cosares S., A bound of 4 for the diameter of the symmetric traveling salesman polytope, \textit{SIAM J. Discrete Math.}, \textbf{11} (1998), 373–380.
Differential Geometry and Applications
On PNDP-manifolds
Dr. Alexander Pigazzini, Mathematical and Phisical Science Foundation
\baselineskip3.6ex %\baselineskip5ex %TOPMATTER \title{} \author{} %\keywords{} %AMSART %\2010 Mathematics Subject Classification{} %AMSART %\subjclass{} %AMSART %\date{\today} \date{} % %END TOPMATTER %-------------------------------------------------------------------- %\maketitle %article % % %-------------------------------------------------------------------- \makeatletter \def\@makefnmark{} \makeatother We provide a possible way of constructing new kinds of manifolds which we will call Partially Negative Dimensional Product manifold (PNDP-manifold for short). \\ In particular a PNDP-manifold is an Einstein warped product manifold of special kind, where the base-manifold $B$ is a Remannian (or pseudo-Riemannian) product-manifold $B=\Pi_{i=1}^{q'}B_i \times \Pi_{i=(q'+1)}^{\widetilde q} B_i$, with $\Pi_{i=(q'+1)}^{\widetilde q} B_i$ an Einstein-manifold, and the fiber-manifold $F$ is a derived-differential-manifold (i.e., $F$ is the form: smooth manifold ($\mathbb{R}^d$)+ obstruction bundle, so it can admit "virtual" negative dimension). \\ Since the dimensions of a PNDP-manifold is not related with the usual geometric concept of dimension (we consider they as "virtual" dimensions), from the speculative and applicative point of view, we use special projections/desuspensions, to identify the PNDP with another type of "object" of the same dimension, thus introducing a new type of "hidden" dimensions. \\ We have consider three kinds of PNDP-manifolds: \\ \\ {\bfseries Type I)} the $(n, -n)$-PNDP manifold that has overall, zero-dimension ($dim M = dim B + dim F = n + (-n) = 0$). The speculative result may be interpreted as an "invisible" manifold but made up of two manifolds with $n$ and $-n$ dimensions, respectively, then we try to consider it as a kind of "point-like manifold" (zero-dimension), but with "hidden" dimensions, and \\ \\ {\bfseries Type II)} the $(n,-d)$-PNDP manifold, where $n$ (the base-manifold dimension) is different from $d$ (with $d$ a positive integer number and the fiber-manifold dimension is $m=-d$) such that $dim=n+(-d)>0$. The particular speculative feature of this manifold is that it appears as another Einstein-manifold with $(n + (-d))$-dimension. \\ \\ {\bfseries Type III)} it is like the \textit{Type II}, but $ dim = n + (- d) <0 $. It has the speculative feature of being considered, through our projection, like $||(n-d)||$-th desuspension of a point. \\ \\ These manifolds, introducing this new concept of "hidden" dimensions, could have some applications, in particular in the description of "point-like" structures, as a structure of superconducting graphene or in the MOG theory without anomalies, but recently also in the econophysical field, in the description of financial markets influenced by ghost fields such as dark volatility. \vspace{2mm} \noindent\textsc{2010 Mathematics Subject Classification:} 53C25 \vspace{2mm} \noindent\textsc{Keywords and phrases:} PNDP-manifolds; Einstein warped product manifolds; negative dimensional manifolds; derived-manifold; desuspension; point-like manifold; virtual dimensions.
Dynamical Systems and Ordinary Differential Equations and Applications
The minimality of Sturm-Liouville problems with a boundary condition depending quadratically on the eigenparameter
Dr. Yagub Aliyev, ADA University
We study minimality of root functions of the following Sturm-Liouville problem $$ -y^{\prime \prime }+q(x)y=\lambda y,\ 0<x<1, \eqno(0.1) $$ $$ y(0)\cos \beta=y^{\prime }(0)\sin \beta ,\ 0\leq \beta <\pi , \eqno(0.2) $$ $$ y(1)=(a\lambda ^2+b\lambda +c)y^{\prime }(1),\ a\ne 0, \eqno(0.3) $$ where $\lambda $ is the spectral parameter, $q(x)$ is a real valued and continuous function on the interval $[0,1]$, and $a,b,c$ are real. It is known that the eigenvalues of (0.1)-(0.3) form an infinite sequence, accumulating only at $+\infty$, and one of the following cases are possible: \textbf{(a)} All the eigenvalues are real and simple; \textbf{(b)} All the eigenvalues are simple and all, except a conjugate pair of non-real, are real; \textbf{(c)} All the eigenvalues are real and all, except one double, are simple; \textbf{(d)} All the eigenvalues are real and all, except one triple, are simple. By constructing the biorthogonal system explicitly, it is possible to show that in the case (a) the system of eigenfunctions with any two eigenfunctions excluded forms a minimal system in space $L_2(0,1)$. It is also possible to give similar results in the cases (b), (c) and (d). In particular, for the case (b) one can prove that the system of eigenfunctions without the eigenfunction, corresponding to the double eigenvalue, is a minimal system. Similarly, in the case (c) one can prove that the system of eigenfunctions with no excluded functions is a minimal system. Finally, in the case (d) it is possible to prove that the system of eigenfunctions, without the two eigenfunctions, corresponding to non-real eigenvalues, is minimal. These minimality results can then be extended to basis properties in $L_2(0,1)$ and $L_{p}(0,1)$ $(1<p<\infty )$. But the study of these properties wouldn't be complete if we will not consider similar questions for the system of root functions which contain beside eigenfunctions some associated functions. If the associated functions are not excluded then these minimality properties do not always hold true. There are associated functions which when included make the system of root functions not minimal anymore. In the current work, we will give necessary and sufficient conditions for the system of root functions to be minimal in $L_2(0,1)$ for all possible choices of the two excluded functions, including the cases when the excluded functions are eigenfunction and when they are associated functions.
Non-uniformly hyperbolic dynamics for some classes of piecewise smooth systems.
Dr. Sergey Kryzhevich, Saint Petersburg State University
We consider piecewise smooth systems (PWS) of ordinary differential equations. The phase space is supposed to be split into two (or more) subsets with a threshold being a peicewise smooth surface. For such systems, smoothly depending on a parameter, the so-called grazing bifurcation is considered. Roughly speaking, this bifurcation corresponds to the existence of a periodic solution tangent to the threshold. Basing on the approaches of previous author's works and some new ideas on estimating Lyapunov exponents for near-grazing periodic solutions, we describe non-uniformly hyperbolic invariant sets for some classes of PWS. In other words, local coexistence of infinitely many periodic solutions with distinct dimensions of stable/unstable manifolds is proved for those classes of PWS. In addition, we discuss how the developed theory may be applied to study dynamics of strongly nonlinear systems e.g. the van der Pol equation.
