Mathematical Sciences: Connections of Modern Analysis with Geometry and Topology

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Douglas will study the connection between problems in operator theory on one hand and algebraic and analytic geometry in the presence of a hermitian metric. The goal is to understand multi- variate phenomena and to provide techniques to study more general cases. He will also investigate various invariants for elliptic operators on manifolds. Starting from the case of elliptic operators on compact, smooth manifolds without boundary which is now classical, one is seeking to understand what happens when these hypothesis are relaxed. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics.