Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the two-dimensional Gaussian free field, are conjectured to form universality class of extreme value statistics (notably in the work of Carpentier & Le Doussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables, and models where correlations start to affect the statistics. In this talk, I will describe a general approach based on rigorous works in spin glass theory to describe features of the Gibbs measure of these Gaussian fields. I will focus on the two-dimensional discrete Gaussian free field. At low temperature, we show that the normalized covariance of two points sampled from the Gibbs measure is either 0 or 1. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable. (with L.-P. Arguin, 2015).

In a second work (with M. Pain, 2021), we prove absence of temperature chaos for the two-dimensional discrete Gaussian free field using the convergence of the full extremal process, which has been obtained by Biskup and Louidor. This means that the overlap of two points chosen under Gibbs measures at different temperatures has a nontrivial distribution. Whereas this distribution is the same as for the random energy model when the two points are sampled at the same temperature, we point out here that they are different when temperatures are distinct: more precisely, we prove that the mean overlap of two points chosen under Gibbs measures at different temperatures for the DGFF is strictly smaller than the REM's one. Therefore, although neither of these models exhibits temperature chaos, one could say that the DGFF is more chaotic in temperature than the REM.

Finally, I will discuss in detail (depending on the time left) recent works with B. Bonnefont (ex-PhD student, now Post-Doc at University of Geneva) and M. Pain (CR @Toulouse), on questions suggested by B. Derrida.

In this talk we will explain a computation describing Hochschild-Kostant-Rosenberg isomorphism theorems as exponential maps. This computation uses the construction of a filtered circle obtained in collaboration with Moulinos and Toën. As applications we will describe motivic Donaldson-Thomas invariants in positive characteristic and an extension of Hochschild homology for elliptic curves.

A classical topic in spectral theory is Weyl’s law describing the asymptotics of the eigenvalues of the Laplacian on a bounded open set. We are interested in these asymptotics in low regularity situations. Both in the Dirichlet and in the Neumann case we show two-term asymptotics for Riesz means of any positive order under the assumption that the boundary is Lipschitz continuous. For convex sets we obtain universal, nonasymptotic bounds. Tools in our proof are universal heat kernel bounds, as well as Tauberian Remainder Theorems.

Les surfaces d'Alexandrov à courbure intégrale bornée sont les surfaces les plus générales telles que le théorème de Gauß-Bonnet existe et admettant des coordonnées isothèrmes. Une fois cette introduction faite nous discuterons, suivant la théorie de Sturm, l'existence d'inégalité de Poincaré, de mesure doublante, d'inégalité d'Harnack et enfin de noyau de la chaleur.

Cette présentation explore les variétés toriques, où géométrie et combinatoire se rencontrent. Nous montrons comment des éventails (objets géométriques simples) définissent des variétés algébriques riches, en établissant un dictionnaire précis entre leurs propriétés. La correspondance est illustrée par des exemples classiques (espaces projectifs, cônes) puis étendue aux variétés horosphériques, révélant des applications en géométrie birationnelle. Un pont entre visualisation concrète et théorie profonde.

In this presentation, we will construct regular solutions to linear and nonlinear elliptic-parabolic equations in which the natural direction of parabolicity reverses along a critical line. To prevent the emergence of singularities, we will impose orthogonality conditions on the source terms, and follow them during the execution of the nonlinear schemes.

This is a joint work with Anne-Laure Dalibard and Jean Rax, motivated by recirculation problems in boundary layer theory for fluid mechanics, and based on the preprint https://arxiv.org/abs/2203.11067

Splitting schemes are a natural and easy to implement approach to integrate numerically in time differential equations. However, high order splitting methods suffer in general from an order reduction phenomena when applied to the integration of partial differential equations with non-periodic boundary conditions. In this talk, inspired by recent corrector techniques for the second order Strang splitting method, we present a new splitting method of order three for a class of semilinear parabolic problems that avoids order reduction. We prove the third order convergence of the method in a simplified linear setting and confirm the result by numerical experiments. Moreover, we observe numerically that the high order convergence persists for an order four variant of a splitting method, and also for a nonlinear source term.

This presentation focuses on the Vlasov-Poisson system with and without collisions. This kinetic model encodes the multiple scales that arise in a plasma, ranging from fluid-like behavior when collisions dominate, to wave interactions in weakly collisional regimes. We present a numerical method for the Vlasov-Poisson system which preserves its structure in both collision and collisionless regimes. We explain the key ideas in order to preserve the energy structure of the system and its large time behavior in collisional settings. We also show that the method adapts to higher dimensional frameworks.

A differential graded algebraic structure A (e.g. an associative algebra, a Lie algebra, an operad, etc.) is formal if it is related to its homology H(A) by a zig-zag of quasi-isomorphisms preserving the algebraic structure. Kaledin classes were introduced as an obstruction theory fully characterizing the formality of associative algebras over a characteristic zero field. In this talk, I will present a generalization of Kaledin classes to any coefficients ring, to other algebraic structures (encoded by operads, possibly colored, or by properads), and to address a more general problem: the existence of homotopy equivalences between algebraic structures. I will prove new formality and homotopy equivalence results based on this obstruction theory, presenting applications in several domains such as algebraic geometry, representation theory and mathematical physics.