- 8h00: Breakfast
- 9h30 : Gautier Schanzenbacher -
An Introduction to Hyperbolic Geometry: surfaces, geodesics and entropy.
For centuries, mathematicians tried to prove Euclid’s fifth axiom (the parallel postulate) using only the first four. In the 19th century, it was discovered that the parallel postulate is independent of the others and that consistent geometries can be developed in which it does not hold. One such geometry is hyperbolic geometry.
In this talk, I will start from these foundations to explain the world of curves, surfaces and explain the concept of entropy in the simplest way possible.
- 10h00 : Léo Delage -
The structure of free-by-cyclic groups.
A free-by-cyclic group is a semidirect product of a free group with the integers. It is defined by an automorphism of the free group and its geometry encodes the behaviour of the automorphism. Some of them arise as special cases of one-relator groups and they are closely related to the conjugacy problem for the Outer automorphism group of a free group.
In this talk, I will focus on structural results from the last two decades that elucidate the structure of free-by-cyclic groups. As an application, I will mention my result about new constructions of nonpositively curved 2-complexes whose fundamental groups are free-by-cyclic.
- 10h30 : Break
- 11h00 : Nhat Thang Le -
A forward approach to optimal stopping problems and optimal stopping games.
Optimal stopping theory, developed through seminal works of Dynkin, Shiryaev, and others, is a fundamental topic in probability, stochastic control, and mathematical finance. Classical approaches typically rely on dynamic programming principles and the Snell envelope, which characterize the value function through backward recursive procedures. While these methods provide a powerful theoretical framework, alternative computational approaches have also been investigated.
In this talk, I will first review some classical optimal stopping problems and discuss Irle's forward algorithm for finite-state Markov processes. In contrast to traditional backward methods, Irle's approach constructs the value function through a forward iterative procedure, leading to an efficient computational framework and new insights into the structure of optimal stopping problems.
Motivated by these ideas, I will then consider a two-player zero-sum stopping game driven by a homogeneous Markov process on a finite state space. Such games extend optimal stopping problems to a setting with competing players and strategic interactions. I will present a new algorithm for computing the value function of the game, which can be viewed as a game-theoretic extension of Irle's forward algorithm. The convergence of the method, the number of iterations required, and several numerical examples illustrating its performance will also be discussed.
- 11h30 : Thomas Agugliaro -
Numerical verification of the Riemann hypothesis.
The zeta function is one of the most famous holomorphic functions. It was named after Riemann because he was the first to notice that the behavior of its zeroes is intimately linked to the distribution of prime numbers. What is less known is that Riemann had to perform tedious computation to verify its hypothesis for the first non-trivial zeroes. In this talk, we will see which computations led Riemann to see that the first zeroes lies *exactly* on the critical line.
- 12h30 : Lunch
- 15h30 : Ludovic Felder -
The parity conjecture.
The Birch-Swinnerton-Dyer conjecture is one of the most difficult open questions in algebraic geometry, for now. It concerns the rank of the rational points on abelian varieties defined over number fields, that is far from easy to compute in general.
In this talk, I will define abelian varieties in a very informal way, and try to explain some of the difficulties arising while trying to solve this conjecture. In order to give insight into a problem that seems more manageable, I will talk about the parity conjecture and explain how it fits into this context.
- 16h00 : Juan Mardomingo-Sanz -
Slow-fast limits of stochastic particle systems arising in telomere biology.
The ends of linear chromosomes, called telomeres, shorten at each cell replication, eventually driving the cells to a senescent state when they become too short. The enzyme telomerase, present in cancerous cells and some unicellular organisms, elongates the telomeres and allows cells to continue replicating. Recent experiments show that if this enzyme is inactivated some rare survivors (ALT), which elongate their telomeres without telomerase, will appear and will eventually invade the cultures. I will present a simple stochastic particle system which accounts for the emergence and invasion of these ALT cells under an appropriate scaling with different speeds for each cell type.
- 16h30 : Break
- 17h00 : Roméo Troubat -
(G,X)-structures and discrete subgroups of Lie groups.
A (G,X)-structure on a manifold M where G is a Lie group and X a homogenous space associated to G is a geometric structure on M for which M locally looks like the space X. This includes for instance flat or conformally flat Riemannian manifolds, projective manifolds, etc. Due to works of Thurston, the study of those "model geometries" have been deemed a good way to study geomtric structures as a whole. We will establish a link between (G,X)-structures and discrete subgroups of G and will exhibit interesting examples of (G,X)-structures.
- 19h00 : Dinner