Studies in Noncommutative Geometry and Index Theory

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Abstract Gorokhovsky This proposal belongs to the area of noncommutative geometry. It is devoted to three projects. The goal of the first project is to obtain construction of characteristic classes for the important class of noncommutative spaces, arising naturally in the contexts of discrete group actions on manifolds and foliations. The approach is based on the recently developed by A. Connes and H. Moscovici theory of cyclic cohomology of Hopf algebras. The goal of the second project (joint work with R. Nest and A. Uribe) is to obtain index theorem for algebras of Fourier integral operators, and study applications of these results. The third project (joint work with J. Lott) is devoted to the proof of the local version of Connes' index theorem for etale groupoids, as well as various extensions of this theorem. Noncommutative geometry is a field of mathematics situated on the crossroads between analysis, geometry, and mathematical physics. Discovery of quantum mechanics has shown that the classical principles of geometry are not applicable to the world of microscopic particles. Noncommutative geometry unifies geometry with analysis for the solution of the problems of quantum mechanics. Different parts of the present work deals both with solving internal problems of noncommutative geometry and with applying methods of noncommutative geometry to other parts of mathematics and mathematical physics.