On Linear Inhomogeneous Boundary-Value Problems for Differential Systems in Sobolev Spaces
Ms. Olena Atlasiuk, Institute of Mathematics of the National Academy of Sciences of Ukraine
For the systems of ordinary differential equations of an arbitrary order on a compact interval, we study a character of solvability of the most general li\-near boundary-value problems in the Sobolev spaces $W_{p}^{n}$, with $n \in\mathbb{N}$ and $1\leq p\leq \infty$. We find the indices of these Fredholm problems and obtain a criterion of their well-posedness. Each of these boundary-value problems relates to a certain rectangular numerical characteristic matrix with kernel and cokernel of the same dimension as the kernel and cokernel of the boundary-value problem. The condition for the sequence of characteristic matrices to converge is found. We obtain a constructive criterion under which the solutions to these~problems depend continuously on the~small parameter $\varepsilon$ at $\varepsilon=0$, and find the degree of convergence of the solutions. Also applications of these results to multipoint boundary-value problems are obtained. \begin{enumerate} \item O.\,M. Atlasiuk, V.\,A. Mikhailets, \textit{Fredholm one-dimensional boundary-value problems in Sobolev spaces}. Ukr. Math. J. \textbf{70} (2019), no. 10, 1526\,--\,1537. \item O.\,M. Atlasiuk, V.\,A. Mikhailets, \textit{Fredholm one-dimensional boundary-value problems with parameter in Sobolev spaces}. Ukr. Math. J. \textbf{70} (2019), no. 11, 1677\,--\,1687. \item O.\,M. Atlasiuk, \textit{Limit theorems for solutions of multipoint boundary-value problems in Sobolev spaces}. Journal of Mathematical Sciences \textbf{247} (2020), no.~2, 238\,--\,247. \end{enumerate}
Limit states of multi-component discrete dynamical systems
Mrs. Oksana Satur, Institute of Mathematics of the National Academy of Sciences of Ukraine
\textbf{Object}. We study models of multicomponent discrete dynamic conflict systems with attractive interaction, which are characterized by a positive value that is called the attractor index. Consider the set of discrete probability measures $\mu_i \in M^{+}_{1}(\Omega)$ on finite space $\Omega= \{\omega_1, \ldots, \omega_n\}$, $i =\overline{1, m}$. Each of these measures $\mu_i$ can be identified with a stochastic vector ${\bf p}_i = (p_{ij})_{j=1}^n$, where $$p_{ij} = \mu_i(\omega_j), \quad i =\overline{1, m}, \quad j =\overline{1, n}.$$ Consider the mapping $\divideontimes$ \begin{equation}\label{tds} \{{\bf p}^t_1, {\bf p}^t_2, \ldots, {\bf p}^t_m\} \stackrel{\divideontimes, t}{\longrightarrow } \{{\bf p}_1^{t+1}, {\bf p}_2^{t+1}, \ldots, {\bf p}_m^{t+1}\}, \end{equation} which generates multi-component discrete dynamical systems with trajectories (1), where the coordinates of each vector ${\bf p}^t_i = (p^t_{ij})_{j=1}^n$ are changed according to equations \begin{eqnarray}\label{kk} p^{t+1}_{ij} = \frac{1}{z^t} \big(p^t_{ij}(\theta^t+1)+\tau^t_{j} \big), \quad t =0, 1, \ldots. \end{eqnarray} Here $\theta^t = \theta({\bf p}^t_1, {\bf p}^t_2, \ldots, {\bf p}^t_m)$ is a finite positive function, $\mathcal{T}^t = (\tau^t_j)_{j=1}^n$ is a vector with non-negative coordinates (attractor index), and $z^t=\theta^t+1+W^t$ is normalizing denominator, ${W^t = \sum_{j=1}^{n} \tau^{t}_j}$. \textbf{Main results}. \emph{\textbf{Theorem 1.} Let all coordinates of vector ${\bf w}^t=(w_j^t)_{j=1}^n$, $w_j^t:= \frac{\tau^t_j}{W^t}$ be bounded and monotonic (increase or decrease independently one to other). Then for all $i=\overline{1, m}$ there exist %each trajectory of the dynamical system with the initial state $\{{\bf p}_1, {\bf p}_2, \ldots, {\bf p}_m\}$ to the fixed limit state $\{{\bf p}^{\infty}_1, {\bf p}^{\infty}_2, \ldots, {\bf p}^{\infty}_m\}$ $${\bf p}_{i}^\infty = \lim_{t\to \infty} {\bf p}_{i}^t$$ and all limit vectors ${\bf p}_{i}^\infty $ coincide with the vector ${\bf w}^\infty$, i.e. $$p_{ij}^{\infty} = \frac{\tau_j^{\infty}}{W^\infty} \ \ \forall j. $$} Let us consider the different variants of attractor index $\mathcal{T}^t$: \begin{eqnarray} % \nonumber % Remove numbering (before each equation) \tau^t_j := \tau_{j, \min}^t = \min\limits_{i} \{p^t_{ij}\}, \\ \tau^t_j := \tau_{j, \max}^t = \max\limits_{i} \{p^t_{ij}\}, \\ {\tau}^t_{j}:= \overline{\tau}^t_{j}= \frac{1}{m}\sum_{i=1}^{n}p^t_{ij} , \\ \tau^t_{j_1} = \tau^t_{j_2} > 0, \quad j_1, j_2 = \overline{1,n}. \end{eqnarray} \emph{\textbf{Theorem 2.} Let coordinates of the attractor index $\mathcal{T}^{t}$ be given by one of the equations (3), (4), (5), (6). Then each trajectory of a dynamic system~\eqref{tds} with an initial state $\{{\bf p}_1, {\bf p}_2, \ldots, {\bf p}_m\}$ converges to the fixed point $\{{\bf p}^{\infty}_1, {\bf p}^{\infty}_2, \ldots, {\bf p}^{\infty}_m\}$ $${\bf p}_{i}^\infty = \lim_{t\to \infty} {\bf p}_{i}^t, \ \forall \ i=\overline{1, m},$$ where the coordinates of ${\bf p}^{\infty}_i$ have a view: \begin{equation}\label{twstar}p_{ij}^{\infty} = \frac{\tau_j}{W} \ \ \forall \ i=\overline{1, m}, \ \ j=\overline{1, n}. \end{equation}} \textbf{Stability}. The limit state is unstable in cases (3)-(5), however it is stable only in the case (6), when all limit coordinates are equal to $\frac{1}{n}$. \textbf{Application}. Such model of dynamic systems can describe the dynamics of real processes. Attractor index can describe a real external influence on a certain system (for example, information influence on a society). System behavior can be controlled or described by setting attractor index which can be exposed to such influence. \begin{center} {REFERENCES} \end{center} 1. Koshmanenko, V.D., Satur, O.R. Sure Event Problem in Multicomponent Dynamical Systems with Attractive Interaction. Journal of Mathematical Sciences, (2020), 249, pp. 629–646. https://doi.org/10.1007/s10958-020-04962-3 2. Satur O.R. Limit states of multicomponent discrete dynamical systems. Nonlinear Oscillations, (2020), 23, No. 1, pp. 77--89. 3. Koshmanenko V. D., Spectral Theory of Dynamical Conflict Systems [in Ukrainian]. Naukova Dumka, Kyiv (2016).
