Our main research interest is in the mathematical theory of general relativity. We are currently investigating the spherically symmetric collapse of an anisotropic fluid (one possible model of a star) into a black hole. In the solution, we would like to satisfy some of the energy conditions (weak, dominant, or strong). We would like to obtain an analytic equation for the collapsing boundary, and satisfy one of three jump conditions (Synge, Synge-O'Brien, or Darmoir) across the surface of discontinuity. We wish to match the interior solution to an exterior Schwarzschild (or Kruskal) or to a radiating Vaidya metric. Our goal is to obtain a reasonable, mathematically rigorous model for which the collapse of a spherically symmetric star into the final singularity can be pursued analytically in a doubly-null coordinate chart.
We have another research interest in the mathematical foundation of quantum theory of interacting fields. The present formulation is plagued by severe divergence difficulties. In 1960, we eliminated divergence problems by using partial difference equations (which presuppose a fundamental length). Unfortunately, the theory was not Lorentz-covariant. Recently, we have been formulating relativistic lattice wave field equations (in terms of partial difference equations) in order to obtain divergence-free S-matrix expansions.