IMPAN Colloquium
Wednesdays at 14:15 in Room 321, before the lecture there are cookies and tea in room 409 at 13:45
Organizers:
Marcin Lara, Grigor Sargsyan, Mateusz Wasilewski, Aneta Wróblewska-Kamińska
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UPCOMING MEETINGS
UPCOMING MEETINGS
NOVEMBER 12
Damian Osajda (University of Copenhagen/University of Wrocław)
The Cannon Conjecture
A finitely generated group is called "hyperbolic" (in the sense of Gromov) if all geodesic triangles in a Cayley graph of the group are uniformly thin. Basic examples are free groups and fundamental groups of closed Riemannian manifolds of negative sectional curvature, e.g. fundamental groups of closed surfaces of genus at least two. For a given hyperbolic group there exists a topological space associated to it - its Gromov boundary. It is the collection of classes of all geodesic rays in a Cayley graph up to a finite Hausdorff distance. The Cannon Conjecture asserts that if the Gromov boundary of a hyperbolic group is homeomorphic to the two-sphere then the group is virtually (i.e. up to a finite index subgroup) the fundamental group of a closed 3-manifold of constant curvature -1.
I will explain all the notions involved in the formulation of the Cannon Conjecture, and say a word or two about our recent work on the subject, joint with Daniel Groves, Peter Haïssinsky, Jason Manning, Alessandro Sisto, and Genevieve Walsh.
NOVEMBER 19, 15:00
Jacek Miękisz (University of Warsaw)
NOVEMBER 26
Wojciech Gajda (Adam Mickiewicz University in Poznań)
PAST MEETINGS
OCTOBER 8
Błażej Miasojedow (University of Warsaw)
Sampling as optimisation over the space of measures
Sampling from probability distributions that are only known up to a constant is a key challenge in computational statistics and machine learning. A common method is to use stochastic processes such as Langevin dynamics, which form the basis of many MCMC (Markov chain Monte Carlo) algorithms. In this talk, we present an alternative view: sampling can be seen as an optimisation problem in the space of probability measures equipped with a specific metric, the so-called Wasserstein space W2. From this perspective, sampling algorithms approximate gradient flows in the W2 metric. This approach provides a clearer interpretation of existing methods, extends their use to a wider range of distributions, and allows for more precise convergence guarantees.
OCTOBER 15
Jakub Byszewski (Jagiellonian University)
Algebraic Approaches to Generalised Polynomials
Generalised polynomials are functions obtained from ordinary polynomials by allowing the use of the floor function, together with addition, multiplication, and composition. Such expressions arise naturally in number theory and dynamics. Despite their simple definition, they exhibit a rich and rather subtle arithmetic, combinatorial, and dynamical behaviour.
In this talk, I will discuss several algebraic methods that are used to study these properties. These include techniques based on nilmanifolds and the algebra of ultrafilters, as well as tools of a more Diophantine nature, such as those related to S-unit equations.
The talk is based on joint work with Jakub Konieczny (Kyiv).
OCTOBER 22
Joachim Jelisiejew (University of Warsaw)
Moduli spaces, tensors and complexity theory
The best answer to a classification problem is a finite list. When this is not possible, one can hope that the set in question has more structure, for example, topology, or even some structure of a variety. This is called a moduli space. It comes with a wealth of invariants and additional structure.
In the talk I will illustrate how this works for finite-dimensional algebras, which are impossible to classify, yet basic enough to appear in various contexts, including outside mathematics, for example in complexity theory. The integrals, geometry and singularities of the moduli space yield interesting information about the algebras themselves and applications to computer science. The fact that the moduli space appears "naturally" makes it possible to study using a range of methods, from combinatorics to derived geometry.
The talk is aimed at non-geometers not familiar with the notions above. The pace will depend on the audience. You are free to derail the talk.
OCTOBER 29
Barbara and Jaroslav Zemánek Prize Award Ceremony
14:15–15:15 Introductory lecture
Rafał Latała (University of Warsaw)
Selected Results of Alexandros Eskenazis with a Probabilistic Perspective
Alexandros Eskenazis is the author of many deep results in metric geometry, high-dimensional convex geometry, probability theory, and discrete analysis. During the talk we will discuss a selection of his results from the standpoint of a probabilist.
15:45–16:45 Lecture of the laureate
Alexandros Eskenazis (Sorbonne)
Metric characterizations in the Ribe program
A key goal of the Ribe program is to provide metric characterizations of classical linear properties of norms in an attempt to create a nonlinear analogue of the local theory of Banach spaces. In this talk, we shall explore some recent advances in this direction, including the development of new bi-Lipschitz invariants for metric spaces, the refutation of conjectured metric analogues of classical results from the linear theory, and the emergence of new, purely metric phenomena within the Ribe program dictionary. The talk is based on joint works with Manor Mendel and Assaf Naor.
NOVEMBER 5
Krzysztof Oleszkiewicz (University of Warsaw)
On the asymptotic behavior of the optimal constants in the Khinchine and Khinchine-Kahane inequalities
The lecture will center around tail estimates for Rademacher sums, i.e. linear combinations of independent symmetric +/-1 random variables. More precisely, we will focus on estimates of the form P(|S|>t) < P(|G|+K>t) for every t>0, where G is a Gaussian random variable with distribution N(0,1) and S is a Rademacher sum normalized in such a way that |S| has the same expectation as |G|. While it cannot hold for K=0, we will show that there exists a universal positive constant K for which this inequality is satisfied; in other words, |G| + K stochastically dominates |S|. Some similar results will also be discussed, as well as their relation to the Khinchine inequalities.
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