## Mathematical Sciences: K-Theory and Cyclic Cohomology Related to Operator Algebras

##### Abstract

Professor Natsume's project is in the area of K-theory and cyclic cohomology in noncommutative geometry. He will study operator algebras arising from geometry. Solutions to the problems he poses will illuminate the area of interaction of the two fields. Some of the specific problems he will consider are: (i) determine the structure of the group of diffeomorphisms of a twisted group C*-algebra of the fundamental group of a closed Riemann surface, (ii) determine whether the Baum-Connes conjecture is valid for codimension one foliations which are almost without holonomy, (iii) generalize the vanishing theorem of the Godbillon-Vey map in analytical K-theory to higher codimension cases, and (iv) establish a Fourier inversion formula for the canonical cyclic cocycle of a simply connected solvable Lie group. The general area of this project is operator algebras and geometry. Operators can be thought of as infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. Among other problems, Professor Natsume will study operator algebras which arise in the analysis of closed surfaces.