# Analysis Seminar

## Observable events" and "typical trajectories" in finite and infinite dimensional dynamical systems

## On dynamical spectral rigidity and determination

## When do interacting organisms gravitate to the vertices of a regular simplex?

## A rigorous derivation of the kinetic wave equation

In this talk I will outline recent work in collaboration with Pierre Germain, Zaher Hani and Jalal Shatah regarding a rigorous derivation of the kinetic wave equation. The proof presented will rely of methods from PDE, statistical physics and number theory.

## On the gradient-flow structure of multiphase mean curvature flow

Due to its importance in materials science where it models the slow relaxation of grain boundaries, multiphase mean curvature flow has received a lot of attention over the last decades.

## The singular set in the fully nonlinear obstacle problem

For the Obstacle Problem involving a convex fully nonlinear elliptic operator, we show that the singular set of the free boundary stratifies. The top stratum is locally covered by a $C^{1,\alpha}$-manifold, and the lower strata are covered by $C^{1,\log^\eps}$-manifolds. This essentially recovers the regularity result obtained by Figalli-Serra when the operator is the Laplacian.

## Displacement convexity for point processes and an application

## The Surface Quasigeostrophic equation on the sphere

In this talk I will describe joint work with D. Alonso-Orán and A. Córdoba where we extend a result, proved independently by Kiselev-Nazarov-Volberg and Caffarelli-Vasseur, for the critical dissipative SQG equation on a two dimensional sphere. The proof relies on De Giorgi technique following Caffarelli-Vasseur intermingled with a nonlinear maximum principle that appeared later in the approach of Constantin-Vicol. The final result can be paraphrased as follows: if the data is sufficiently smooth initially then it is smooth for all times.

## Strong ill-posedness of the logarithmically regularized 2D Euler equations in the borderline Sobolev space

The well-posedness of the incompressible Euler equations in borderline spaces has attracted much attention in recent years. To understand the behavior of solutions in these spaces, the logarithmically regularized Euler equations were introduced. In borderline Sobolev spaces, local well-posedness was proved Chae-Wu when the regularization is sufficiently strong, while strong ill-posedness of the unregularized case was established by Bourgain-Li. In this talk, I will discuss the strong ill-posedness of the remaining intermediate regime of regularization.