Paul Zinn-Justin (University of Melbourne)
Title T.B.A.
10:00 • EPF Lausanne
Dr. Tariq Osman (Universität Zürich)
Title T.B.A.
13:30 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 28
Serge Cantat (Université de Rennes)
On the product replacement algorithm in compact Lie groups abstract
Abstract:
Let G be a compact Lie group. Let n be an integer. Given n elements g1, …, gn in G, one can choose one of the gi and replace it into its inverse, or into the product gigj for some j distinct from i; these basic operations do not change the subgroup of G generated by the gi. Doing so, one gets a group of transformations of Gn (generated by all these basic operations). The problem I will discuss is: what can be said on the dynamics of this group action on Gn? For instance, can we describe the closures of orbits, or the invariant probability measures? As we shall see, the answer is yes when n becomes large, in fact larger than an explicit constant that depends on G. This is based on a joint work with Christophe Dupont and Florestan Martin-Baillon, which complements previous works of Tsachik Gelander.
15:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Prof. Dr. Patricia Alonso Ruiz (Friedrich Schiller University Jena)
Abstract:
With its long history and practical applications, the isoperimetric inequality serves as a geometric tool whose proof has connected geometry with many areas of mathematics. For instance, given a certain amount of material, you would like to find a shape that maximizes the volume you can wrap with it. However, if the material shows fractal features, such as being highly porous and symmetric... Which volume measure do you use? Which perimeter measure? And what kind of relationship (!) between these measurements can one expect?... and where is Brownian motion, by the way? This talk will address the questions above, revisiting a beautiful connection between Brownian motion, functions of bounded variation and the isoperimetric inequality originally due to Ledoux. We will see how these ideas may provide an approach to study isoperimetric problems on fractal spaces, and in particular on some whose micro- and macrostructure present different scaling. The talk is based on joint work with Fabrice Baudoin (Aarhus University).
16:15 • Universität Bern, Hörsaal B78, ExWi, Sidlerstrasse 5, 3012 Bern
Prof. Dr. Matthew Colbrook (University of Cambridge)
Beyond the circle: Schiffer’s conjecture and computation abstract
Abstract:
When does the behaviour of a single eigenfunction determine the shape of a domain? Schiffer’s conjecture predicts that a bounded, simply connected planar domain must be a disc if it admits a nonconstant Neumann eigenfunction that is constant on the boundary. I will present joint work with George Stepaniants disproving this conjecture by constructing a noncircular domain with real-analytic boundary. The same example disproves the planar Pompeiu conjecture, connecting spectral geometry to a problem about averages over translated and rotated domains dating back to 1929. The challenge is not merely to find a convincing numerical picture, but to prove that an exact domain and eigenfunction exist. I will explain how conformal mapping, spectral analysis and a contraction argument turn a numerical candidate into a theorem. The crucial step is to control everything a finite computation leaves out, including the infinitely many omitted modes, while rigorously accounting for numerical error. I will then draw on recent work on eigenvalue certification and the conjectures of Forsythe and Kenig to discuss the changing role of computers and AI in mathematical discovery. These examples range from assistance with code to the discovery of proof mechanisms and formal verification in Lean. They also expose persuasive arguments that fail. I will distinguish the roles of exploration, mathematical judgment and independent verification, and discuss how these tools are changing which problems are practical to pursue. No background in computer-assisted proof or AI will be assumed.
16:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2