Jan Draisma (Universität Bern)
Integer sequences associated to matroids (after Huh and collaborators) abstract
Abstract:
June Huh and collaborators proved remarkable theorems about various integer sequences associated to simple combinatorial objects such as graphs or point configurations in vector spaces. Matroids are the right common generalisation of these combinatorial objects. I will introduce matroids, the fundamental operations of deletion and contraction, and several integer sequences associated to matroids. After this introduction, I will formulate some of Huh\'s theorems and sketch some of the ideas that go into their proofs.
10:15 • Université de Neuchâtel, Rue Emile Argand 11, Auditoire B217
Robert Tichy (TU Graz)
Quasi Monte Carlo methods in numerical analysis and insurance abstract
Abstract:
For high-dimensional numerical integration Quasi Monte Carlo methods are an appropriate tool. The approximation error can be estimated by the Koksma-Hlawka inequality which involves a certain concept of total variation of the integrand and the so-called discrepancy, measuring the distribution of the used point set in the unit cube. We discuss various useful concepts of variation and of discrepancy as well as applications to integral equations of the Fredholm type. Furthermore, we consider applications for an insurance surplus process of the Sparre-Andersen type with surplus-dependent premiums. Analytical properties are studied and QMC methods are applied for obtaining solutions to relevant quantities. Numerical examples show that the speed of convergence is satisfactory. This is joint work M. Preischl and S. Thonhauser.
11:00 • EPF Lausanne, UniL campus, Extranef 110
Stefano Abbate (Universität Basel)
"Non-local" calculus for beginners abstract
Abstract:
Differential calculus, as developed by Newton and Leibniz, determines the concept of a derivative from the definition of limit. Since this approach depends only "locally" on the point where it is defined, classical differential calculus can be narrow to represent many physical phenomena. Fractional derivatives are one of the oldest and most known approaches to overcome this issue. In the last century, substantial progress has been developed in establishing "non-local" alternatives.
The aim of this seminar is to present a rigorous approach to define "non-local"operators and a "non-local" vector calculus, introduced by Du et al. in 2011. The goal is to show the analogy with classical differential operators and classical formulas of differential calculus. In this setting, we arrange some models corresponding to the ones known in elementary physics. In particular, we introduce a "non-local" balance law which leads to the study of a basic elliptic equation. In the last part, we give a brief overview on the study of a more complex model, specifically a "non-local" version of the Cahn-Illiard-Oono equation.
12:15 • Universität Basel, Spiegelgasse 5, SR 05.002
Austin J. Stromme (University of Washington)
New statistical phenomena for entropic optimal transport abstract
Abstract:
Optimal transport (OT) is a popular framework for comparing and interpolating probability measures, which has recently been used in diverse application areas throughout science, including in generative modeling, cellular biology, graphics, and beyond. Unfortunately, recent work has shown that OT suffers from a severe statistical curse of dimensionality. In practice, however, the un-regularized OT problem is less common than entropically regularized approximations, known as entropic OT, which afford the use of simpler and more scalable algorithms.
Motivated by its ubiquity in practice, as well as the curse of dimensionality for its un-regularized counterpart, in this talk we identify two new statistical phenomena for entropic OT in the form of bounds on the convergence rate of empirical quantities to their population counterparts. Our first set of bounds are for high-dimensional settings, and give totally dimension-free convergence, albeit with exponential dependence on the regularization parameter. And our second set of bounds are for data distributions with potentially small intrinsic dimension, in which case we show that the dimension-dependence is not only intrinsic to the data distributions, but in fact is automatically the minimum of the intrinsic dimensions of the two distributions at stake. We show that these phenomena hold for entropic OT value estimation, and more generally for the problems of entropic OT map and density estimation. We conclude with applications to transfer learning and trajectory reconstruction. Our simple proof techniques are inspired by convex optimization, and notably avoid empirical process theory almost entirely.
Based on joint work with Philippe Rigollet.
13:15 • EPF Lausanne, Salle GA 3 21
Dr. Lucas Mann (University of Munster)
A p-adic 6-Functor Formalism in Rigid-Analytic Geometry abstract
Abstract:
Using Clausen-Scholze\'s theory of condensed mathematics, we construct a full 6-functor formalism for p-adic sheaves on rigid-analytic varieties. As a special case of this formalism we obtain Poincaré duality for the étale F_p-cohomology of smooth proper rigid-analytic varieties. By applying the formalism to classifying stacks of p-adic groups, we obtain new insights into the p-adic Langlands program.
13:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 25
Dr. Frédéric Paulin (Université Paris-Saclay)
Partial equidistribution of Farey rays in negative curvature abstract
Abstract:
In the unit tangent bundle of a finite volume Riemannian manifold with negative curvature, a closed strong unstable leaf pushed by the geodesic flow equidistributes towards the maximal entropy measure. Fixing a family of discrete points with geometric origin (intersection with divergent orbits of the geodesic flow) on these unstable leaves, and having care of taking neither too many nor too few points (using a prescribed density), we prove that the family of points equidistributes towards a measure supported on a truncated weak stable leaf. We give arithmetic applications by varying arithmetic hyperbolic manifolds. This is a joint work with Jouni Parkkonen.
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
João Pedro Ramos (ETH Zürich)
Time-frequency localization operators, their eigenvalues and relationship to elliptic PDE abstract
Abstract:
In the classical realm of time-frequency analysis, a classical object of interest is the short-time Fourier transform of a function. This object is a modified Fourier transform of a signal f(x), modified by a certain \'window function\', in order to make joint time-frequency analysis of functions more feasible. Since the pioneering work of Daubechies, time-frequency localisation operators have been of extreme importance in that analysis. These are defined through V^∗1_Ω V f=P_Ω f, where V denotes the short-time Fourier transform with some fixed window. These operators seek to measure how much a function concentrates in the time-frequency plane, and thus the study of their eigenvalues and eigenfunctions is intimately connected to the previous questions.
In this talk, we will explore the case of a Gaussian window function φ(x)=e^{-πx^2}, and the operators thus obtained. We will discuss some classical and recent results on domains of maximal time-frequency concentration, their eigenvalues, and stability/inverse problems associated with such properties. During this investigation, we shall see that many of these problems possess some rather unexpected connections with calculus of variations, overdetermined elliptic boundary value problems and free boundary problems in general.
14:15 • Université de Genève, Conseil Général 7-9, Room 1-05
Sergey Finashin (METU Ankara)
Strong Invariants in Real Enumerative Geometry - Lecture 1 abstract
Abstract:
In the first lecture I will discuss a signed count of real lines on real projective hypersurfaces, which is independent of thechoice of real structures and in that sense is “strong invariant”. The simplest examples: a signed count of real lines on a real cubic surface gives 3,while a similar count on a real quintic 3-fold gives 15. In the other lectures I will stick to the case of real del Pezzo surfaces and discussa generalization of the signed count of lines to a signed count of rational curves (involving some combinations of the Welschinger numbers).
All the results are joint with V.Kharlamov.
15:00 • Université de Genève, Conseil Général 7-9, Room 6-13
Reto Kaufmann (ETH Zürich)
Symplectic Toric Submanifolds and Faces of the Moment Polytope abstract
Abstract:
By Delzant\'s Theorem, symplectic toric manifolds are classified by their unimodular moment polytopes. In this talk, we will discuss the bijection between the faces of the moment polytope and the symplectic toric submanifolds of the corresponding symplectic toric manifold. To study recursive aspects like this, it is useful to generalise the notion of a symplectic toric manifold. First we present how this generalisation works in the local picture, given by symplectic toric representations and unimodular polyhedral cones. Then, we will show how this can be used to establish the global correspondence between submanifolds and faces.
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Prof. Dr. Laura Ciobanu (Heriot-Watt University)
Abstract:
Post’s Correspondence Problem is a classical problem in computer science that is famously undecidable, and that acts as a source of undecidability in many areas of algebra, combinatorics, and computer science.Post’s Correspondence Problem (PCP), in its most algebraic formulation, asks whether for a pair of free monoid morphisms g,h : Σ*→ Δ* there is any non-trivial element x ∈ Σ* such that g(x) = h(x). One can similarly phrase a PCP for general groups, rather than free monoids, by asking whether pairs g,h of group homomorphisms agree on any non-trivial inputs. This leads to interesting and unexpected (un)decidability results for PCP in groups, and connections to important topics in geometric group theory and topology.In this talk I will describe some of the recent advances on the Post Correspondence Problem in group theory, its connections to computer science, and to geometry.
17:15 • Universität Bern, Sidlerstrasse 5, 3012 Bern, Lecture Room B6