Mathematical Sciences: K-Theory, C*-algebras and Index Theory of Elliptic Operators

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9401457 Kaminker This project will pursue several directions involving the use of operator algebras to obtain invariants for elliptic operators, algebraic K-theory and dynamical systems. a notion of higher spectral flow will be developed. Index theoretic invariants will be defined which will detect elements of algebraic K-theory of the complex numbers. Operator algebras associated to hyperbolic dynamical systems will be studied and a K-theoretic duality relation will be used to obtain dynamical information. Work will also be done on some questions related to the Novikov conjecture and generalized homology theories on C*- algebras. 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. ***