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0 C*-代数同态的扰动理论。这将提供C*-代数的几何性质与其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。

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.