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

On s'intéresse aux marches aléatoires en milieu aléatoire. La difficulté principale dans l'étude de ces systèmes est l'interaction très forte entre la marche et son environnement. Dans cet exposé, on propose un critère qui, lorsqu'il est satisfait, permet de décomposer la trajectoire de la marche en incréments iid, et à terme de démontrer des théorèmes limites. Le critère porte sur l'environnement et implique la construction d'un champ aléatoire qui satisfait une certaine propriété de Markov ainsi que des estimées de décorrélation. On applique ce critère à des environnements corrélés en temps comme la percolation booléenne sur Z^d x N et des chaînes de renouvellement iid en espace. Basé sur https://arxiv.org/abs/2409.12515 (travail réalisé avec J. Allasia, R. Baldasso et A. Teixeira).