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Activités scientifiques du département

Le DMA est à la fois un département d'enseignement et un département de recherche. Cette structuration originale vise notamment à mettre très tôt les élèves au plus près de la recherche en train de se faire.

Publications

L'essentielle de publications des membres du département, des thèses et des HDR qui y sont soutenues sont disponibles sur le serveur HAL.

  • 30 September 2025 hal-05288954 publication

    We prove the existence and uniqueness of strong solutions to the equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, subject to perturbations of the source term and lateral boundary conditions. Since the solutions we consider have opposite signs in the lower and upper half of the domain, this is a quasilinear forward-backward parabolic problem, which changes type across a critical curved line within the domain. In particular, lateral boundary conditions can be imposed only where the characteristics are inwards. There are several difficulties associated with this problem. First, the forward-backward geometry depends on the solution itself. This requires to be quite careful with the approximation procedure used to construct solutions. Second, and more importantly, the linearized equations solved at each step of the iterative scheme admit a finite number of singular solutions, of which we provide an explicit construction. This is similar to well-known phenomena in elliptic problems in nonsmooth domains. Hence, the solutions to the equation are regular if and only if the source terms satisfy a finite number of orthogonality conditions. A key difficulty of this work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow. To tackle this deceivingly simple problem, we develop a methodology which we believe to be both quite natural and adaptable to other situations in which one wishes to prove the existence of regular solutions to a nonlinear problem for suitable data despite the existence of singular solutions at the linear level. This paper is a shorter version of [3].

    Anne-Laure Dalibard, Frédéric Marbach, Jean Rax

  • 23 September 2025 hal-05272270 pré-publication

    We study the convergence to equilibrium in high dimensions, focusing on explicit bounds on mixing times and the emergence of the cutoff phenomenon for Dyson-Laguerre processes. These are interacting particle systems with non-constant diffusion coefficients, arising naturally in the context of sample covariance matrices. The infinitesimal generator of the process admits generalized Laguerre orthogonal polynomials as eigenfunctions.

    Our analysis relies on several distances and divergences, including an intrinsic Wasserstein distance adapted to the non-Euclidean geometry of the process. Within this framework, we employ tools from Riemannian geometry and functional inequalities. In particular, we establish exponential decay and derive a regularization inequality for the intrinsic Wasserstein distance via comparison with relative entropy.

    Samuel Chan-Ashing

  • 22 September 2025 tel-05273265 thèse

    Here are seemingly unrelated problems: computing rational homotopy groups of spheres in rational homotopy theory, purity in algebraic geometry, Koszul duality for the category of a reductive group in representation theory, splitting Drinfeld space's de Rham complex in the p-adic Langlands program, deformation quantization of Poisson manifolds in mathematical physics. And yet, all of them boil down to the same question: formality. A differential graded algebraic structure A (e.g. an associative algebra, a Lie algebra, a Pre-Calabi-Yau algebra, etc.) is formal if it is related to its homology H(A) by a zig-zag of quasi-isomorphisms preserving the algebraic structure. This thesis develops obstruction classes allowing to prove formality results. On the one hand, it incorporates aforementioned results into a single theory. On the other hand, it provides tools to study these questions in cases little studied hitherto: over any coefficient ring and for algebraic structures with several outputs: algebras encoded by properads.

    Coline Emprin

Les actualités de la recherche

Annonce de conférences, congrès et autres événements scientifiques.

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Annales de l’ENS

Les Annales scientifiques de l’École normale supérieure publient 6 fascicules par an. Elles sont éditées par la Société mathématique de France depuis 2008.