Joel Smoller has done research in shock-wave theory, Navier-Stokes equations, systems of reaction-diffusion equations, dynamical systems (Conley Index Theory), and bifurcation theory (symmetry-breaking bifurcations). For the last several years he has been working in General Relativity (GR). Specifically, with JB Temple he has studied shock waves in GR, including new cosmological models, and expansion waves in GR, which give possible explanations for the anomalous acceleration of the Universe, wholly within classical GR and Einstein's equations.

With F. Finster, and S.-T. Yau he has worked on the coupled Einstein-Dirac-Yang/Mills equations. He has studied the stability of various fields in a background rotating (Kerr), black-hole geometry, with F. Finster, N. Kamran, and S.-T. Yau. He has also worked with T. Luo on existence and stability of rotating Newtonian stars. Some of these problems have led to various interesting offshoots; for example a mathematical proof of the Penrose proposal for energy extraction from a rotating BH, and rigorous error bounds for approximate solutions for both the Riccoti equation with real or complex potentials, and WKB approximations to Schrödinger equation. Smoller continues to be active in these areas.

http://www.math.lsa.umich.edu/~smoller/