Linear and nonlinear wave equations model many phenomena, including electromagnetic and gravitational waves. Their study is central to at least two areas of mathematics: scattering theory and general relativity. This project focuses on the study of partial differential equations on singular spaces with an emphasis on the scattering theory of waves. The long-time behavior of wave equations on smooth spaces is well-understood in many regards, but the presence of singularities raises many unresolved problems. The singularities studied include boundaries, cone points, and large-scale structures "at infinity." These projects are of interest to mathematicians and physicists working in partial differential equations, geometry, general relativity, and numerical analysis. The investigator intends to foster communication among researchers in these communities through collaboration and by organizing conferences in these fields.This research will provide new insights into the long-time behavior of waves in novel contexts. Other than on perturbations of Euclidean and hyperbolic spaces, very little fine information is known about the long-time behavior of waves. Many facts that are known to hold for the linear wave equation in flat space are largely unknown when the underlying spacetime on which the evolution takes place is instead time-dependent or singular. The investigator will employ microlocal, or phase space, techniques to explore problems in these areas. In particular, the investigator will address problems concerning (1) wave decay on curved spacetimes such as those arising in general relativity, (2) wave maps, or nonlinear sigma models in the language of quantum field theory, on conic surfaces, and (3) boundary value problems for the Helmholtz equation arising in numerical analysis.
