Analysis Seminar

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...

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...

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 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...

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.

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...

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...

In spin systems, the existence of a spectral gap has far-reaching consequences. So-called "frustration-free" spin systems form a subclass that is special enough to make the spectral gap problem amenable and, at the same time, broad enough to include...

For initial datum of finite kinetic energy Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this talk, I will discuss very recent joint work with Vlad Vicol in...