Professor Bayliss works in the numerical solution of partial differential equations, especially those modeling combustion, solid mechanics, wave propagation, and fluid dynamics.  Development of adaptive numerical procedures for both finite differences and spectral methods. Computation and analysis of pattens in combustion, including both gaseous combustion and solid fuel combustion, a procedure for the synthesis of materials usig highly exothermic reactions.  Analysis and computations of integro-partial dfferential equations modeling nonloal interactions.