Institut de recherche mathématique avancée

Résumé : Les variétés de caractères sont des espaces classifiant les représentations d’un groupe dans un autre, à conjugaison près. Ils apparaissent naturellement dans de nombreux problèmes d’origines géométriques et dynamiques.Ces variétés de caractères ont un groupe de symétries naturelles (le groupe modulaire pour les groupes fondamentaux des surfaces), et on peut s’intéresser à l’action de ce groupe. Cette action est riche d’un point de vue dynamique. Nous expliquerons d’abord que cette dynamique est très chaotique, et assez bien comprise, quand le groupe d’arrivé est compact, puis nous présenterons un programme et des résultats qui visent à comprendre cette dynamique quand le groupe d’arrivé est PSL(2,R), en utilisant des techniques qui viennent de la renormalisation des échanges d’intervalles.

Résumé : For a graph $G$ and integer $k$, the Ramsey number $R(G; k)$ is the smallest integer $n$ such that any edge-coloring of a complete graph $K_n$ on $n$ vertices with $k$ colors results in a monochromatic copy of $G$. Determining Ramsey numbers even in the case of two colors has been a stubborn problem since its introduction in 1930. Except for finding the classical Ramsey number $R(t) = R(K_t;2)$ for cliques, one of the main open problems in the area is to determine multicolour Ramsey numbers for cycles. For positive integers $k$ and $\ell$, we show that $R(C_{2 \ell + 1}; k) \leq (4 \ell)^k \cdot k^{k/\ell}$, where $C_{2\ell +1}$ is a cycle of length $2\ell+1$. This is the first improvement for fixed $\ell$ and large $k$ since the bound $2\ell (k+2)!$ by Erd\H{o}s et al. from 1973. This is a joint work with Wouter Cames van Batenburg, Oliver Janzer, Lukas Michel, and Mathieu Rundstr\"{o}m.

Résumé : Traditionally, Langlands functoriality refers to the identification of automorphic forms whose parameters take a special shape. In this talk, we explain how to ask analogous questions on the Hitchin moduli space using perspectives from the relative Langlands program. We gain, in this setting, the advantage of working with a version of Langlands duality which is readily computable, and if time permits we will discuss the ramifications of these calculations for automorphic periods and L-functions.

Résumé : Non-associative algebras arise in many mathematical settings, with the Lie algebra being a significant example. A powerful tool to study a Lie algebra is its universal enveloping algebra, which is an associative algebra, whose representation coincides with the representation of that Lie algebra. The classical Poincaré-Birkhoff-Witt (PBW) theorem plays a role by giving a vector space basis of the enveloping algebra. We will generalise the notion of enveloping algebra to arbitrary (non-associative) algebra and present the generalised PBW theorem via giving examples. If time permits, we could talk about a more generalised perspective — universal enveloping Operads.

Résumé : This presentation will have two parts. The first part introduces an extension of spectral clustering on data streams. Assuming observations are made in an online fashion, we show how spectral clustering can be performed on the fly with a fixed amount of available memory. Studying this problem amounts to the spectral analysis of a particular Gram matrix in the high-dimensional regime, which we perform using tools from Random Matrix Theory. Based on our results, we describe the optimal memory policy and the corresponding clustering performance.
The second part of the presentation tackles the estimation of a planted low-rank signal in a large-dimensional tensor. This problem reveals a statistical-to-computational gap: a regime in which the maximum likelihood estimator is efficient, yet no polynomial-time algorithm can compute it. We study the performance of a procedure based on unfoldings, which is known to achieve the best algorithmic threshold, thereby revealing insights into the computational barrier.