Global Methods in the Analysis on Singular Spaces and Partial Differential Equations

Abstract

Nistor

This project is devoted to the analysis and spectral theory of differential operators on non-compact manifolds. The investigator is especially interested in generalizing the classical results of elliptic theory for compact manifolds, including index theory. Little can be said about all non-compact manifolds in general, but results of Bismut, Beunning, Cordes, Mazzeo, Melrose, Meuller, Shubin, and others have singled

out a class of manifolds that is more amenable to study: the class of manifolds with a uniform structure at infinity. The local theory (regularity, local existence) for these manifolds is the same as for compact manifolds, so our methods will necessarily be global. Thus, in addition to the methods of Partial differential equations and Differential geometry used by the above mentioned authors, methods from Operator algebras have come to play an increasingly important role in the study of non-compact manifolds with

a uniform structure at infinity, as is seen from the work of Connes, V.F.R. Jones, Lauter, Monthubert, Skandalis, Taylor, and the investigator.

Manifolds with a uniform structure at infinity appear naturally in Scattering theory, Differential geometry, Representation theory, Mathematical physics, and certain problems of Applied mathematics. The results of the proposed research will have, in the long run, applications to all these domains. The main methods that are proposed belong to Analysis, especially Partial differential equations, Operator algebras, Spectral theory, and K-theory. A main technical tool will be provided by algebras generated by differential

operators on non-compact manifolds with a uniform structure at infinity. By using Sobolev spaces, one can reduce many of our basic questions to questions about algebras of bounded operators. A novel feature of this proposal is the study of boundary value problems for manifolds with a uniform structure at infinity that have Lipschitz boundaries, as in the work of Mitrea and Taylor on such compact domains.