Conditions of global solvability, Lyapunov stability, Lagrange stability and dissipativity for time-varying semilinear differential-algebraic equations, and applications
Dr. Maria Filipkovska, B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
Theorems on the existence and uniqueness of global solutions, Lagrange stability and instability, dissipativity (ultimate boundedness), Lyapunov stability and instability, and asymptotic stability for time-varying semilinear DAEs (nonautonomous degenerate ordinary differential equations) will be presented, and mathematical models of nonlinear time-varying electrical circuits will be considered in order to demonstrate the application of the presented theorems. The features and advantages of the obtained theorems will also be discussed. The talk is based on the results published in the journals “Differential Equations”, “Global and Stochastic Analysis”, and “Proceedings of the Institute of Mathematics and Mechanics”.
Logic and Mathematical Aspects of Computer Science
Towards Non-Presentable Models of Homotopy Type Theory
Mr. Nima Rasekh, EPFL
One important aspect of homotopy type theory is the construction of {\it models}: $(\infty,1)$-categories in which we can interpret the axioms of our type theory. Studying various models can help us discern which statements can and cannot be proven with our given axiomatization. \par Due to a result by Shulman, we already know that every Grothendieck $(\infty,1)$-topos is a model for homotopy theory. However, we do not expect all models to be a Grothendieck $(\infty,1)$-topos and in particular, we anticipate non-presentable models of homotopy type theory. \par The goal of this work is to take a first step towards showing the existence of non-presentable models of homotopy type theory, by constructing a non-presentable {\it elementary $(\infty,1)$-topos} . Elementary $(\infty,1)$-toposes share many features with Grothendieck $(\infty,1)$-toposes (such as descent, universes, natural number objects, ... ), but are not required to be presentable and thus can include examples that are not Grothendieck $(\infty,1)$-toposes. \par We will construct such examples via the {\it filter construction}. Generalizing a result from $1$-category theory, we prove that for every elementary $(\infty,1)$-topos $\mathcal{E}$ and filter of subobjects $\Phi$, we can construct an elementary $(\infty,1)$-topos $\prod_\Phi \mathcal{E}$, which is in fact not presentable if the filter $\Phi$ is not principal. We will apply this result to the $(\infty,1)$-category of Kan complexes to construct non-presentable examples of elementary $(\infty,1)$-toposes.
Mathematical Physics
Constructive Method to Solving 3D Navier - Stokes Equations
Prof. Alexander Koptev, Admiral Makarov state university of maritime and inland Shipping, Saint-Petersburg
\abstract{ We consider 3D Navier – Stokes equations for motiom of incompressible media. This equations are of mathematical interest and have a lot of applications to practical problems. For today many aspects connected with the Navier – Stokes equations have been studied not enough and need more profound investigation [1-2]. Main unresolved problem is the lack of a constructive method of solution. How to resolve the Navier – Stokes equations while preserving all nonlinear terms is the question that needs to be addressed. \\ \indent The author proposes an approach to this problem the essence of which is to reduce the basic problem of solving the Navier – Stokes equations to a set of simple tasks. We face to five more simple tasks that you need to consistently allow. Each of the individual Navier – Stokes equations must be reduced to a free divergent form and integrated. Resulting equality can be converted so as to exclude some nonlinear and non-divergent terms. As the result we arrive to nine equations which link main unknowns $u$, $v$, $w$, $p$, associated unknowns $\Psi_{i}$ where $i=1,2, …,15$ and an arbitrary additive functions in three variables $\alpha_{i}$, $\beta_{i}$, $\gamma_{i}$, $\delta_{i}$. Considered together these nine ratios provide the first integral of 3D Navier – Stokes equations [3]. The well-known integrals of Bernoulli, Euler - Bernoulli and Lagrange - Cauchy are its special cases [4]. Four of the nine obtained equations represent expressions for basic unknowns. So they determine the overall structure of the solutions. The remaining five equations can be resolved relative to six associated unknowns $\Psi_{j}$ where $j=10,11,…,15$ only if two conditions of compatibility are hold. They reduce to two fifth order equations with respect to nine unknown $\Psi_{k}$ where $ k=1,2,…,9$. Each set of functions $\Psi_{k}$ satisfying a given system determines exact solution of the Navier - Stokes equations. To complete the solution you need to find the six remaining associated unknowns $\Psi_{j}$, where $k=10,11,…,15$. Three of the last functions can be set arbitrarily whereas the remaining three are defined as solution of linear inhomogeneous equations.\\ \indent As a result all values presents of the structure formula for main unknowns are defined and these unknowns are easy to find. Some exact solutions constructed in this way are given in [5-6]. A similar approach can be applied to construct solutions of the Euler equations for case of motion for inviscid incompressible media. In all relationships it is enough to put $\frac{1}{Re}=0$. } \begin{thebibliography}{99} \bibitem{Lodizhenskaya} Lodizhenskaya O.A., { The Mathematical Theory of Viscous Incompressible Fluid }, {\it Gordon and Breach, New York, USA\rm}, 1969. \bibitem{Fefferman} Charls L. Fefferman, { Existence and Smoothness of the Navier --- Stokes Equation}, {\it Preprint, Princeton Univ., Math. Dept., Princeton, New Jzersy, USA\rm}, 2000, pp. 1-5. \bibitem{Koptev1} Koptev A.V., {Integrals of Motion of an Incompressible medium Flow. From Classic to Modern. In book: Handbook on Navier --- Stokes Equations. Theory and Applied Analysis }, Nova Science Rublishers, New York, USA 2017, pp.443-460. \bibitem{Koptev2} Koptev A.V., {Systematization and analysis of integrals of motion for an incompressible fluid flow}, {\it Jour. of Siberian Federal Univ., Math. and Phys.}, {\bf 2018\rm}, 11(3), P. 370-382. \bibitem{Koptev3} Koptev A.V., {Exact solutions of 3D Navier --- Stokes equations}, {\it Jour. of Siberian Federal Univ., Math. and Phys.}, {\bf 2020\rm}, 13(3), P. 306-313. \bibitem{Koptev4} Koptev A.V., {Method for Solving the Navier --- Stokes and Euler Equations of Motion for Incompressible Media}, {\it Jour. of Mathematical Sciences, Springer, New York, USA\rm.}, {\bf 2020\rm}, Vol. 250(1), P. 254-382. \end{thebibliography}
Number Theory
Generalization of proofs and codification of graph families
Mr. Lorenzo Sauras Altuzarra, TU Wien
In this talk, I will explain how to apply Baaz's generalization method and a recent graph-theoretical technique to, respectively, formal proofs from elementary number theory and certain increasing families of simple graphs. These procedures have resulted to have a considerable potential in revealing arithmetical patterns, that got reflected in theorems (for example, sufficient conditions for a value to be a divisor of an arbitrary Fermat number) and conjectures (mainly about integer sequences).
Numerical Analysis and Scientific Computing
Numerical computation of the complex zeros of Bessel and Hankel functions
Mr. Diego Ruiz-Antolín, University of Cantabria
The complex zeros of cylinder functions appear in several problems of applied mathematics and theoretical physics. For example, the complex zeros of Hankel functions are involved in quantum scattering problems by spheres and cylinders. An algorithm (with a Matlab implementation) for computing the complex zeros of the Bessel function of first kind $J_{\nu}(z)$, second kind $Y_{\nu}(z)$, Hankel functions $H^{(1)}_{\nu}(z)$, $H^{(2)}_{\nu}(z)$ and general combinations of Bessel functions $\alpha J_{\nu}(z)+ \beta Y_{\nu}(z)$ and Hankel functions $\alpha H^{(1)}_{\nu}(z)+ \beta H^{(2)}_{\nu}(z)$ is described in this presentation. The algorithm, based on the results obtained in [1], [2], allows to obtain with certainty and accuracy all the zeros of the selected function inside a box in the complex plane. The performance of the algorithm is illustrated with numerical examples. This is a joint work in collaboration with Amparo Gil and Javier Segura. \newline \textbf{References} [1] Computing the complex zeros of special functions. J. Segura, Numerische Mathematik 124(4), 2013, pp 723-752 [2] On the complex zeros of Airy and Bessel functions and those of their derivatives. A. Gil, J. Segura. Anal. Appl. 12(5) (2014) 573-561 [3] An algorithm for computing the complex zeros of Bessel and Hankel functions A. Gil, D. Ruiz-Antolin, J. Segura In progress
Approximation of the two variables Discontinuous Functions by Discontinuous Interlination Splines using Triangular Elements
Prof. Iuliia Pershyna, Ukrainian Engineering Pedagogics Academy
The problem of approximation of smooth functions by continuous construction with sufficient completeness was considered in the work of many researchers. However, as a rule, all these problems are only pleasant exceptions or the result of excessive idealization. More often, the objects of study are mathematically described by functions with discontinuities, kinks, and other violations of smoothness. Such nonsmooth functions include profile shock waves generated by powerful acoustic emitters, or the shape of dunes in the desert, which provides characteristic sharpening. Discontinued objects also very often arise in tasks using remote methods. Thus, the detection of cracks in industrial products using non-destructive testing is an important task in flaw detection, as is the determination of deviations of the internal structure of the product from the standard. In many tasks of geophysics, the determination of the location of bound aries separating blocks with different physical properties is the first step in further studies aimed at determining the physical quantities characterizing the internal structure of the Earth. In computed tomography, when studying the internal structure of a body, it is useful to take into account its heterogeneity, that is, different densities in different parts of the body. In general, remote methods are one of the most promising areas of environmental studies. They serve as the most important source of objective and operational information in various phenomena occurring in the geographic shell of the Earth, and are an effective tool for monitoring the state of the environment and solving environmental management problems. It is obvious that further progress in the development of natural sciences is inextricably linked with the use of remote research methods. This work belongs to a series of works by the authors aimed at the study and improvement of mathematical models in computed tomography. To date, tomography has developed many computational methods, algorithms and software tools aimed at restoring the internal properties of an object. They perform well when restoring objects with smooth properties, but give unsatisfactory results for objects with discontinuous characteristics. Therefore, there is a need to create mathematical methods for approximating discontinuous functions for a more accurate idea of the structure of the studied object. The mathematical foundations of tomography were laid at the beginning of the last century in the works of the German scientist J. Radon, who developed the theory of the transformation of functions of many variables (Radon transformation). According to these transformations, the function of many variables can be characterized not only by its values at points of multidimensional space, but also by integrals from this function taken over an infinite set of lines or planes. A series of works by authors [18-20] devoted to solving the flat problem of radon computed tomography using the heterogeneity of the internal structure of a two-dimensional body. For this purpose, it is advisable to use function interlination operators, since these operators restore (possibly approximated) functions on their known traces on a given system of lines. They provide an opportunity to construct operators whose integrals from these lines (linear integrals) will be equal to integrals from the most renewable function. That is, interlination is a mathematical apparatus, naturally related to the task of restoring the characteristics of objects according to their known projections. This article is a continuation of this article series. Paper is devoted to the development of a method for approximating two variables discontinuous functions by discontinuous interlination splines using arbitrary triangular elements. Experimental data are one-sided traces of a function along a system of given lines, such data are used in remote methods, in particular in tomography. The paper is also devoted to the development a method for approximating of two variables discontinuous functions by triangular elements that comprise one curved side. These methods make it possible to approximate the discontinuous function, using its more complex domains of definition and avoiding the Gibbs phenomenon
Approximate Calculation of Triple Integrals of Rapidly Oscillating Functions using Different Types of Information about Functions
Mrs. Olesia Nechuiviter, Ukrainian Engineering and Pedagogical Academy
Modern methods of digital signal processing are characterized by new approaches to obtaining, processing and analyzing information. There is a need to build mathematical models in which information can be given not only by the values of the function at points, but also as a set of traces of the function on the planes and as a set of traces of the function on the lines. Report is dedicated to the improvement of mathematical models of digital signal processing and imaging by the example of constructing cubature formulas of approximate calculation of integrals of highly oscillatory functions of three variables. The feature of the proposed cubature formulas is using the input information about function or as a set of traces of function on planes or a set of traces of function on lines and of course as a set of values of the function in the points. The cubature formulas are correlated with a formula of the Filon type. The error estimations of the approximation of the integrals from the highly oscillating function by the cubature formulas on the class of differential functions are obtained. The main advantages of methods are high exactness of approximation, the opportunity to decrease the amount of information about function during the calculation.
