School of Mathematics

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 (possibly flat everywhere else) whose future development contains a trapped surface. This extends considerably the previous result of Christodoulou which required instead a uniform condition along all null geodesic generators.

A continuous representation of a profinite group induces a continuous pseudorepresentation, where a pseudorepresentation is the data of the characteristic polynomial coefficients. We discuss the geometry of the resulting map from the moduli formal groupoid of representations to the moduli formal scheme of pseudorepresentations. We find that, relative to the moduli of pseudorepresentations, the groupoid of continuous representations is algebraizable and universally closed, being representable by a finitely presented algebraic stack.

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).