Institut de recherche mathématique avancée

Résumé : La catégorie de flot d'une paire de Morse-Smale est une catégorie topologique dont les objets sont les points critiques de la fonction, et les espaces de morphismes sont les espaces de trajectoires brisées de pseudo-gradient reliant ces points critiques. La construction du complexe de Morse classique utilise les espaces de trajectoires reliant des points critiques d'indices consécutifs. Récemment, Barraud, Damian, Humilière et Oancea ont étendu la construction du complexe de Morse à la catégorie des infini systèmes locaux sur la variété, en faisant intervenir les espaces de trajectoires de toutes dimensions. Cette construction s'étend immédiatement à des objets plus généraux appelés modules sur la catégorie de flot. Cela conduit à s'intéresser à la catégorie de flot en tant qu'infini-catégorie. Je parlerai d'un résultat que j'ai obtenu, affirmant que cette infini-catégorie est équivalente à l'infini-catégorie des chemins sortants pour la stratification de la variété par les variétés stables du pseudo-gradient.

Résumé : Proof-of-Work blockchains rely on miners who continuously spend resources in exchange for uncertain rewards. To reduce this uncertainty, many miners join Pay-Per-Share mining pools where they receive regular payments while the pool manager absorbs the risk. This creates a natural question: how should miners choose between competing pools, and how should pools set their fees and payout policies? This talk discusses a stochastic model for this problem in which the surplus of a mining pool is described by a two-sided jump process. The first part presents a mean-variance approach to miners' allocation across pools and explains how this leads to a Nash equilibrium between competing pool managers. The second part turns to the pool manager's dividend problem, formulated in an expected discounted dividend framework. Using ideas from actuarial risk theory, it is shown how barrier strategies arise naturally, under which conditions they are optimal, and how the optimal barrier can be computed numerically. The aim is to understand the economic incentives behind mining pools, and their possible impact on the decentralization of Proof-of-Work blockchains.

Résumé : In this talk we discuss the convergence of adaptive FEM for strongly monotone nonlinear partial differential equations arising from the minimization of a convex energy functional. Rather than focusing on specific algorithmic implementations, we establish a generalized theoretical framework that identifies sufficient conditions for an adaptive scheme to contract the energy difference at an optimal rate relative to the number of degrees of freedom. The presented abstract point of view is subsequently shown to include many existing error estimation approaches (local residuals, flux equilibration, linear residual liftings, energy descent), thereby providing a unified proof for their optimal convergence. A key contribution of our recent work is the introduction of computable local equivalence factors yielding a computable a posteriori bound on the contraction rate which can then be used for an improved local refinement procedure. Finally, we will discuss technical challenges posed by higher-order discretizations and show how the arising oscillations can be managed through local higher-order flux-equilibration.

Résumé : In this talk, I will introduce the nilpotent cohomological Hall algebra COHA(S, Z) of coherent sheaves on a smooth quasi-projective complex surface S that are set-theoretically supported on a closed subscheme Z. This algebra can be viewed as the "largest" algebra of cohomological Hecke operators associated with modifications along a subscheme Z of S. When S is the minimal resolution of an ADE singularity and Z is the exceptional divisor, I will describe how to characterize COHA(S, Z) in terms of the Yangian of the corresponding affine ADE quiver Q (based on joint work with Emanuel Diaconescu, Mauro Porta, Oliver Schiffmann, and Eric Vasserot, arXiv:2603.03386). More generally, I will discuss nilpotent COHAs arising from Bridgeland stability conditions on the bounded derived category of nilpotent representations of the preprojective algebra of Q, following joint work with Olivier Schiffmann and Parth Shimpi (arXiv:2511.08576).

Résumé : Abstract: In this talk, I'll give a gentle introduction to both traditional and neural numerical methods for the simulation of partial differential equations (PDEs). On the one hand, traditional numerical methods (finite differences, finite elements, ...) have been successfully used for the last 50 years to obtain approximate solutions to PDEs. On the other hand, new methods based on neural networks (e.g. PINNs, Physics-Informed Neural Networks) have recently been introduced for the same purpose. While they are often (mistakenly!) seen as black-box solvers, I'll show that both (traditional and neural numerical) approaches can be defined as oblique projections on suitable function spaces. This unified framework offers a better way to compare their specific strengths and weaknesses. If time permits, I will also briefly conclude with some research directions undertaken in the MACARON team.

Bio: Victor Michel-Dansac is a permanent researcher (ISFP, INRIA Starting Faculty Position) in the project-team MACARON of the Inria Strasbourg research center, located at IRMA (Institut de Recherche Mathématique Avancée), in Strasbourg. His research areas encompass several topics, including scientific computing, the development of numerical methods, and, more recently, Scientific Machine Learning, enriching numerical schemes with techniques from machine learning.

Résumé : The Gromov non-squeezing theorem is a famous result in symplectic geometry that illustrates the notion of symplectic rigidity. It states that, to symplectically embed a closed ball into a cylinder, the radius of the ball must be smaller than that of the cylinder. In this talk, I will present the proof for linear symplectomorphisms and explain why the Gromov non-squeezing theorem is surprising when compared to the linear case and the volume-preserving diffeomorphism case.