Mathematical Sciences: "Invariants of Operator Algebras"

Dadarlat将研究C*-代数的渐近同态的非稳定理论，次齐次C*-代数的*-同态的同伦分类，以及实秩零C*-代数的同态的扰动理论。这将提供C*-代数的几何性质与它们的代数不变量之间的联系，并将为核简单可分C*-代数的分类理论奠定基础。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。

Dadarlat will investigate a nonstable theory of asymptotic homomorphisms of C*-algebras, a homotopy classification of *- homomorphisms of subhomogeneous C*-algebras, and a perturbation theory of homomorphisms into real rank zero C*-algebras. This will provide connections between the geometrical properties of C*- algebras and their algebraic invariants, and will lay the foundation for a classification theory of nuclear simple separable C*-algebras. 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.