Exceptionnellement, l'accueil de la Cellule Informatique au bureau 225

• ne sera ouvert que de 10h à 12h et de 14h à 16h
• et sera fermé vendredi

en raison de la participation d'une partie de l'équipe informatique aux JRES (journées réseaux de l'enseignement et de la recherche).

• -> Pensez à anticiper les retraits de matériel de prêt.

In this talk, we will define the class group of a number field, which is an important object in number theory and in some areas of cryptography.

Given a number field K, there exists a classic algorithm to compute its class group Cl(K) : the Buchmann algorithm. However, the time complexity is exponential in the degree d of the number field K.

The goal in the second part will be to describe some algorithms that can compute the class group of a large degree number field K inductively, by reducing the problem to the same computation over others smaller degree number fields.

These methods make use of the properties of norm relations and generalised norm relations, which are relations in the G-module Q[G], where G is the Galois group of the Galois closure of K.

Real-world industrial applications frequently confront the task of decision-making under uncertainty. The classical paradigm for these complex problems is to use both machine learning (ML) and combinatorial optimization (CO) in a sequential and independent manner. ML learns uncertain quantities based on historical data, which are then used used as an input for a specific CO problem. Decision-focused learning is a recent paradigm, which directly trains the ML model to make predictions that lead to good decisions. This is achieved by integrating a CO layer in a ML pipeline, which raises several theoretical and practical challenges.

In this talk, we aim at providing a comprehensive introduction to decision-focused learning. We will first introduce the main challenges raised by hybrid ML/CO pipelines, the theory of Fenchel-Young losses for surrogate differentiation, and the main applications of decision-focused learning. As a second objective, we will present our ongoing works that aim at developing efficient algorithms based on the Bregman geometry to address the minimization of the empirical regret in complex stochastic optimization problems.

Statistical estimation in a geometric context has become an increasingly important issue in recent years, not least due to the development in ML of non-linear dimension reduction techniques, which involve projecting high-dimensional data onto a much lower-dimensional sub-manifold. The search for statistical guarantees justifying both the use and the effectiveness of these algorithms is now a much-studied area. In this talk, we will take a geometric view of the issue, and see how some usual curvature quantities are translated into algorithmic guarantees. First, we will see that upper bounds on sectional curvatures give good properties for barycenter estimation, and then we will see that a lower bound on Ricci curvature implies the existence of depth points, giving rise to robust statistical estimators. Those works are based on joint works with Victor-Emmanuel Brunel (ENSAE Paris) and Shin-ichi Ohta (Osaka University).

Given a functional inequality whose extremizers are known, the question of stability is as follows: If a function almost saturates the inequality, is it close to some extremizer? The most famous example is perhaps that of the isoperimetric inequality: The extremizers are the balls, and the question of stability comes down to showing that the isoperimetric deficit controls a certain distance from the ball. There are 4 methods for obtaining such results: the symmetrisation method, the transport mass approach, the selection principle and the ABP method. In this talk, I will present a recent work in which I introduce a fifth method, Stein's method. In particular, I will show how it proves a stability result for the first Steklov eigenvalue.

In addition, I will also present a stability result for the Brascamp-Lieb inequality, which is a functional inequality encoding certain weighted isoperimetric properties. This last result is based on joint work in progress with Bonnefont (IMB Bordeaux) and Joulin (IMT Toulouse).

On dit qu'une classe de groupes de type fini satisfait une alternative de Tits si chacun de ces groupes est soit "petit" (le sens peut dépendre du contexte), soit contient un groupe libre. L'alternative de Tits originelle concerne les groupes linéaires (et dans ce cas petit signifie virtuellement résoluble). Depuis, elle a été démontrée dans de nombreux contextes géométriques, souvent en courbure négative : groupes agissant sur des espaces hyperboliques, sous-groupes de groupes modulaires de surfaces ou de Out(F_N), groupes agissant sur des complexes simpliciaux avec des bonnes propriétés de courbure, etc.

Je présenterai une nouvelle preuve de l'alternative de Tits pour les groupes agissant sur des immeubles de type Ã_2 (objets que j'introduirai). La nouveauté de notre approche est qu'elle se base sur des marches aléatoires. On démontre également au passage un théorème "local-global" : un groupe dont tous les éléments fixent un point a un point fixe global. C'est un travail en commun avec Corentin Le Bars et Jeroen Schillewaert.

Dans cet exposé nous étudierons la taille du groupe de Tate-Shafarevich de certaines surfaces abéliennes sur le corps de fonctions $\mathbb{F}_q(t)$. Hindry et Pacheco ont montré que, pour les variétés abéliennes sur des corps de fonctions, la taille du Sha (dès que finie) est majorée par la hauteur exponentielle. Nous montrerons qu'en dimension 2 leur borne est optimale. Pour cela, on construira une suite de Jacobiennes vérifiant la conjecture de BSD, puis nous calculerons explicitement leur fonction L à l'aide de sommes de caractères. Grâce à des méthodes analytiques, nous estimerons la taille de la valeur spéciale, pour retrouver finalement la borne souhaitée sur le cardinal de leur groupe de Sha.