Caltech/UCLA joint Analysis seminar

Spring 2020

Organizers: Terence Tao, Polona Durcik

Tuesday Apr 7
4:00-4:50pm (https://caltech.zoom.us/j/747242458) Mohammad Ghomi (Georgia Tech) Total curvature and the isoperimetric inequality The classical isoperimetric inequality states that in Euclidean space spheres provide enclosures of least perimeter for any given volume. According to the Cartan-Hadamard conjecture, this inequality may be generalized to spaces of nonpositive curvature. In this talk we discuss an approach to proving this conjecture via a comparison formula for the total curvature of level sets of functions on nonpositively curved manifolds. This is joint work with Joel Spruck.
5:00-5:50pm (https://caltech.zoom.us/j/747242458) Terence Tao (UCLA) Almost all Collatz orbits attain almost bounded values Define the Collatz map Col on the natural numbers by setting $Col(n)$ to equal $3n+1$ when $n$ is odd and $n/2$ when $n$ is even. The notorious Collatz conjecture asserts that all orbits of this map eventually attain the value $1$. This remains open, even if one is willing to work with almost all orbits rather than all orbits. We show that almost all orbits $n$, $Col(n)$, $Col^2(n)$, ... eventually attain a value less than $f(n)$, for any function $f$ that goes to infinity (no matter how slowly). A key step is to obtain an approximately invariant (or more precisely, self-similar) measure for the (accelerated) Collatz dynamics.
Tuesday Apr 21
3:00-3:50pm (https://ucla.zoom.us/j/9264073849) Hong Wang (IAS) Distinct distances for well-separated sets Given a set $E$ of dimension $s>1$, Falconer conjectured that its distance set $\Delta(E)=\{\|x-y\|: x, y\in E\}$ should have positive Lebesgue measure. Orponen, Shmerkin and Keleti-Shmerkin proved the conjecture for tightly spaced sets, for example, AD-regular sets. In this talk, we are going to discuss the opposite type: well-separated sets. This is joint work with Larry Guth and Noam Solomon.
4:00-4:50pm (https://ucla.zoom.us/j/9264073849) Ioannis Angelopoulos (Caltech) Semi-global constructions of vacuum spacetimes I will describe some techniques for constructing semi-global solutions to the characteristic initial value problem for the vacuum Einstein equations with different types of data, and will also mention some applications as well as some open problems in the area.
Tuesday May 5
3:00-3:50pm (https://caltech.zoom.us/j/747242458) David Beltran (U. Madison Wisconsin) Regularity of the centered fractional maximal function I will report some recent progress regarding the boundedness of the map $f \mapsto \|\nabla M_\beta f\|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a pointwise relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.
4:00-4:50pm (https://caltech.zoom.us/j/747242458) Luca Spolaor (UCSD) Regularity of the free boundary for the two-phase Bernoulli problem I will describe a recent result obtained in collaboration with G. De Philippis and B. Velichkov concerning the regularity of the free boundaries in the two phase Bernoulli problems. The novelty of our work is the analysis of the free boundary at branch points, where we show that it is given by the union of two $C^1$ graphs. This completes the work started by Alt, Caffarelli, and Friedman in the 80's.
Tuesday May 19
3:00-3:50pm (https://ucla.zoom.us/j/9264073849) Kevin Hughes (U. of Bristol) Discrete restriction estimates We will discuss Wooley's Efficient Congruencing approach to discrete restriction estimates for translation-dilation invariant systems of equations. Then we will discuss recent estimates for the curve $(X,X^3)$ which lie just outside of this framework as well as that of Decoupling.
4:00-4:50pm (https://ucla.zoom.us/j/9264073849) Dmitry Khavinson (U. of South Florida) Classical Potential Theory from the High Ground of Linear Holomorphic PDE "Between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain." P. Painleve, 1900. Newton noticed that the gravitational potential of a spherical mass with constant density equals, outside the ball, the potential of the point-mass at the center. Rephrasing, the gravitational potential of the ball with constant mass density continues as a harmonic function inside the ball except for the center. Fairly recently, it was noted that the latter statement holds for any polynomial, or even for entire densities. If a harmonic in a spherical shell function vanishes on one piece of a line through the center piercing the shell, then it must vanish on the second piece of that line. Yet, the similar statement fails for tori. If we solve the Dirichlet problem in an ellipse with entire data, the solution will always be an entire harmonic function. Yet, if we do that in a domain bounded by the curve $x^4 + y^4 =1$, with the data as simple as $x^2+y^2$, the solution will have infinitely many singularities outside the curve. Where and why do eigenfunctions of the Laplacian in domains bounded by algebraic curves start having singularities? We shall discuss these and some other questions under the unified umbrella of analytic continuation of solutions to analytic pde in $C^n$.
Tuesday June 2
9:00-9:50am (https://caltech.zoom.us/j/747242458) Mihailis Kolountzakis (U. of Crete) Orthogonal Fourier analysis on domains: methods, results and open problems We all know how to do Fourier Analysis on an interval, on ${\mathbb R}^d$, or other groups. But what if our functions live on a subset of Euclidean space, let's say on a regular hexagon in the plane? Can we use our beloved exponentials, functions of the form $e_\lambda(x) = \exp(2\pi i \lambda\cdot x)$ to analyze the functions defined on our domain? In other words, can we select a set of frequencies $\lambda$ such that the corresponding exponentials form an orthogonal basis for $L^2$ of our domain? It turns out that the existence of such an orthogonal basis depends heavily on the domain. So the answer is yes, we can find an orthogonal basis of exponentials for the hexagon, but if we ask the same question for a disk, the answer turns out to be no. Fuglede conjectured in the 1970s that the existence of such an exponential basis is equivalent to the domain being able to tile space by translations (the hexagon, that we mentioned, indeed can tile, while the disk cannot). In this talk we will track this conjecture and the mathematics created by the attempts to settle it and its variants. We will see some of its rich connections to geometry, number theory and harmonic analysis and some of the spectacular recent successes in our efforts to understand exponential bases. We will emphasize several problems that are still open.
10:00-10:50am (https://caltech.zoom.us/j/747242458) Yakov Shlapentokh-Rothman (Princeton) Naked Singularities for the Einstein Vacuum Equations: The Exterior Solution We will start by recalling the weak cosmic censorship conjecture. Then we will review Christodoulou's construction of naked singularities for the spherically symmetric Einstein-scalar field system. Finally, we will discuss joint work with Igor Rodnianski which constructs spacetimes corresponding to the exterior region of a naked singularity for the Einstein vacuum equations.
Last update: May 27, 2020