Applied Math

A widely accepted conjecture in the physical literature states that classical wave-fields propagating in random media over large distances eventually follow a complex circular Gaussian distribution. In this limit, the wave intensity becomes exponentially distributed, which corroborates the…

I will present a new proof of the stability of flat spacetime in general relativity. The proof is based on a center-normalized Newman-Unti gauge, constructed by fixing a central timelike geodesic and considering the outgoing null cones emanating from it. I will compare the choice of gauge with…

We study the open KPZ equation, a prototypical one-dimensional random growth model subject to boundary conditions. Using stochastic analytic tools, we show that a suitably resampled Brownian motion describes its long-time behavior.

The Anderson tight-binding model describes the transport of quantum particles in disordered environments.  A central question in this model is to understand whether, in the long-time limit, particles diffuse out to infinity or else remain trapped in some bounded region.  In this talk,…

Waveform inversion seeks to estimate an inaccessible heterogeneous medium by using sensors to probe the medium with signals and measure the generated waves. It is an inverse problem for a hyperbolic system of equations, with the sensor excitation modeled as a forcing term and the heterogeneous…

The Toda lattice is a system of classical mechanics discovered by Toda in 1967, which describes interacting particles on a line. Due to its integrability, the Toda lattice with N particles possesses N independent conserved quantities. In this work, we show that under a certain class of random…

In cosmology, a basic explanation of the observed concentration of mass in singular structures is provided by the Zeldovich approximation, which takes the form of free-streaming flow for perturbations of a uniform Einstein-de Sitter universe in co-moving coordinates. The adhesion model…

Random band matrices are Hermitian matrices with random entries supported in a band of width W around the diagonal. The eigenfunctions of such matrices are expected to decay exponentially at the scale W^2 in dimension one, and exp(CW^2) in dimension two. Remarkably, the same…

Polar sea ice is a composite material structured on length scales ranging over many orders of magnitude. A principal challenge in modeling sea ice is how to use microstructural data to find effective or homogenized behavior relevant to large-scale geophysical and ecological models. Similar…

We study the equilibrium temperature distribution in a model for strongly magnetized plasmas in dimension two and higher. Provided the magnetic field is sufficiently structured (integrable in the sense that it is fibered by co-dimension one invariant tori, on most of which the field lines…