Mathematical Sciences: Connections of Modern Analysis with Geometry and Topology

道格拉斯将在厄米度规的存在下，一方面研究算子论中的问题与代数几何和解析几何之间的联系。其目的是了解多变量现象，并提供研究更一般情况的技术。他还将研究流形上椭圆算子的各种不变量。从现在经典的紧致光滑流形上的椭圆算子出发，人们试图理解当这些假设放松时会发生什么。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。

Douglas will study the connection between problems in operator theory on one hand and algebraic and analytic geometry in the presence of a hermitian metric. The goal is to understand multi- variate phenomena and to provide techniques to study more general cases. He will also investigate various invariants for elliptic operators on manifolds. Starting from the case of elliptic operators on compact, smooth manifolds without boundary which is now classical, one is seeking to understand what happens when these hypothesis are relaxed. 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.