# Caltech/UCLA joint Analysis seminar

## Fall 2019

Past: Winter 2018, Spring 2018, Fall 2018, Winter 2019, Spring 2019
Future: Winter 2020

## Friday Oct 4

4:30-5:20 Caltech, Linde 310
Svetlana Jitormirskaya (UC Irvine)
Fractal properties of the Hofstadter's butterfly and singular continuous spectrum of the critical almost Mathieu operator

Harper's operator - the 2D discrete magnetic Laplacian - is the model behind the Hofstadter's butterfly. It reduces to the critical almost Mathieu family, indexed by phase. We discuss the proof of singular continuous spectrum for this family, for all phases, finishing a program with a long history. We also present a result (with I. Krasovsky) that proves one half of the Thouless' one half conjecture from the early 80s: that Hausdorff dimension of the spectrum of Harper's operator is bounded by 1/2 for all irrational fluxes.

5:30-6:20 Caltech, Linde 310
Khang Huynh (UCLA)
Hodge-theoretic analysis on manifolds with boundary, heatable currents, and Onsager's conjecture in fluid dynamics

We use Hodge theory and functional analysis to develop a clean approach to heat flows and Onsager's conjecture on Riemannian manifolds with boundary, where the weak solution lies in the trace-critical Besov space $B_{13}^{3,1}$. We also introduce heatable currents as the natural analogue to tempered distributions and justify their importance in Hodge theory.

## Friday Oct 18

4:00-4:50 UCLA, MS 6627
Tarek Elgindi (UCSD)
Singularity formation in incompressible fluids

We will discuss recent results on singularity formation for solutions to the incompressible Euler equation in various settings. We will discuss in some detail a construction of self-similar blow-up for C^{1,a} solutions to the 3D Euler equation as well as a proof of stability that yields finite-energy local self-similar blow-up. Some of the results are joint with I. Jeong, T. Ghoul, and N. Masmoudi.

5:00-5:50 UCLA, MS 6627
Eyvindur Palsson (Virginia Tech)
Falconer type theorems and integral operators

Finding and understanding patterns in data sets is of significant importance in many applications. One example of a simple pattern is the distance between data points, which can be thought of as a 2-point configuration. Two classic questions, the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem, explore that simple pattern. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as various 3-point configurations. In this talk I will explore such generalizations. The key focus will be integral operators, that arise naturally from such point configuration questions. In addition, a novel group-theoretic viewpoint, which has allowed for much progress recently, will be highlighted.

## Friday Nov 1

4:30-5:20 Caltech, Linde 310
Random quasiconformal mappings and Delauney triangulations

I shall discuss my very recent work with Oleg Ivrii regarding two problems dealing with the homogenization of random media. We show that a random quasiconformal mapping is close to an affine mapping, while a circle packing of a random Delauney triangulation is close to a conformal map. Moreover, on a Riemann surface equipped with a conformal metric, a random Delauney triangulation is close to being circle packed.

5:30-6:20 Caltech, Linde 310
Danielle Hilhorst (CNRS and U. Paris-Sud)
A reaction-diffusion-ODE model in anthropology

A reaction-diffusion-ODE model for the Neolithic spread of farmers in Europe has been recently proposed by Elias, Kabir and Mimura. This model involves hunter-gatherers and farmers, which are divided into two subpopulations, namely sedentary and migrating farming populations. The conversion between the farming subpopulations depends on the total density of the farmers and it is superimposed on the classical Lotka-Volterra competition model; it is therefore described by a three-component reaction-diffusion-ODE system. In this talk, we study its singular limit as the conversion rate tends to infinity and prove that solutions of the three-component system converge to solutions of a two-component system with linear diffusion in one of the equations and nonlinear degenerate diffusion in the other. This is joint work with Jan Elias, Masayasu Mimura and Yoshihisa Morita.

We then study the limiting two component system. From an ecological viewpoint, the nonlinear diffusion takes into account the population density pressure of the farmers on their dispersal. The interaction between farmers and hunter-gatherers is of the Lotka-Volterra prey-predator type. We present an alternative method for proving the existence and uniqueness of the global-in-time solution and study its asymptotic behavior as time tends to infinity. This is joint work with Jan Elias and Masayasu Mimura.

## Friday Nov 15

4:00-4:50 UCLA, MS 6627
Florian Richter (Northwestern)
A sumset conjecture of Erdős

A longstanding conjecture of Erdős stated that every subset of the integers with positive density contains a sumset $B+C=\{b+c: b\in B, c\in C\}$ for two infinite sets $B$ and $C$. I will talk about joint work with Joel Moreira and Donald Robertson in which we resolve this conjecture. Our proof utilizes ideas and methods coming from Ergodic Theory, including an intersectivity lemma of Bergelson and a novel decomposition theorem for arithmetic functions into structured and random components.

5:00-5:50 UCLA, MS 6627
Susan Friedlander (USC)
The Joy of Small Parameters

Many equations that model fluid behavior are derived from systems that encompass multiple physical forces. When the equations are written in non dimensional form appropriate to the physics of the situation, the resulting PDEs often involve multiple non-dimensional parameters. Frequently some of these parameters are very small and they enter into the analysis in different ways. We will discuss one such system which has been proposed as a model for magnetostrophic turbulence and describe results that can be obtained in several different small parameter limits. In this talk we will concentrate on a forced drift-diffusion equation for the temperature where the fluid viscosity enters via the drift velocity. We examine the convergence of solutions in the limit as the viscosity goes to zero. We introduce a natural notion of ”vanishing viscosity ” weak solutions and prove the existence of a compact global attractor for the critical drift-diffusion equation.
This is joint work with Anthony Suen.

## Friday Dec 6

4:30-5:20 Caltech, Linde 310
Vjekoslav Kovac (U. Zagreb/Georgia Tech)
On maximal and variational Fourier restriction

The problem of restricting the Fourier transform of a non-integrable function to a hypersurface gained attention in the 1970s and is an active research field today. On the other hand, maximal Fourier restriction estimates were inaugurated recently by Müller, Ricci, and Wright in order to give a pointwise meaning to the restriction of the Fourier transform of a function. We will discuss variational Fourier restriction estimates, which quantify the pointwise convergence in question. Moreover, we will sketch a simple abstract argument that deduces certain maximal and variational Fourier restriction estimates from ordinary a priori estimates.

5:30-6:20 Caltech, Linde 310