Smoller and E. Conway began the analysis of numerical difference schemes for conservation laws in several space dimensions by introducing the Tonnelli-Cesari variation norm into shock-wave theory.

With C. Conley he introduced new topological methods into PDE’s, brought the shock structure problem from the physics-engineering community to center focus in PDE’s, and pioneered the analysis of spatially dependent systems into mathematical biology. Their multi-dimensional analysis of the Hodgkin-Huxley and other biological equations, and definitive invariant regions theorem (essential in DiPerna’s important existence proof in compressible flow theory), are landmarks in reaction-diffusion equations.

In a tour de force with A. Wasserman, he developed an equivariant version of the Conley Index and applied it to prove (the first) general symmetry-breaking result in elliptic PDE’s. With B. Temple he extended the celebrated Oppenheimer-Snyder (1939) model to non-zero pressure, and introduced a physically plausible finite mass shock-wave refinement of the Standard Model of Cosmology. Recently, they proposed a new family of expansion waves in General Relativity leading to an alternative to Dark Energy.

Following numerical simulations by R.Bartnik and J. McKinnon, Smoller with collaborators F. Finster, A. Wasserman, and S.-T. Yau gave a definitive mathematical analysis of particle-like and black hole solutions of the Einstein-Yang/Mills equations, established the important result that solutions are stabilized by adding a Dirac field, and demonstrated that colored black holes do not satisfy the no-hair theorem.

More recently, Smoller and his collaborators Finster, Yau and N.Kamran proved decay of the scalar wave equation and Dirac’s equation in a Kerr background geometry, and provided a rigorous proof of Penrose energy extraction for linearized rotating black holes.

He has more recently introduced new mathematical ideas extending the Auchmuty-Beals theory of rotating Newtonian stars.

Smoller’s book, Shock Waves and Reaction-Diffusion Equations, (1983, 2nd ed 1993), has been instrumental in teaching the fundamental works of the J. Glimm and P.Lax shock wave theory. His book also enabled students to learn the basics of Reaction-Diffusion equations, as well as the Conley Index theory.