Mathematical Sciences: Index Theory and Cyclic Cohomology

点击翻译按钮获取中文摘要

Nistor will continue his work in that part of the theory of operator algebras relating to noncommutative geometry, and particularly index theory and cyclic cohomology. His immediate goals include a formula for the bivariant Chern-Connes character and the identification of the Atiyah-Singer integrand, computation of the cyclic cohomology of foliation algebras using Bott's simplicial methods, and a combination of these two into an index theorem for foliations. 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.