In this talk we consider the four-waves spatially homogeneous kinetic equation arising in weak wave turbulence theory from the microscopic Fermi-Pasta-Ulam-Tsingou (FPUT) oscillator chains. This equation is sometimes referred to as the Phonon Boltzmann Equation. I will discuss the global existence and stability of solutions of the kinetic equation near the Rayleigh-Jeans (RJ) thermodynamic equilibrium solutions. This is a joint work with Pierre Germain (Imperial College London) and Joonhyun La (KIAS).

Je vous présenterai un résultat récent, en collaboration avec Benoît Grébert et Tristan Robert, concernant l'existence de tores invariants de dimension infinie pour les équations de Schrödinger non-linéaires sur le cercle (sans paramètres externes). En particulier, nous montrons l'existence de solutions presque-périodiques qui ne sont pas quasi-périodiques. Notre preuve repose notamment sur la théorie KAM et des effets régularisants dispersifs que je décrirai.

This talk will investigate the asymptotic behavior of evolution equations, including non-conservative ones, that are expected to converge to equilibrium at exponential speed. After reviewing some of the most classical methods used to derive this type of convergence, we will introduce an original approach to address this question. The key idea is to transform the problem into an equivalent conservative one, where solutions are probability measures, enabling the use of tools from optimal transport. Two examples in 1D -namely, the inhomogeneous heat equation with growth, and growth-fragmentation- will help us illustrate the method. This work is a collaboration with V. Calvez.

We consider the problem of learning a graph modeling the statistical relations of the d variables of a dataset with n samples. Standard approaches amount to searching for a precision matrix representative of a Gaussian graphical model that adequately explains the data. However, most maximum likelihood-based estimators usually require storing the d^2 values of the empirical covariance matrix, which can become prohibitive in a high-dimensional setting. In this talk, we adopt a “compressive” viewpoint and aim to estimate a sparse precision matrix from a sketch of the data, i.e., a low-dimensional vector of size m≪d^2 carefully designed from the data using nonlinear random features (e.g., rank-one projections). Under certain spectral assumptions, we show that it is possible to recover the precision matrix from a sketch of size m=Ω((d+2k)log(d)), where k is the maximal number of edges of the underlying graph. These information-theoretic guarantees are inspired by the compressed sensing theory. We investigate the possibility of achieving practical recovery with an iterative algorithm based on the graphical lasso, viewed as a specific denoiser. We compare our approach and the graphical lasso on synthetic datasets, demonstrating its favorable performance even when the dataset is compressed. Joint work with : Titouan Vayer, Rémi Gribonval and Paulo Gonçalves.

It is known that a manifold diffeomorphic to a K3 surface admits an almost toric fibration (ATF). However, given a specific symplectic form on a K3 it is unclear if it admits an almost toric fibration with lagrangian fibres for the given form. In this talk, we prove that when a Kähler K3 surface admits a Type III Kulikov degeneration with a symplectic form taming the complex structure, the symplectic form admits an ATF whose base is the intersection complex of the degenerate fibre. Furthermore, we shall show that a smooth anti-canonical hypersurface in a smooth toric Fano threefold, equipped with a toric Kähler form, admits such a symplectic Kulikov model.

This is based on joint work with Yoel Groman.

This talk is about learning the parameter-to-solution map for systems of partial differential equations (PDEs) that depend on a potentially large number of parameters covering all PDE types for which a stable variational formulation (SVF) can be found. A central constituent is the notion of variationally correct residual loss function, meaning that its value is always uniformly proportional to the squared solution error in the norm determined by the SVF, hence facilitating rigorous a posteriori accuracy control. It is based on a single variational problem, associated with the family of parameter dependent fiber problems, employing the notion of direct integrals of Hilbert spaces. Since in its original form, the loss function is given as a dual test norm of the residual, a central objective is to develop equivalent computable expressions. A first critical role is played by hybrid hypothesis classes, whose elements are piecewise polynomial in (low-dimensional) spatio-temporal variables with parameter-dependent coefficients that can be represented, for example, by neural networks. Second, working with first order SVFs, we distinguish two scenarios: (i) the test space can be chosen as an L2-space (such as for elliptic or parabolic problems) so that residuals can be evaluated directly as elements of L2; (ii) when trial and test spaces for the fiber problems depend on the parameters (as for transport equations), we use ultraweak formulations. In combination with discontinuous Petrov-Galerkin concepts, the hybrid format is then instrumental to arrive at variationally correct computable residual loss functions. Our findings are illustrated by numerical experiments representing (i) and (ii), namely elliptic boundary value problems with piecewise constant diffusion coefficients and pure transport equations with parameter-dependent convection fields.

Je présenterai quelques résultats récents obtenus pour le processus d'exclusion facilitée, en une dimension. Ce modèle appartient à la famille des "stochastic lattice gases", et il est soumis à de fortes contraintes cinétiques qui créent une transition de phase continue vers un état absorbant à une valeur critique de la densité de particules. Si la dynamique microscopique est symétrique, son comportement macroscopique, sous des conditions aux limites périodiques et une échelle de temps diffusive, est régi par une EDP non linéaire appartenant aux problèmes à frontières libres (ou problèmes de Stefan). L'un des ingrédients est de montrer que le système atteint typiquement une composante ergodique en temps sous-diffusif. Lorsque le système de particules est mis en contact avec des réservoirs de particules (qui peuvent soit détruire, soit injecter des particules aux deux extrémités), nous observons un impact habituel sur les valeurs limites de la densité empirique. Basé sur des travaux communs avec O. Blondel, H. Da Cunha, C. Erignoux, M. Sasada et L. Zhao.

L'étude de la distribution des zéros de polynômes aléatoires, et plus généralement de sections holomorphes aléatoires de fibrés, est un sujet classique. Je discuterai d'un travail en commun avec Michele Ancona (Université Côte d'Azur), dans lequel nous avons obtenu des résultats sur le lieu des zéros de l'image par un opérateur de Berezin-Toeplitz d'une section holomorphe aléatoire d'une grande puissance d'un fibré en droites complexes au-dessus d'une variété kählérienne. Mon but sera d'introduire gentiment toutes ces notions et d'expliquer nos motivations et nos résultats, ainsi qu'une suite naturelle qui constitue un travail en cours.

We report on on-going work in collaboration with P. Hislop on the eigenvalue statistics of the fractional Anderson model, part of a research program on long range Hamiltonians with diagonal disorder. We review some recent results on the spectral properties on long range random models and the conjectures involving the spectral and dynamical transition expected for these models.