Asymptotics and Spectral Theory. Approximate solutions when some parameter becomes small are important not only in grinding out numbers, but also in understanding the qualitative features of complicated exact solutions. Of special interest today are the relationships among the spectral properties (eigenvalues, etc.) of a partial differential operator, the asymptotic properties of the Green functions associated with that operator, and the coefficient functions in the operator itself and the geometry of the region where it acts. Applications range from renormalization in quantum field theory, to the index theorems that relate the topology, geometry, and analysis of manifolds, to very practical problems in all fields that use differential equations. Research in this area is done by Stephen Fulling, whose work has been motivated by mathematical physics (quantum field theory in curved space-time and semiclassical approximations in quantum mechanics).