# Analysis & PDE

Organizers: Jonathan Luk, Eugenia Malinnikova, John Anderson, and Ryan Unger

## Past Events

Abstract

Abstract: This is a talk about concavity and convexity of trace functionals. In a celebrated paper in 1973, Lieb proved what we now call Lieb's Concavity Theorem and resolved a conjecture of Wigner, Yanase and Dyson in 1963. This result, together with its many extensions, has found plenty of…

Abstract: Carleson proved in 1966 that the Fourier series of any square integrablefunction converges pointwise to the function, by establishing boundednessof the maximally modulated Hilbert transform from L^2 into weak L^2. Thistalk is about a generalization of his result, where the Hilbert…

Abstract

Metric properties of harmonic measure is a perennial topic having much attention in the 80’ after works of Makarov, Jones, Wolff, Bourgain. However, certain questions of Peter Jones and Chris Bishop were left unsolved. Some of them concern free boundary problems for 1, 2, 3…

Abstract: In this talk, I will present a method to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. We first present a conditional result. Assuming that a global solution to the geometric reduced system exists and satisfies several well-chosen…

Abstract: : In this talk, we study initial value problem for the Einstein equation with null matter fields, motivated by null shell solutions of Einstein equation. In particular, we show that null shell solutions can be constructed as limits of spacetimes with null matter fields. We also study…

Abstract

We introduce a new way to look at level sets of eigenfunctions by viewing the value of the eigenfunction as an independent time variable, with successive level surfaces evolving over time. The evolution obeys a variational principle analogous to mean curvature flow, and this…

Abstract: In Newtonian gravity, a self-gravitating gas around a massive object such as a star or a planet is modeled via Vlasov Poisson equation with an external Kepler potential. The presence of this attractive potential allows for bounded trajectories along which the gas neither falls in…

Abstract: The conjecture broadly asserts that small data should yield global solutions for 1D defocusing dispersive flows with cubic nonlinearities, in both semilinear and quasilinear settings. The aim of the talk will be to present some very recent results in this direction. This is joint…

Abstract: We discuss recent works studying the sharp mapping properties of weighted X-ray transforms and weighted normal operators. These include a C^\infty isomorphism result for certain weighted normal operators on the Euclidean disk, whose proof involves studying the spectrum of a…