Mathematical Sciences: Noncommutative Geometry

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ABSTRACT Wodzicki In this project Wodzicki shall study K-theory and cyclic homology of operator ideals, and also the Hochschild homology of the algebra of bounded operators on a Hilbert space with coefficients in an ideal. In particular, he will study: determinants, regulators, higher analytic and topological index invariants, the commutator structure of bounded linear operators on a Hilbert space, and a connection with the K-theory of algebraic varieties. This combination of topics naturally results from the previous work of the principal investigator. The approach joins together several areas of mathematics with a special focus on homological study of (noncommutative) algebras of analytic and geometric significance and on the development of suitable homological and homotopical tools. The project touches upon the fundamental directions in which modern mathematics is developing and addresses questions of fundamental importance and interest in mathematics. As such, it is a "pure research". However, it may have a considerable impact in the future in such areas as, e.g. High-Energy and Nuclear Physics. The current project is a natural continuation and development of the principal investigator's research program of last 15 years. His earlier results obtained in this program already find application in the recent work of several physicists (especially those associated with CERN -- European Laboratory of Particle Physics in Geneve).