Manifolds and C*-Algebraic Index Theory

University Of Maryland, College Park, College Park MD

Investigators

Abstract

DMS-0103647 Johathan M. Rosenberg Professor Jonathan Rosenberg will study the topology and geometry of manifolds and manifold-like spaces, as well as C*-algebraic index theory. By this we mean a combination of C*-algebra theory, index theory of elliptic operators, K-theory, geometry, and topology. One main focus of the proposal will be the use of invariants coming from C*-algebras (especially Kasparov's KK-theory) to study the geometry and topology of manifolds. For example, KK-classes coming from the classical elliptic operators (the Euler characteristic operator, the signature operator, and the Dolbeault operator) will be intensively studied. Attention will also be paid to the equivariant case, where a finite group acts on the manifold in question. These studies will deepen the link between topological and analytic approaches to manifold theory. In addition, Professor Rosenberg will study the classification of manifolds via their Yamabe invariants, and the classification of metrics of positive scalar curvature, problems which involve a quite subtle blend of differential topology, differential geometry, and analysis. He will also study the applications of K-theory to the study of C*-algebras, and various related problems on algebraic K-theory, especially on algebras of operators or on algebras coming from quantization of geometrical systems. The context of this project is the use of new methods in analysis, based on noncommutative operator algebras, for attacking problems in geometry and topology. On the one hand, these methods are useful as a new source of tools for studying geometry, especially in high dimensions. And on the other hand, the methods are motivated (and more or less forced) by the quantization of classical physical systems. The study of the scalar curvature properties of manifolds is also motivated by problems in gravitation. Professor Rosenberg will continue to train graduate students in analysis, geometry, and topology, and will also work toward integration of mathematical software into the undergraduate mathematics curriculum.