Friday Jan 12 

4:305:30 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 BeurlingMalliavin 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 BeurlingMalliavin Theorem and proving a weaker but "more effective" version of it. 

5:306:30 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 nonlinear elliptic PDE. It is well known that up to postcomposition with a Mobius transformation, a finite Blaschke product may be uniquely described by the set of its critical points. I will discuss an infinitedegree 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 ''nearlymaximal solutions'' of the Gauss curvature equation and subspaces of $\kappa$Beurlingtype of a weighted Bergman space. 

Friday Jan 26 

4:005:00 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. 

5:006:00 UCLA, MS 6627 Alexey Miroshnikov (UCLA) TBA 

Friday Feb 9 

4:005:00 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 numbertheoretic considerations can themselves be approached via methods from harmonic analysis. 

5:006:00 Caltech, Downs 103 Max Engelstein (MIT) A Epiperimetric approach to singular points in the AltCaffarelli functional We prove a uniqueness of blowups result for isolated singular points in the free boundary of minimizers to the AltCaffarelli 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 areaminimizing hypersurfaces. This is joint work with Luca Spolaor (Princeton/MIT) and Bozhidar Velichkov (Universite Grenoble Alpes). 

Friday Feb 23 

4:005:00 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 multilinear operators and their (multiple) vector valued extensions. Joint work with Cristina BENEA. 

5:006:00 UCLA, MS 6627 TBA TBA 

Friday Mar 9 

4:005:00 Caltech, Downs 103 Joris Roos (U WisconsinMadison) TBA 

5:006:00 Caltech, Downs 103 TBA TBA 

Last update: Jan 22, 2018 