### Caltech Analysis Seminar 2019-2020

Friday, October 11th, 2019
3-4pm, Linde 255
speaker: Cosmin Pohoata (Caltech)
title: Expanding polynomials for sets with additive or multiplicative structure
abstract: Given an arbitrary set of real numbers A and a two-variate polynomial f with real coefficients, a remarkable theorem of Elekes and R\'onyai from 2000 states that the size |f(A,A)| of the image of f on the cartesian product A x A grows asymptotically faster than |A|, unless f exhibits additive or multiplicative structure. Finding the best quantitative bounds for this intriguing phenomenon (and for variants of it) has generated a lot of interest over the years due to its intimate connection with the sum-product problem. In this talk, we will first review some of the results in this area, and then discuss some new bounds for |f(A,A)| when the set A has few sums or few products.

Friday, October 25th, 2019
3-4pm, Linde 255
speaker: Benjamin Harrop-Griffiths (UCLA)
title: Sharp well-posedness for some integrable PDEs
abstract: Despite its innocuous appearance, the 1d cubic NLS is a truly remarkable PDE. Not only does it arise as a model in numerous physical scenarios, for example fluid dynamics and nonlinear optics, but it is also part of the select group of integrable equations, in the sense that it possesses a Lax pair and infinitely many conserved quantities. Building on the work of Killip and Visan on the KdV equation, in this talk we present a proof of well-posedness for the cubic NLS that combines its deep mathematical structure with robust PDE techniques to obtain a sharp result in Sobolev spaces. We will also discuss the corresponding results for an intimately related equation, the mKdV. This is joint work with Rowan Killip and Monica Visan.

Friday, November 8th, 2019
3-4pm, Linde 255
speaker: Jack Burkart (Stony Brook)
title: Dimension in Holomorphic Dynamics
abstract: Holomorphic dynamics studies the iteration of rational, polynomial, or general entire functions. The Julia set of such a function can informally be though of as the set of all points where the sequence determined by the function and its iterates fails to be equicontinuous, so that nearby points follow different trajectories under iteration. Computer images suggest the Julia set has a rich fractal structure. In this talk, we will define various notions of dimension (Hausdorff, Minkowski, and packing) used to study fractals. We will discuss relevant dimension results for Julia sets of polynomial/rational functions, and compare these results to what is known about the iteration of (transcendental) non-polynomial entire functions. We will conclude by a discussion of my recent result constructing the first known examples of transcendental entire functions with fractional packing dimension.

Friday, November 22nd, 2019
3-4pm, Linde 255
speaker: João Pedro Ramos (Bonn)
title: Fourier uncertainty principles, interpolation and uniqueness sets.
abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that $$f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}.$$ This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers. We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$ In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.

Friday, December 13th, 2019
3-4pm, Linde 255
speaker: Peter Lin (Stony Brook)
title: TBD
abstract: TBD

(2018-2019 Caltech analysis seminar)