The principal investigator will study pseudodifferential techniques in the study of degenerate elliptic equations with applications to geometry. He will extend techniques to determine a spectral analysis of the Hodge-Laplace operator acting on differential forms on complete asymptotic hyperbolic manifolds of infinite volume and on geometrically finite quotients of hyperbolic space. Related projects involve the study of the relationship between unique continuation theorems for the Laplacian at infinity and the presence of negative curvature. For many years mathematicians have studied the behavior of certain differential operators on hypersurfaces which extend off to infinity. This study has its origins in the calculus of Newton. In this project the principal investigator will analyze operators which have large variations over very large domains "near" infinity. In the past, only progress on operators which have small local variations was made.