Partial Differential Equations and Applications
Multiple entire solutions to the curl-curl problem with critical exponent
Mr. Jacopo Schino, Polish Academy of Sciences
We prove the existence of infinitely many solutions with diverging energy to the problem \[ \nabla\times\nabla\times\mathbf{U} = |\mathbf{U}|^4\mathbf{U} \quad \text{in }\mathbb{R}^3. \] We consider vector fileds of the form \[ \mathbf{U}(x) = \frac{u(x)}{r}\Bigl(\begin{smallmatrix} -x_2\\ x_1\\ 0 \end{smallmatrix}\Bigr) \] with $r=\sqrt{x_1^2+x_2^2}$ and $u(x)=u(r,x_3)$ to reduce the curl-curl operator to the vector Laplacian; at the same time we consider an isometric isomorphism between $\mathcal{D}^{1,2}(\mathbb{R}^3,\mathbb{R}^3)$ and $H^1({\mathbb{S}^3,\mathbb{R}^3})$ to recover compactness.
Method of energy estimates for studying of singular boundary regimes in quasilinear parabolic equations
Mrs. Yevgeniia Yevgenieva, Institute of Applied Mathematics and Mechanics of NAS of Ukraine
In the cylindrical domain $Q=(0,T)\times\Omega,\ 0<T<\infty$, where $\Omega\subset {R}^n$ is a bounded domain such that $\partial \Omega\in C^2$, the following problem is considered: \begin{equation}\label{probl1} \begin{aligned} &(|u|^{q-1}u)_t-\Delta_pu=0,\quad p\geqslant q>0,\, \\&u(0,x)=u_0\ \ \text{in} \ \Omega,\ \ u_0\in L^{q+1}(\Omega), \\&u(t,x)\Big|_{\partial\Omega}=f(t,x), \end{aligned} \end{equation} where $f$ generates boundary regime with singular peaking, namely, \begin{equation}\label{bound1} f(t,x)\to\infty\quad\text{as }t\to T,\quad \forall\,x\in \ K\subset\partial\Omega,K\neq\emptyset. \end{equation} Function $f$ is called a localized boundary regime (S-regime) if \begin{equation*} \overline{\Omega}\setminus\Omega_0\neq\emptyset,\quad\text{where }\Omega_0:=\left\{x\in\overline{\Omega}:\sup_{t\to T}u(t,x)=\infty\right\} \end{equation*} for an arbitrary weak solution $u$ of problem \eqref{probl1}. Sharp conditions of localization of boundary regime were obtained by some version of local energy estimates (see \cite{GSh1} and references therein). Papers \cite{ShYev1,ShYev2} are devoted to investigation of the behavior of weak solutions in the case when the blow-up set $\Omega_0\subset\partial\Omega$ (LS-regime). The precise estimates of the limiting profile of solutions were obtained, namely, \begin{equation*} \sup_{t\to T}u(t,x)\leqslant\psi(x),\quad x\in\Omega, \end{equation*} where function $\psi$ is determined by characteristics of peaking of the boundary regime $f$. As an application of these results we study the following parabolic quasilinear equation with a nonlinear absorption term: \begin{equation}\label{eq2} (|u|^{q-1}u)_t-\Delta_pu=-b(t,x)|u|^{\lambda-1}u,\quad(t,x)\in Q,\quad \lambda>p\geqslant q>0, \end{equation} Here $b(t,x)\geqslant0$ is a degenerate absorption potential: $b(t,x)\to0$ as $t\to T$ $\forall\,x\in\Omega$. Precise upper estimates for all weak solutions of equation \eqref{eq2} near to $t=T$ (limiting profile of solution), depending on the behavior of function $b$, were obtained in the papers \cite{ShYev1,Yev2}. It is important to underline that the obtained estimates don't depend on initial and boundary values and hold for large solutions of the equation \eqref{eq2} (if they exist). \begin{thebibliography}{100} \bibitem{GSh1} Galaktionov V.A., Shishkov A.E. {\it Self-similar boundary blow-up for higher-order quasilinear parabolic equations}, Proc. Roy. Soc. Edinburgh. 135A (2005), 1195--1227. \bibitem{ShYev1} Shishkov~A.E., Yevgenieva~Ye.A. {\it Localized peaking regimes for quasilinear parabolic equations}, Mathematische Nachrichten. 292 (2019), no.~6, 1349--1374. \bibitem{ShYev2} Shishkov~A.E., Yevgenieva~Ye.A. {\it Localized blow-up regimes for quaislinear doubly degenerate parabolic equations}, Mathematical Notes, 106 (2019), 639–-650. \bibitem{Yev2} Yevgenieva~Ye.A. {\it Propagation of singularities for large solutions of quasilinear parabolic equations}, Journal of Mathematical Physics, Analysis, Geometry. 15 (2019), no.~1, 131--144. \end{thebibliography}
Some problems for a mixed type equation fractional order with non-linear loaded term
Dr. Obidjon Abdullaev, V.I.Romanovskiy Institute of Mathematics . Uzbek Academy of Science
Note, that with intensive research on problems of optimal control of the agro-economical system, regulating the label of ground waters and soil moisture, it has become necessary to investigate BVPs for a loaded partial differential equations. Integral boundary conditions have various applications in thermo-elasticity, chemical engineering, population dynamics, etc. In this work we consider parabolic-hyperbolic type equation fractional order involving non-linear loaded term: $$ 0=u_{xx}-{}_cD_{oy}^\alpha u+f_1(x,y;u(x,0)),\,\,\,\,\,x>0,\,y > 0$$ $$0=u_{xx}-{u_{yy}}+f_2(x,y;u(x+y,0)),\,\,\,\,\,\,x>0,\,y<0, \eqno(1) $$ $$ 0=u_{xx}-{u_{yy}}+f_3(x,y;u_y(0,x+y)),\,\,\,\,\,\,\,x<0,\, y>0 $$ where $f_i(x,y,u),\,\ (i=1,2,3)$ are given functions, $_{C}D_{oy}^\alpha u$ Caputo operator fractional order $\alpha$ $(0 < \alpha < 1)$ (see {[1]}.p.92): $$ \left( {_CD_{ax}^\alpha f} \right)x = \frac{1}{{\Gamma(1 - \alpha)}}\int\limits_a^x \frac{f'(t)}{{{{\left( {x - t} \right)}^{\alpha }}}}dt ,\,\,\ x > a.$$ Let $\Omega\subset R^2,$ be domain bounded with segments $B_2A_2,$ $A_2{A_1}$ on the lines $x=l,$ $y=h$ at $x>0,\,\ y>0;$ and $A_1C_2,$ $C_2{B_1}$ on the characteristics $x-y=l,$ $x+y=0$ of the Eq. (1) at $x>0,\,\ y<0,$ also with the segments ${B_1}C_1,$ $C_1B_2$ on the characteristics $y-x=h,$ $x+y=0$ of the Eq. (1) at $x < 0,\,\ y>0.$ We denote as$\Omega_0$ parabolic part of the mixed domain $\Omega,$ and hyperbolic parts through $\Omega_1$ at $x>0$ and $\Omega_2$ at $x<0.$ In the domain $\Omega$ (${\Omega} = \Omega^+ \cup {\Omega^-}\cup (A_1B_1),$) we will investigate following \textbf{ Problem I.