Mathematical Sciences: Geometric Invariants, Pseudodifferential Operators and Several Complex Variables

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The principal investigator will study global invariants on strictly pseudoconvex manifolds that arise as renormalized Chern classes for complete Kaehler metrics. He will continue his study of pseudodifferential operator calculi adapted to the study of degenerate operators similiar to the Bergman-Laplace operator. A goal will be to obtain index formulae relating the renormalized Chern numbers to "analytic" indices for degenerate elliptic complexes. And he will continue an investigation of the problem of embedding three dimensional CR-manifolds. 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.