Conférence à venir :

The area of stochastic partial differential equations has seen rapid progress over the past decade, spurred by the introduction of the theory of regularity structures and of para-controlled calculus. Despite the close connections of singular SPDEs to physical phenomena, the theory of singular SPDEs has largely been developed homogeneous settings, involving constant-coefficient operators.
First I shall motivate the study of SPDEs in heterogeneous environments using the examples of Φ^4 as a model of ferromagnetism and the parabolic Anderson model as it relates to branching processes.
After recalling the renormalisation of the classical homogeneous versions of these equations, I will describe what changes in the variable-coefficient setting. In particular, I will present a natural choice of renormalisation functions that is local and, for sufficiently covariant regularisations of the noise, explicit.
The talk will be non-technical and focused on the main ideas.

The problem of estimating the maximal number H(m) of limit cycles that planar polynomial vector fields of degree m can exhibit has long been a central question in the qualitative theory of planar dynamical systems. A natural extension to the threedimensional space is to study the maximum number N(m) of limit tori that can occur in spatial polynomial vector fields of degree m. In this work, we focus on normally hyperbolic limit tori and show that the corresponding maximum number N_h(m), if finite, increases strictly with m. More precisely, we prove that N_h(m+1) ⩾ N_h(m)+ 1. Our proof relies on two central results established in this paper. The first is that the normal
hyperbolicity of compact invariant manifolds is preserved under time reparametrizations.

Despite the fundamental nature of this statement, a complete proof has, surprisingly, not previously appeared in the literature, except under rather restrictive assumptions on the flow restricted to the invariant manifold. The second result concerns the torus bifurcation phenomenon near Hopf–Zero equilibria in spatial vector fields. While the conditions for such bifurcations are typically expressed in terms of higher-order normal form coefficients, we derive explicit and verifiable criteria for the occurrence of torus bifurcation, assuming only that the linear part of the unperturbed vector field is in Jordan normal form.

This approach not only circumvents intricate computations involving higher-order normal forms but also ensures the normal hyperbolicity of the bifurcated torus.

Dans cet exposé, je présenterai des résultats sur le phénomène de cutoff pour le mouvement brownien sur certaines variétés compactes de grande dimension, notamment les tores plats, les sphères et les espaces projectifs.

Le cutoff décrit une transition brutale vers l’équilibre : avant un temps critique, le processus reste loin de l’équilibre, tandis qu’immédiatement après, il devient proche de sa mesure invariante. Ici, la convergence est étudiée en distance de séparation.

Nous verrons l’existence du cutoff ainsi que le profil asymptotique associé lorsque la dimension tend vers l’infini. Les méthodes utilisées reposent sur les temps stationnaires forts, les relations d’entrelacement et la construction de processus duaux.

Le théorème de Bour (1862) constitue un résultat fondamental de la géométrie différentielle classique, mettant en évidence l’existence de familles de surfaces isométriques non congruentes dans ℝ³ illustrant la distinction profonde entre géométrie intrinsèque et extrinsèque.
Dans cet exposé, nous proposons une généralisation de ce théorème au cadre des surfaces invariantes dans les variétés riemanniennes tridimensionnelles. Plus précisément, nous considérons des surfaces invariantes sous l’action d’un sous-groupe d’isométries, engendré par un champ de Killing. En exploitant la structure géométrique du quotient associé, nous montrons que toute surface invariante admet des paramètres naturels dans lesquels la métrique induite prend une forme canonique.
Ce cadre permet de construire explicitement une famille à un paramètre de surfaces isométriques, généralisant ainsi la construction classique de Bour. Nous discutons également des applications à l’étude des surfaces minimales invariantes et mettons en évidence les relations avec les travaux de do Carmo et Dajczer sur les surfaces hélicoïdales.

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.

Il existe plusieurs notions de borne de courbure de Ricci pour les graphes et les groupes. Y. Ollivier a proposé une approche via le transport optimal à la Lott-Villani-Sturm. D. Bakry et M. Emery ont proposé une autre approche via la théorie des diffusions et la formule de Bochner en géométrie riemmanienne. Le but de l’exposé sera de montrer, via cette deuxième approche, comment donner des versions discrètes de résultats classiques en analyse sur les variétés (Bisoop-Gromov, Li-Yau, Harnack, Poincaré, …).

This is joint work with Bonthonneau-Chhaibi-Rivière-To on the continuum construction of the 2d Yang Mills measure and Nohra on scaling limits from discrete models towards the continuum measure.
In this talk, we will discuss the recent construction of the 2d Yang Mills measure on surfaces as random
distributional connections using ideas from Morse theory. This generalizes previous works of Chevyrev on the flat torus to the case of arbitrary closed compact surfaces of any genus. I will also discuss a universality result: how our measure arises as scaling limits of certain piecewise affine connections arising from a large family of lattice gauge theories.