} To find a solution $u(x,y)$ of Eq. (1) from the class of functions: $u(x,y) \in C(\bar \Omega )\cap C^1\left(\{\overline{\Omega}_2 \setminus \overline{A_1C_2}\}\cup\{\overline{\Omega}_1 \setminus \overline{C_1B_2}\}\right)\cap {C^2}({\Omega_2}\cup\Omega_1);$ ${u_{xx}}, {}_CD_{oy}^\alpha u \in C\left( {{\Omega_0 }} \right);$ satisfying boundary conditions: $$ u(l,t)= {\varphi _{1}}(y),\,\ 0 \leq y \leq h; \eqno(2) $$ $$ \frac{d}{dx}u\left(\theta_1(x)\right)=a_1(x)u_y(x,0)+a_2(x)u_x(x,0)+a_3(x)u(x,0)+a_4(x),\,\ 0\leq x < l; \eqno(3) $$ $$ \frac{d}{dy}u\left(\theta_2(y)\right)=b_1(y)u_x(0,y)+b_2(y)u_y(0,y)+b_3(y)u(0,y)+b_4(y),\, 0\leq y < h; \eqno(4) $$ and integral gluing condition: $$ \mathop{\lim}\limits_{y \to + 0} y^{1-\alpha}u_y(x,y) = {\lambda_1}(x){u_y}(x, -0) + \lambda_2(x)u_x(x,-0)+ $$ \begin{equation}\label{Igc.1} +\lambda_3(x)\int\limits_0^xr_1(t)u(t,0)dt+\lambda_3(x)u(x,0)+\lambda_4(x),\, 0<x<l \end{equation} $$ u_x(-0,y) = {\mu_1}(y){u_x}(+0,y) + \mu_2(y)u_y(0,y)+\mu_3(y)\int\limits_0^yr_2(t)u(0,t)dt+ $$ \begin{equation}\label{Igc.2} +\mu_4(y)u(0,y)+\mu_5(y),\, 0<y<h \end{equation} where $\theta_1(x)=\theta_1\left(\frac{x}{2};-\frac{x}{2}\right),$ $\theta_2(y)=\theta_2\left(-\frac{y}{2};\frac{y}{2}\right),$ $a_i(x),\,b_i(y),\, (i=1,2,3),$ $ \lambda_j(x),\,\ \mu_j(y)\,\ (j=1,2,3,4,5),$ $\varphi_1(y)$ are given functions, besides $\sum\limits_{k = 1}^4 {\lambda_k^2(x)}\ne 0,$ $\sum\limits_{k = 1}^4 {\mu_k^2(x)}\ne 0,$ $\sum\limits_{k = 1}^3 {a_k^2(x)}\ne 0$ and $\sum\limits_{k = 1}^3 {b_k^2(x)}\ne 0.$ On the certain conditions to given function, we can prove uniqueness of solution of the \textbf{Problem I} applying the method of integral energy. Existence of solution, reduced to the Volterra and Fredholm type non linear integral equation respected to $u_y(0,y)=\tau'_2(y)$ and $u(x,0)=\tau_1(x)$ accordingly. %\footnotesize Please use the following commands to format of funding acknowledgements (can be omitted) \textit{Keywords:} Loaded equation, Caputo operator, non-local condition, integral gluing condition, non-linear integral equations.
Poster Session
Positive periodic solutions for nonlinear delay dynamic equations on time scales
Mr. Kamel ALI KHELIL, University of Annaba
In this work, we use fixed point theorem to study the existence of positive periodic solutions for delay dynamic equation on time scales. Transforming the equation to an integral equation enables to show the existence of positive periodic solutions by appealing to Krasnoselskii's fixed point theorem. The obtained integral equation is the sum of two mappings; one is a contraction and the other is compact.
Integration the loaded KdV equation in the class of steplike function
Dr. Iroda Baltaeva, Urgench State University
2010 Mathematics Subject Classification: 39A23, 35Q51, 34K13, 34K29. {\bf {\it Keywords :}} loaded Korteweg-de Vries equation, inverse scattering problem, the class of steplike function, scattering data, Lax pair, eigenvalue, eigenfunction. It is known, that the Korteweg-de Vries equation can be integrated with Inverse Scattering Method [1]. In the works [2,3], the Korteweg-de Vries equations with a self-consistent source were integrated for a class of initial data of ``step'' type; in particular, laws of evolution of the scattering data were established. In applications of the method of inverse scattering transformation one looks for pairs of operators $B$ and $L$ such that the equation has some interesting nonlinear evolution equation for functions $u(x,t)$ that occur as potentials in the operator $L$. For the successful application of the method two further ingredients are needed: 1. the inverse scattering problem must be solved so that the potentials $u(x,t)$ can be reconstructed from scattering data; 2. and that one must be able to determine the evolution of the scattering data with$t$. In this paper, we will consider the loaded Korteweg-de Vries equation $$u_{t} - 6uu_{x} + u_{xxx} + \gamma (t)u(0,t)u_{x} = 0, \eqno(1)$$ where $u = u(x,t), \,x \in R, \,\, t \geq 0, \,\, \gamma (t)$ - is an arbitrary, continuous function. The function $u = u(x,t)$ is a sufficiently smooth and tending to its limits steplike $(c > 0)$ $${\int\limits_{ - \infty} ^{0} {(1 - x){\left| {u(x,t)} \right|}dx}} + {\int\limits_{0}^{\infty} {(1 + x){\left| {u(x,t) - c^{2}} \right|}dx}} + {\sum\limits_{k = 1}^{3} {{\int\limits_{ - \infty} ^{\infty} {{\left| {{\frac{{\partial ^{k}u(x,t)}}{{\partial x^{k}}}}} \right|}dx}}} } < \infty \eqno(2) $$ The equation (1) is considered with initial condition \newline $${\left. {u} \right|}_{t = 0} = u_{0} (x), \quad x \in R^{1},\eqno(3)$$ where $u_{0} (x)$ function satisfies the conditions $(c > 0)$: 1. ${\int\limits_{ - \infty} ^{0} {(1 - x){\left| {u_{0} (x)} \right|}}} dx < \infty ,\;\;\;\;\;\;\;{\int\limits_{0}^{\infty} {(1 + x){\left| {u_{0} (x) - c^{2}} \right|}}} dx < \infty ,\;$ 2. Suppose that, the equation $ - {y}'' + u_{0} (x)y = \lambda y,\;\;x \in R^{1}$ has $\lambda _{1} (0)$, $\lambda _{2} (0),\;...,\;\lambda _{N} (0)$ negative eigenvalues. In this work the solution $u(x,t)$ of the loaded Korteweg-de Vries equation (1) in the class of steplike function (2) with initial condition (3) is obtained via Inverse Scattering Method. \begin{center} {\bf References} \end{center} [1] C.S.Gardner, I.M. Green, M.D. Kruskal Method for solving the Korteweg-de Vries equations.J.Phys.Rev.Lett.-USA,1967.-V.19.P.p.1095-1097. [2] G. U. Urazboev and A. B. Khasanov, ``Integration of the KdV equation with a self-consistent source for step-like initial data,'' in:Proc. Intern. Conf. ``Symmetry and Differential Equations,''Krasnoyarsk (2000),pp. 248--251. [3] G. U. Urazboev and A. B. Khasanov, ``Integration of the KdV equation with a self-consistent source for step-like initial data,''Teor. Mat. Fiz., 129, 38--64 (2001). [4] G. U. Urazboev and A. B. Khasanov, ``Solution of the general KdV equation in the class of step functions'' Journal of Mathematical Sciences, Vol. 136, No. 1, 2006 [5] A.B. Yakhshimuratov, M.M. Matyokubov Integration of loaded Korteweg-de Vries equation in a class of periodic functions Proceedings of Higher Education, Mathematics. - Russia -2016, V.2,pp.8792.
