Mathematical Relativity
In collaboration with Blake Temple, I worked extensively on the question when singularities are removable, i.e., when a connection can be mapped to optimal regularity by coordinate transformation. This led recently to the discovery of the Regularity Transformation equations, a system of elliptic PDE's which determines whether connections on the tangent bundles of arbitrary manifolds can be smoothed to optimal regularity by coordinate transformation. By developing an existence theory for these equations we proved in particular that unphysical singularities associated with shock wave solutions of the Einstein Euler equations can always by removed.
Theory of Shock Waves
I worked on an extension of Lax's existence theory for the Riemann problem and of Glimm's random choice method for the Cauchy problem to an existence theory for states in Riemannian manifolds.
Quantum Mechanics and Quantum Field Theory
In collaboration with Felix Finster, I worked on a construction of a distinguished ground state (vacuum state), required for construction of the Fock space in canonical Quantum Field Theory. The basic object underlying the construction of this ground state is the fermionic signature operator.
Stellar Structure and Black Holes
I am interested in stability of static solutions of the Einstein Euler equations and in the question of gravitational collapse of fluid spheres (stars) to black holes.