Mathematical Sciences: Index Theory and Cyclic Cohomology

Nistor将继续他在算子代数理论中与非交换几何有关的部分的工作，特别是指数理论和循环上同调。他近期的目标包括建立双变量Chern-Connes特征标的公式和Atiyah-Singer被积函数的确定，用Bott的单纯形方法计算叶代数的循环上同调，以及将这两者结合成关于叶代数的指数定理。这个项目的一般数学领域以希尔伯特空间算子的代数理论为基础。运算符可以被认为是复数的有限或无限矩阵。特殊类型的运算符通常放在一个代数中，自然称为运算符代数。这些抽象对象具有令人惊讶的各种应用。例如，它们在纽结理论中扮演着关键的角色，而纽结理论目前正被用于研究DNA的结构，并且它们在非对易几何中具有基本的重要性，这在物理学中变得越来越重要。

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.