Probability
Markov chains in stationary and ergodic random environment
Dr. Attila Lovas, Alfréd Rényi Institute of Mathematics
Markov chains in stationary random environments (MCREs) with a general (not necessarily countable) state-space appear in several branches of applied probability including mathematical finance, queuing models with non-i.i.d. service times and statistical learning theory. Assuming suitable versions of the standard drift and minorization conditions, we prove the existence of limiting distributions for MCREs in cases when the system dynamics is contractive on the average with respect to the Lyapunov function and large enough small sets exist with large enough minorization constants. We also establish that a law of large numbers holds for bounded functionals of the process. Applications to queuing systems and to machine learning algorithms are also presented.
Joint functional convergence of partial sum and maxima processes
Dr. Danijel Krizmanić, University of Rijeka
For a strictly stationary sequence of random variables we study functional convergence of the joint partial sum and partial maxima process under joint regular variation with index $\alpha \in (0,2)$ and weak dependence conditions. The convergence takes place in the space of $\mathbb{R}^{2}$--valued c\`{a}dl\`{a}g functions on $[0,1]$ with the Skorohod weak $M_{1}$ topology, and the limiting process consists of an $\alpha$--stable L\'{e}vy process and an extremal process. We also show that the weak $M_{1}$ topology in general can not be replaced by the standard $M_{1}$ topology.
Level densities for general $\beta$-ensembles: An operator-valued free probability perspective
Dr. Andrej Srakar, IER
Random point processes corresponding to $\beta$-ensembles for arbitrary $\beta$ $>$ 0, or, equivalently, log gases at inverse temperature $\beta$, are being subject to intense study. The orthogonal, unitary, and symplectic ensembles ($\beta$ = 1, 2, or 4, respectively) are now well understood, but other values of $\beta$ are believed to also be relevant in theory (e.g. relevant for the study of self-adjoint and Schrödinger operators) and applications (e.g. in logistics). For certain rational values of $\beta$, $\beta$-ensembles are related to Jack polynomials, but for general $\beta$ much less is known. In a seminal article, Dumitriu and Edelman (2002) constructed tridiagonal random matrix models for general $\beta$-Hermite and $\beta$-Laguerre ensembles and defined open problems for research on general $\beta$-ensembles, including finding a unified formula for the level density in general $\beta$-case. In general, level density is defined as distribution of a random eigenvalue of an ensemble (by the Wigner semicircular law, the limiting distribution of the eigenvalue is semicircular). We derive the formula for the level density in the general $\beta$-case depending on the multivariate Fuss-Narayana polynomials (related to generalized Fuss-Narayana numbers) and homogenous polynomials from operator-valued free probability theory. In an alternate way, we derive the general $\beta$-case level density formulas using Malliavin-Stein fourth moment-based asymptotic calculus and study their perturbation invariability (Wang and Yan, 2005; Kozhan, 2017). Derivations allow us large possibilities of additional work, and we present extensions to problems of sampling general $\beta$-ensembles (referring to Li and Menon, 2012; Olver et al., 2013; Srakar and Verbic, 2020) and limiting entropy in $\beta$-ensembles related point processes (Mészaros, 2020).
Statistics and Financial Mathematics
Consistency of Bayesian inference with Gaussian priors in an elliptic nonlinear inverse problem
Mr. Matteo Giordano, University of Cambridge
We consider nonparametric Bayesian inference for nonlinear inverse problems based on Gaussian process priors. We present posterior consistency results for the problem of recovering the unknown conductivity in an elliptic PDE in divergence form from noisy discrete observations of its solution, and give a convergence rate for the reconstruction error of the associated posterior mean estimator. The analysis is based on a contraction rates theory for the induced regression problem, combined with a stability estimate for the solution map of the PDE.
Spatial-Temporal Modelling of Temperature for Pricing Temperature Index Insurance
Dr. Che Mohd Imran Che Taib, Universiti Malaysia Terengganu
This paper discusses the pricing methodology of the temperature index insurance based on spatial temporal modelling of temperature. The crucial problem here is the location of the potential insurance buyer relative to the station where index is calculated. Since the observed temperatures at particular station are not always correlated to the temperature where the insurance holder lives, it is important to consider spatial issues in the pricing methodology. Thus, we model the temperature using spatial temporal stochastic processes and employ the universal Kriging method to predict the future temperature at some specific locations. Based on temperature index, we may then price the temperature insurance. We illustrate the pricing methodology using 20 years data from five stations in Malaysia. The findings are important for the development of weather index insurance.
