Mathematical Sciences: Operator Algebras

洛林工作的中心焦点将是算子上的关系和相关的全称C*-代数。这些将与k理论、摄动问题、提升问题和表示理论联系起来进行研究。Gilfeather将致力于扩展最近关于在代数中取值的代数的hochschild上同的结果。特别地，他将研究非自伴代数的完全有界上同调。一个相关的领域涉及由代数拓扑的各种构造引起的自同构。这个项目的一般数学领域在希尔伯特空间算子的代数理论中有其基础。运算符可以被认为是有限或无限的复数矩阵。特殊类型的运算符通常放在一个代数中，自然地称为运算符代数。这些抽象对象有各种各样的应用。例如，它们在结理论中发挥着关键作用，而结理论目前正被用于研究DNA的结构，它们在非交换几何中具有重要的基础意义，而非交换几何在物理学中正变得越来越重要。

The central focus of Loring's work will be relations on operators and the associated universal C*-algebras. These will be studied in connection with K-theory, perturbation problems, lifting problems and representation theory. Gilfeather will work on extending recent results on Hochshild cohomology of an algebra taking values in that algebra. In particular, he will investigate completely bounded cohomology for non-selfadjoint algebras. A related area involves automorphisms motivated by various constructions from algebraic topology. 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.