Caltech/UCLA joint Analysis seminar

Friday Jan 12

4:30-5:30 Caltech, Downs 103
Ruixiang Zhang (IAS)
The (Euclidean) Fractal Uncertainty Principle and its proof

Recently Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which roughly says that: Assuming a function on $\mathbb{R}$ has its Fourier support contained in a fractal set $Y$. Then its $L^2$ norm on a fractal set $X$, whose scale is dual to that of $Y$, cannot be close to its $L^2$ norm over the whole $\mathbb{R}$ as long as $\dim X, \dim Y < 1$. The proof seems quite interesting and unusual to me. I will talk about the ingredients in the proof, including the Beurling-Malliavin Theorem. In the original work of Bourgain and Dyatlov the FUP was ineffective. I will also talk about why we can obtain an effective version (joint work with Long Jin) by looking closely into the proof of the Beurling-Malliavin Theorem and proving a weaker but "more effective" version of it.

5:30-6:30 Caltech, Downs 103
Oleg Ivrii (Caltech)
Describing Blaschke products by their critical points

In this talk, I will discuss a question which originates in complex analysis but is really a problem in non-linear elliptic PDE. It is well known that up to post-composition with a Mobius transformation, a finite Blaschke product may be uniquely described by the set of its critical points. I will discuss an infinite-degree version of this problem posed by Dyakonov. Let J be the set of inner functions whose derivative lies in the Nevanlinna class. I will explain that an inner function in J is uniquely determined by the inner part of its derivative (its critical structure), and describe all possible critical structures of inner functions in J. I will also give a concrete description of the natural topology on J which respects the convergence of critical structures. Similar results hold for ''nearly-maximal solutions'' of the Gauss curvature equation and subspaces of $\kappa$-Beurling-type of a weighted Bergman space.

Friday Jan 26

4:00-5:00 UCLA, MS 6627
Semyon Dyatlov (MIT)
Lower bounds on eigenfunctions on hyperbolic surfaces

I show that on a compact hyperbolic surface, the mass of an $L^2$-normalized eigenfunction of the Laplacian on any nonempty open set is bounded below by a positive constant depending on the set, but not on the eigenvalue. This statement, more precisely its stronger semiclassical version, has many applications including control for the Schrödinger equation and the full support property for semiclassical defect measures. The key new ingredient of the proof is a fractal uncertainty principle, stating that no function can be localized close to a porous set in both position and frequency. This talk is based on joint works with Long Jin and with Jean Bourgain.

5:00-6:00 UCLA, MS 6627
Alexey Miroshnikov (UCLA)
TBA

Friday Feb 9

4:00-5:00 Caltech, Downs 103
Jonathan Hickman (U Chicago)
Factorising $X^n$

How many ways can the polynomial $X^n$ be factorised as a product of linear factors? Well, of course, the answer depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo $N$. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.

5:00-6:00 Caltech, Downs 103
Max Engelstein (MIT)
A Epiperimetric approach to singular points in the Alt-Caffarelli functional

We prove a uniqueness of blowups result for isolated singular points in the free boundary of minimizers to the Alt-Caffarelli functional. The key tool is a (log-)epiperimetric inequality, which we prove by means of two Fourier expansions (one on the function and one on its free boundary). If time allows we will also explain how this approach can be adapted to (re)prove old and new regularity results for area-minimizing hypersurfaces. This is joint work with Luca Spolaor (Princeton/MIT) and Bozhidar Velichkov (Universite Grenoble Alpes).

Friday Feb 23

4:00-5:00 UCLA, MS 6627
Camil Muscalu (Cornell)
The Helicoidal Method

Not long ago we discovered a new method of proving vector valued inequalities in Harmonic Analysis. With the help of it, we have been able to give complete positive answers to a number of questions that have been circulating for some time. The plan of the talk is to describe (some of) these, and to also explain how this method implies sparse domination results for various multi-linear operators and their (multiple) vector valued extensions. Joint work with Cristina BENEA.

5:00-6:00 UCLA, MS 6627
TBA
TBA

Friday Mar 9

4:00-5:00 Caltech, Downs 103