Mathematical Sciences: Index Theory on Noncompact Manifolds

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9500724 Haskell This project involves the study of the index theory of Dirac-Schrodinger operators on noncompact manifolds. In this area of index theory both incomplete and complete noncompact manifolds can be considered from a single perspective. The perspective combines techniques of regular singular theory and operator algebras. It encompasses self-adjoint Dirac-Schrodinger operators as well as those with nonzero Fredholm indices. It addresses problems in the index theory of invariant operators elliptic in directions transverse to actions of compact Lie groups as well as problems involving singular Riemannian foliations and related questions on manifolds with conical singularities. These problems all involve Hilbert space operators and their relation to the geometric properties of manifolds, the higher dimensional analogues of surfaces. Partial differential operators play a large role in this work. These operators occur naturally in the study of applied mathematics and mathematical physics. The overall goal is to compute invariants associated with these operators which simultaneously yield geometric information and information about the nature of solutions to important partial differential equations. ***