# Analysis & PDE

Organizers: Jonathan Luk, Eugenia Malinnikova, and John Anderson

## Upcoming Events

Abstract: It is well-known that in three space dimensions, smooth solutions to the equations describing a compressible gas can break down in finite time. One type of singularity which can arise is known as a "shock", which is a hypersurface of discontinuity across which the integral forms of…

## Past Events

Abstract: Compressible Euler solutions develop jump discontinuities known as shocks. However, physical shocks are not, strictly speaking, discontinuous. Rather, they exhibit an internal structure which, in certain regimes, can be represented by a smooth function, the shock profile. We…

Abstract: This talk discusses a problem introduced by Yau on estimating the size of nodal sets of eigenfunction in terms of the eigenvalue. We show that one can obtain improved polynomial upper bounds when the Riemannian manifold has a Gevrey or quasianalytic regularity. Yau's upper bound…

Abstract: It is expected that the Klein-Gordon equation on a Schwarzschild black hole behaves very differently from the wave equation at late-time, due to the presence of stable (timelike) trapping. We present our recent work demonstrating that despite the presence of stable timelike trapping on…

Abstract: We show low regularity local well-posedness for Maxwell equations in media. To this end, we show Strichartz estimates for solutions to Maxwell equations withcoefficients of regularity C^2 and lower. The proof is based on a diagonalization with pseudo-differential operators. In two…

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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…

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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…