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.

In this overview talk, I will present the encounter-based approach to diffusive processes in Euclidean domains and highlight its fundamental relation to the Steklov spectral problem. So, the Steklov eigenfunctions turn out to be particularly useful for representing heat kernels with Robin boundary condition and describing diffusive dynamics with reaction events on the boundary. In the second part of the talk, I will discuss the asymptotic behavior of the Steklov eigenvalues for the exterior Steklov problem. Some open questions related to spectral, probabilistic and asymptotic aspects of this problem will be outlined.

References:

[1] D. S. Grebenkov, Paradigm Shift in Diffusion-Mediated Surface Phenomena, Phys. Rev. Lett. 125, 078102 (2020).

[2] D. S. Grebenkov and A. Chaigneau, The Steklov problem for exterior domains: asymptotic behavior and applications (accepted to J. Math. Phys.; preprint ArXiv 2407.09864v2)

Two smoothly embedded surfaces in a smooth 4-manifold are called exotic if they are topologically isotopic but not smoothly isotopic. The phenomenon of exotic surfaces in dimension 4 is an interesting topic in low-dimensional topology. In this talk, we start with an introduction to equivariant knot concordance theory in the context of exotic disks. Then we recall how involutive Heegaard Floer theory can work as a machinery for equivariant concordance. Then we demonstrate our recent progress on equivariant concordance of Whitehead doubles, which produces exotic disk pairs.

This is joint work with Sungkyung Kang and JungHwan Park.

In this short lecture, I will explain the definition of Lefschetz fibrations on 4-manifolds and basic properties of them. In particular, I will describe a classification theorem of Kas and Matsumoto which asserts that there is one to one correspondence between the isomorphism classes of Lefschetz fibrations and the conjugacy classes of monodromy