Analysis Seminar

The conventional perturbative approach and the nonperturbative lattice approach are the two standard yet very distinct formulations of quantum gauge theories. Since in dimension two Yang-Mills theory has a rigorous continuum limit of the lattice...

I will discuss results on local eigenvalue statistics for uniform random regular graphs. For graphs whose degrees grow slowly with the number of vertices, we prove that the local semicircle law holds at the optimal scale, and that the bulk...

We will mainly report on the progress done recently the connectedness properties of the set of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation. To make the lecture accessible to people with no previous knowledge in...

Numerous manifestations of wave localization permeate acoustics, quantum physics, mechanical and energy engineering. It was used in construction of noise abatement walls, LEDs, optical devices, to mention just a few applications. Yet, no systematic...

Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular...

We discuss our results on global existence and convergence of solutions to the gradient flow equation for the Yang-Mills energy functional over a closed, four-dimensional, Riemannian manifolds: If the initial connection is close enough to a minimum...

Stochastic quantization equations are evolutionary PDEs driven by space-time white noises. They are proposed by physicists in the 80s as the natural dynamics associated to the (Euclidean) quantum field theories. We will discuss the recent progress...

I present a new, fully anisotropic, criterion for formation of trapped surfaces in vacuum obtained in collaboration with J. Luk and I. Rodnianski. We provide conditions on null data, concentrated in a neighborhood of a short null geodesic segment...

I will discuss recent work on the global stability of the Euler-Maxwell equations in 3D (joint work with Guo and Pausader), and of the gravity water-wave system in 2D (joint work with Pusateri).

We study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a...