Initial data analysis for longitudinal data – a general framework
Lara Lusa, University of Primorska
Systematic initial data analysis (IDA) and clear reporting of the findings is an important step towards reproducible research. A general framework of IDA for observational studies was proposed to include data cleaning, data screening, and possible refinements of the preplanned analyses (Huebner 2018). Longitudinal studies, where participants are observed repeatedly over time, have special features that should be taken into account in the IDA steps before addressing the research question. Our aim was to propose a framework for IDA in longitudinal studies. Based on the IDA framework from Huebner et al. (2018) we refined it for use in longitudinal studies and provided guidance on how to prepare an IDA plan for longitudinal studies. The framework includes several steps that are specific to longitudinal data, or bear greater importance when data are longitudinal. Appropriate numerical and graphical tools for longitudinal data allow the researchers to conduct IDA in a reproducible manner to avoid non-transparent impact on the interpretation of model results. For example, in the framework we propose how to summarize the time frame of the study, the characteristics of the participants, the outcome and the time varying covariates, how to summarize longitudinal average trends, how to explore the variation between individuals, how to characterize the correlation and the covariance of the outcome and of selected covariates, methods for the exploration of missing values and the description of drop-out. We provide an example on how to conduct IDA in the context of a longitudinal population cohort study (Boesch-Supan et al., 2013). The paper is presented on behalf of the Topic Group “Initial Data Analysis” of the STRATOS Initiative (STRengthening Analytical Thinking for Observational Studies, http://www.stratos-initiative.org). {\bf References} Huebner, M., le Cessie, S., Schmidt, C.O., Vach, W. A Contemporary Conceptual Framework for Initial Data Analysis. Observational Studies 4 (2018) 171-192.\\ Boesch-Supan, A., Brandt, M., Hunkler, C., Kneip, T., Korbmacher, J., Malter, F., Schaan, B., Stuck, S., Zuber, S. Data resource profile: the Survey of Health, Ageing and Retirement in Europe (SHARE), International Journal of Epidemiology 42 (2013) 992—1001.\\
Topology
Sets with the Baire Property in Topologies Defined From Vitali Selectors of the Real Line
Dr. Venuste NYAGAHAKWA, University of RWANDA
Let $\mathcal{F}$ be the family of all countable dense subgroups of the additive topological group $(\mathbb{R},+)$ of real numbers, and let $V$ be a Vitali selector related to some element $Q$ of $\mathcal{F}$. According to the generalized version of Vitali's theorem, it is well known that all elements of the family $\mathcal{P}:=\{V+q: q\in Q\}$ of translated copies of $V$ by points of $Q$, do not have the Baire property in $\mathbb{R}$, with respect to the Euclidean topology, and they are not measurable in the Lebesgue sense. In this paper, we consider the topological space $(\mathbb{R}, \tau(V))$, where $\tau(V)$ is a topology having $\mathcal{P}$ as a base. Apart from studying the topological properties of $(\mathbb{R}, \tau(V))$, we also look at the relationship between the families of sets with the Baire property in topologies defined from $\tau(V)$, by using distinct ideals of sets on $\mathbb{R}$. Moreover, we show that for any $Q_i\in \mathcal{F}$, the spaces $(\mathbb{R}, \tau(V_i)), i=1, 2$, where $V_i$ is Vitali selector related to $Q_i$, are homeomorphic. We further prove that the families of sets with the Baire property in the spaces $(\mathbb{R}, \tau(V_1))$ and $(\mathbb{R}, \tau(V_2))$ are Baire congruent. \begin{thebibliography}{10} \bibitem[Kh1]{Kh1} A.B. Kharazishvili, \emph{Nonmeasurable sets and functions}, Vol.195, Elsevier, 2004. \bibitem[JH]{JH} D. Jankovi\'c and T.R. Hamlett, \emph{New topologies from old via Ideals}, Amer. Math. Monthly, 1990, Vol.97, No.4, 295 - 310. \bibitem[CN1]{CN1} Vitalij A. Chatyrko and V. Nyagahakwa, \emph{Vitali selectors in topological groups and related semigroups of sets}, Questions And Answers in General Topology, 2015, Vol.33, No. 2, 93 - 102. \bibitem[CN2]{CN2} Vitalij A. Chatyrko and V. Nyagahakwa, \emph{On the families of sets without the Baire property generated by the Vitali sets, p-Adic Numbers, Ultrametric Analysis and Applications}, 2011, Vol.3, No. 2, 100-107. \bibitem[Vi]{Vi} G. Vitali, \emph{Sul problema della misura dei gruppi di punti di una retta}, Bologna, 1905. \bibitem[Ox]{Ox} J.C. Oxtoby, \emph{Measure and Category: Graduate texts in mathematics}, Springer-Verlag, New York, Heldenberg, Berlin, 1971. \end{thebibliography}
Chain connected pair of a topological space and its subspace
Prof. Zoran Misajleski, Ss. Cyril and Methodius University in Skopje
In this talk we introduce the notion of chain connected set in a topological space given in [1,2,3]. A set $C$ in a topological space $X$ is chain connected in $X$ if for every two elements $x$ and $y$ and every open covering ${\cal U}$ of $X$ in $X$, there exists a chain in ${\cal U}$ that connects $x$ and $y$. And by a chain in ${\cal U}$ that connects $x$ and $y$ we understand a finite sequence of elements of ${\cal U}$ such that $x$ belongs to first element, $y$ to the last, and any two consecutive elements from the chain have nonempty intersection. Also we introduced the notions of pair of chain separated sets [1] and weakly chain separated sets [3] in a space. The nonempty sets $A$ and $B$ in a topological space $X$ are chain separated in $X$, if there exists an open covering ${\cal U}$ of $X$ in $X$ such that for every point $a\in{A}$ and every $b\in{B}$, there is no chain in ${\cal U}$ that connects $x$ and $y$. The nonempty sets $A$ and $B$ in a topological space $X$ are weakly chain separated in $X$, if for every point $a\in{A}$ and every $b\in{B}$, there exists an open covering ${\cal U}$ of $X$ in $X$, such that there is no chain in ${\cal U}$ that connects $x$ and $y$. Clearly, if two sets are chain separated in a topological space, then they are weakly chain separated in the same space. We give an example of weakly chain separated sets in a topological space which are not chain separated in the same space. Then we study the properties of these sets. Moreover we give a criteria for chain connected set in a topological space by using the notions of chain separatedness and weakly chain separatedness. A set $C$ is chain connected in a topological space $X$ if and only if it cannot be represented as a union of two chain (weakly chain) separated sets in $X$. Then we prove the properties of chain connected sets in a topological space by using the notions of chain separatedness and weakly chain separatedness. Furthermore, we give the criteria for two types of topological spaces using the notion of chain. The topological space is totally separated if any two different singletons are weakly chain separated in the space, and it is the discrete if they are chain separated. Moreover, we generalize the notion to a set in a topological space called totally chain separated set. A set $C$ in a topological space $X$ is totally chain separated in $X$ if every pair of singletons of $C$ are weakly chain separated in $X$. At the end we prove the properties of totally chain separated sets in a topological space. As a consequence, the properties of totally separated spaces are proven using the notion of chain. [1] Z. Misajleski, N. Shekutkovski, A. Velkoska, Chain Connected Sets In A Topological Space, Kragujevac Journal of Mathematics, Vol. 43 No. 4, 2019, Pages 575-586; [2] N. Shekutkovski, Z. Misajleski, E. Durmishi, Chain Connectedness, AIP Conference Proceedings, Vol. 2183, 030015-1-030015-4, (2019); [3] N. Shekutkovski, Z. Misajleski, A. Velkoska, E. Durmishi, Weakly Chain Connected Set In A Topological Space, In a reviewing process.