Caltech/UCLA joint Analysis seminar

Winter 2018

Organizers: Laura Cladek, Polona Durcik

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)
Weak* Solutions of Conservation Laws: The vacuum in Lagrangian gas dynamics

Co-authors: Robin Young (UMass Amherst)
We introduce a new concept of solution for systems of conservation laws, which allows us to rigorously handle certain measure-valued solutions such as the vacuum in a Lagrangian frame. In particular, the PDEs can be regarded as (weak*) solutions of a nonlinear ODE in Banach space, and inherit certain regularity. Our novel framework makes sense of solutions containing the vacuum in Lagrangian gas dynamics. At and near vacuum, the specific volume becomes infinite and enclosed vacuums are represented by Dirac masses, so they cannot be treated in the usual weak sense. However, the weak* solutions can be extended to include solutions containing vacuums. We present a definition of these solutions and describe solutions with vacuums as an example. We also extend our methods to one-dimensional dynamic elasticity to show that fractures cannot form in an entropy solution. We further discuss approximations to such solutions.

Friday Feb 9

4:30-5:30 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.

5:30-6:30 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
Casey Jao (UC Berkeley)
Strichartz refinements and bilinear restriction quantum harmonic oscillator

We develop refined Strichartz estimates at $L^2$ regularity for a class of Schrödinger operators with time-dependent potentials, most notably the harmonic oscillator. Such refinements play a pivotal part in the global theory of mass-critical NLS. Building on phase space techniques introduced in previous joint work with with Killip and Visan, we reduce to proving certain analogues of (adjoint) bilinear Fourier restriction estimates. Then we extend Tao's bilinear restriction estimate for paraboloids to more general Schrödinger operators.

5:00-6:00 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.

Friday Mar 9

4:30-5:30 Caltech, Downs 103
Joris Roos (U Wisconsin-Madison)
Variation-norm estimates for a Stein-Wainger type oscillatory integral

In this talk I will present recent work on variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. The corresponding maximal operators were first studied by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. The proof relies on local smoothing estimates for a family of Schrödinger-like equations due to Rogers and Seeger and square function estimates for these equations due to Lee, Rogers and Seeger. This is joint work with Shaoming Guo and Po-Lam Yung.

5:30-6:30 Caltech, Downs 103
Tim Austin (UCLA)
The Weak Pinsker Property

This talk is about the structure theory of measure-preserving systems: transformations of a finite measure space that preserve the measure. Many important examples arise from stationary processes in probability, and simplest among these are the i.i.d. processes. In ergodic theory, i.i.d. processes are called Bernoulli shifts. Some of the main results of ergodic theory concern an invariant of systems called their entropy, which turns out to be intimately related to the existence of 'structure preserving' maps from a general system to Bernoulli shifts.
I will give an overview of this area and its history, ending with a recent advance in this direction. A measure-preserving system has the weak Pinsker property if it can be split, in a natural sense, into a direct product of a Bernoulli shift and a system of arbitrarily low entropy. The recent result is that all ergodic measure-preserving systems have this property. Its proof depends on a new theorem in discrete probability: a probability measure on a finite product space such as $A^n$ can be decomposed as a mixture of a controlled number of other measures, most of them exhibiting a strong 'concentration' property. I will sketch the connection between these results and the proof of the latter, to the extent that time allows.

Last update: Feb 28, 2018