In this talk, I will explain how to relate the Novikov ring and microlocal sheaf theory based on the works of Tamarkin and Vaintrob. This relation has been successfully applied to the irregular Riemann-Hilbert correspondence and symplectic geometry, and recently, Scholze revealed certain potential applications of the relation in analytic geometry. Surprisingly, we notice that the almost ring theory invented by Faltings for p-adic Hodge theory plays a central role in the related construction. After that, I will explain some applications of the idea in symplectic geometry.

This talk is based on a joint work with Tatsuki Kuwagaki.

We consider a system of N Brownian particles interacting through a long-range smooth potential. It is known that "propagation of chaos" holds in the mean-field scaling. Assume indeed that the initial distribution of the particles is chaotic, i.e. that the particles are independent and identically distributed. Then, for any given time, and as N becomes large, the distribution of particles remains chaotic. Moreover, the distribution of a typical particle is given by the solution of a Vlasov-Fokker-Planck equation.

In this talk, we will investigate the creation of chaos phenomenon. Starting from an initial distribution of particles which is only exchangeable, we prove that in some weak norm, propagation of chaos holds up to an error stemming from initial correlations, exponentially damped over time. This is a joint work with Armand Bernou and Mitia Duerinckx.

In this talk, we discuss the effects of perturbations on the topology and geometry of nodal sets/zero sets of Laplace eigenfunctions. A conjecture by Payne states that the nodal set of the second Dirichlet eigenfunction on a bounded planar domain intersects the boundary at exactly two points. We will look into certain stability properties of the nodal sets and discuss some recent results concerning the conjecture. Then, utilising the stability properties, we will observe the prescription of nodal data on Riemannian surfaces, focusing on the following two aspects: the construction of eigenfunctions with a prescribed number of nodal intersections at the boundary, and the realisation of Courant-sharp eigenfunctions at arbitrarily high levels.