Seminars Sorted by Series

Given a measure preserving dynamical system, a real-valued observable determines a random process (by composing the observable with the iterates of the transformation). An important topic in ergodic theory is the study of the statistical properties...

A topological dynamical system $(X,T)$ is said to be Möbius disjointnes if \[\tag{$*$} \lim_{N\to\infty}\frac1N\sum_{n\leq N}f(T^nx)\mu(n)=0\] for all $f\in C(X)$ and $x\in X$ ($\mu$ stands for the classical Möbius function).Sarnak's conjecture from...

Zero sets of Laplace eigenfunctions are called nodal sets. The talk will focus on propagation of smallness techniques, which are useful for estimates of the Hausdorff measure of the nodal sets.

We will give an overview of some of the developments in recent years dealing with the description of asymptotic states of solutions to semilinear evolution equations ("soliton resolution conjecture").

New results will be presented on damped...

Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular...

We will show that the appropriately-defined vertical perimeter of a measurable subset of the Heisenberg group is at most a constant multiple of its horizontal (Heisenberg) perimeter. This isoperimetric-type inequality exhibits different behavior in...

The classical Carleson operator, which is intimately related to the Fourier transform, was an oscillatory singular integral operator with a linear phase. Motivated by a question of Eli Stein, recent consideration of Carleson operators has focused on...

In this talk, we will recall the strong openness property of multiplier ideal sheaves (conjectured by Demailly and proved by Guan-Zhou), and then present some recent related progress including some joint work with Professor Xiangyu Zhou.

In this talk, we will recall the strong openness property of multiplier ideal sheaves (conjectured by Demailly and proved by Guan-Zhou), and then present some recent related progress including some joint work with Professor Xiangyu Zhou.

In the early 1920s, Loewner introduced a constructive approach to the Riemann mapping theorem that realized a conformal mapping as the solution to a differential equation. Roughly, the “input” to Loewner’s differential equation is a driving measure...

We discuss two old questions of Landis concerning behavior of solutions of second order elliptic equations. The first one is on propagation of smallness for solutions from sets of positive measure, we answer this question and as a corollary prove an...

A celebrated theorem of C.L. Siegel from 1929 shows that the multiplicity of eigenvalues for the Laplace eigenfunctions on the unit disk is at most two. More precisely, Siegel shows that positive zeros of Bessel functions are transcendental.

We...

We consider the Navier–Stokes equations posed on the half space, with Dirichlet boundary conditions. We give a direct energy based proof for the instantaneous space-time analyticity and Gevrey class regularity of the solutions, uniformly up to the...

We will motivative the conjectures of Kobayashi, Lang, and Yau on various characterizations of positive canonical bundle over a projective manifold. Then we will provide a purely analytic proof of Yau's conjecture that if the manifold has negative...

Given a bounded domain $\Omega$, the harmonic measure $\omega$ is a probability measure on $\partial \Omega$ and it characterizes where a Brownian traveller moving in $\Omega$ is likely to exit the domain from. The elliptic measure is a non...

We introduce a length scale in Plateau’s problem by modeling soap films as liquid with small volume rather than as surfaces, and study the relaxed problem and its relation to minimal surfaces. This is based on joint works with Antonello Scardicchio...

In this talk we discuss the cubic wave equation in three dimensions. In three dimensions the critical Sobolev exponent is 1/2. There is no known conserved quantity that controls this norm. We prove unconditional global well-posedness for radial...

I will discuss some new results for gradient field models with uniformly convex potentials. A connection between the scaling limit of the field and elliptic homogenization was introduced more than twenty years ago by Naddaf and Spencer. In joint...

We consider a system of two interacting one-dimensional quasiperiodic particles as an operator on $\ell^2(\mathbb Z^2)$. The fact that particle frequencies are identical, implies a new effect compared to generic 2D potentials: the presence of large...

We consider a reaction-diffusion equation with a nonlocal reaction term. This PDE arises as a model in evolutionary ecology. We study the regularity properties and asymptotic behavior of its solutions.

In this talk, I will give an overview of some of what is known about solutions to the thin obstacle problem, and then move on to a discussion of a higher regularity result on the singular part of the free boundary. This is joint work with Xavier...

I will discuss random field Ising model on $Z^2$ where the external field is given by i.i.d. Gaussian variables with mean zero and positive variance. I will present a recent result that at zero temperature the effect of boundary conditions on the...

An important question in hydrodynamic turbulence concerns the scaling proprties in the inertial range. Many years of experimental and computational work suggests---some would say, convincingly shows---that anomalous scaling prevails. If so, this...

I will describe joint work with Boris Solomyak, in which we show that the stationary (Furstenberg) measure on the projective line associated to 2x2 random matrix products has the "correct" dimension (entropy / Lyapunov exponent) provided that the...

We describe a recent construction of self-similar blow-up solutions of the incompressible Euler equation. A consequence of the construction is that there exist finite-energy $C^{1,a}$ solutions to the Euler equation which develop a singularity in...

We address the inviscid limit for the Navier-Stokes equations in a half space, with initial datum that is analytic only close to the boundary of the domain, and has finite Sobolev regularity in the complement. We prove that for such data the...

In a recent result, Buckmaster and Vicol proved non-uniqueness of weak solutions to the Navier-Stokes equations which have bounded kinetic energy and integrable vorticity. We discuss the existence of such solutions, which in addition are regular...

In the optimal transport problem, it is well-known that the geometry of the target domain plays a crucial role in the regularity of the optimal transport. In the quadratic cost case, for instance, Caffarelli showed that having a convex target domain...

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...

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...

"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...

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...

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...

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...

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...

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...

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...

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...

Some words in the title are between quotation marks because it is a matter of interpretation. For dynamical systems on finite dimensional spaces, one often equates observable events with positive Lebesgue measure sets, and invariant distributions...

We provide geometric sufficient conditions for Reifenberg flat sets of any integer dimension in Euclidean space to be parametrized by a Lipschitz map with Hölder derivatives. The conditions use a Jones type square function and all statements are...

Consider a (possibly time-dependent) vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindel\"of) Theorem states that, if the vector field $v$ is Lipschitz in space, for every initial datum $x$ there is a...

We present a purely variational approach to the regularity theory for the Monge-Ampère equation, or rather optimal transportation, introduced with M. Goldman. Following De Giorgi’s philosophy for the regularity theory of minimal surfaces, it is...

In 1965, M. Kac proved that discs were uniquely determined by their Dirichlet (or, Neumann) spectra. Until recently, disks were the only smooth plane domains known to be determined by their eigenvalues. Recently, H. Hezari and I proved that ellipses...

We show that a topologically mixing C^\infty Anosov flow on a 3 dimensional compact manifold is exponential mixing with respect to any equilibrium measure with Holder potential. This is a joint work with Masato Tsujii.

Given a Lipschitz map, it is often useful to chop the domain into pieces on which the map has simple behavior. For example, depending on the dimensions of source and target, one may ask for pieces on which the map behaves like a bi-Lipschitz...