Mathematical Sciences: Studies in Semigroups of Endomorphisms, Operator Algebras, and Numerical Quantum Mechanics

Arveson将研究非对易分析的几个领域，涉及von Neumann代数和C*-代数，量子化指数理论，以及Bratteli，Elliott，Evans和Kishimoto的非对易球体在数值量子力学中的作用。研究内容包括因子自同态半群的指数理论和分类理论，与之相关的C*-代数的结构，以及离散化的正则对易关系所对应的C*-代数的性质。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。

Arveson will investigate several areas of non-commutative analysis involving von Neumann algebras and C*-algebras, quantized index theory, and the role of the non-commutative spheres of Bratteli, Elliott, Evans and Kishimoto in numerical quantum mechanics. The research will involve the index theory and classification theory of semigroups of endomorphisms of factors, the structure of the C*-algebras associated to such semigroups, and the properties of C*-algebras associated with the discretized canonical commutation relations. 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.