Analysis & PDE

Starting with the work of Choptuik '92, numerical relativity predicts that naked singularity spacetimes arise on the threshold of dispersion and black hole formation, a phenomenon referred to as critical collapse. In this talk, I will present for 2+1 gravity the first rigorous construction of…

We prove that the Fisher information is monotone decreasing in time along solutions of the space-homogeneous Boltzmann equation for a large class of collision kernels covering all classical interactions derived from systems of particles. For general collision kernels, a sufficient condition for…

In this talk I will discuss the problem of linear stability of traveling Maxwellians for Landau equation with very soft potentials (including the Coloumbic case). I will contrast our case for very soft potentials with the case of moderately soft potentials. In addition, I will discuss how the…

In this talk we consider the four-waves spatially homogeneous kinetic equation arising in weak wave turbulence theory from one-dimensional microscopic oscillator chains. This equation is sometimes referred to as the Phonon Boltzmann Equation. I will discuss the entropy maximisation problem, the…

Abstract: I will explain the relationship between the thin obstacle problem and two "unsigned" versions: 2-valued C^{1,alpha} stationary harmonic functions and Z/2Z harmonic functions. I…

I will talk about joint work with Andras Vasy defining the Feynman propagator for massive scalar fields on asymptotically flat spacetimes general enough to include radiative perturbations of Minkowski space with no symmetries. The Feynman propagator is an inverse of the Klein-Gordon operator…

The classical Calderón problem for the conductivity equation asks if the conductivity of a medium can be determined uniquely by making current and voltage measurements on its boundary. We present an analogous question in the fractional context, called the inverse fractional conductivity problem…

I will present upcoming work proving a forward energy cascade for quasilinear wave equations on Schwarzschild-AdS black hole exteriors. The cascade is driven by a stably trapped 3-mode interaction that transfers energy from low-to high-frequency modes. Our result is motivated by the question of…

Liouville field theory has long been a cornerstone of two-dimensional quantum field theory and quantum gravity, which has attracted much recent attention in the mathematics literature. Timelike (or imaginary) Liouville field theory is a version of Liouville field theory where the coupling…

Abstract: The inverse problem of Calderón, in its geometric formulation, asks if a Riemannian metric in a domain is determined up to isometry by boundary measurements of harmonic functions. Physically this corresponds to determining a matrix electrical conductivity function from voltage and…