### 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: Conformal Welding of Dendrites
abstract: If K is a connected, locally connected, compact subset of the plane, then a Riemann map f: C-D \to C-K extends continuously to the unit circle. Thus K induces an equivalence relation on the circle $x\sim y iff f(x)=f(y)$. One can ask the converse question: given an equivalence relation on the circle, is there a conformal map realizing this equivalence? This is essentially the conformal welding problem. In this talk we discuss some criteria for the existence for such a map. We give example applications of this criteria for equivalence relations arising in dynamics and probability. Joint with Steffen Rohde.

Friday, January 17th, 2020
3-4pm, Linde 255
speaker: Bjoern Bringmann (UCLA)
title: ALMOST SURE SCATTERING FOR THE ENERGY-CRITICAL NONLINEAR WAVE EQUATION
abstract: We will discuss the defocusing energy-critical nonlinear wave equa-tion. For deterministic and smooth initial data, it is widely known thatthe solutions scatter, i.e., they asymptotically behave like solutions tothe linear wave equation. In this talk, we will show that this scatteringbehavior persists under random and rough perturbations of the initialdata. As part of the argument, we will discuss techniques from restric-tion theory, such as wave packet decompositions and Bourgain’s bushargument.

Friday, January 31st, 2020
3-4pm, Linde 255
speaker: Nam-Gyu Kang (KIAS)
title: Conformal field theory for multiple SLEs
abstract: Multiple SLEs describe several random interfaces consistent with conformal symmetries. I will explain a version of conformal field theory constructed from background charge modifications of Gaussian free field and insertion of N-leg operators (with screening) to show that this version produces a collection of martingale-observables for commuting multiple SLEs. Based on joint work with Tom Alberts and Nikolai Makarov.

Friday, February 14th, 2020
3-4pm, Linde 255
speaker: Simon Larson (Caltech)
title: Asymptotic problems in spectral shape optimization
abstract: It is a classical result that the first eigenvalue of the Dirichlet Laplacian amongst open sets of fixed measure is minimal for the ball. For the second eigenvalue it is known that the union of two disjoint balls of equal measure realizes the minimum. For higher eigenvalues little is known. In this talk we will consider a number of problems related to the following question: Does the behaviour of sets minimizing the k-th eigenvalue stabilize as k becomes large? In particular, we shall discuss the problem of minimizing the sum of the first K eigenvalues amongst open sets of fixed measure in the limit as K tends to infinity.

Friday, February 28th, 2020
3-4pm, Linde 255
speaker: Nuria Fagella (UB)
title: Wandering domains in and out.
abstract: In dynamics of holomorphic maps a wandering domain is a component of the normality set whose iterates never meet. In this talk we describe the dynamics of orbits inside simply connected wandering domains related to their distance to each other or to the boundary of the domains. We relate this classification to the presence of singular orbits outside but near the wandering components. Our results is also related to non-autonomous holomorphic dynamical systems on the unit disk.

(2018-2019 Caltech analysis seminar)