Analysis Seminar

Given a planar domain with sufficiently regular boundary, one can study periodic orbits of the associated billiard problem. Periodic orbits have a rich and quite intricate structure and it is natural to ask how much information about the domain is...

Flocking and swarming models which seek to explain pattern formation in mathematical biology often assume that organisms interact through a force which is attractive over large distances yet repulsive at short distances. Suppose this force is given...

In the talk we will focus on certain constructions of weak solutions to the Navier--Stokes inequality (NSI), \[ u \cdot \left( u_t - \nu \Delta + (u\cdot \nabla ) u+ \nabla p \right) \leq 0\] on $\mathbb R^3$. Such vector fields satisfy both the...

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.

In this talk, I want to present two theorems. The first one is...

I will prove the following surprising fact: on a given Kahler manifold (X, J, \omega), a Holder bound on the Kahler potential \phi implies a Holder bound on the distance function of the new Kahler metric \omega+dd^c \phi. Time permitting I will also...

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...

The path between two measures in the sense of optimal transport yields the notion of *displacement interpolation*. As observed by R. McCann, certain functionals that are not convex in the usual sense are nonetheless *displacement convex*. Following...

"I will discuss some history and new results about the forced mean curvature flow in inhomogeneous media.  It is a model for interface propagation in quenched randomness in various physical settings, e.g. contact lines, phase interfaces in porous...

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...

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...