Mathematical Sciences: K-Theory and Other Invariants of C*-Algebras

张将继续研究C*-代数的各种不变量，包括C*-代数的实随机数、拓扑稳定随机数、指数随机数和长度、Grassmann空间的C*-射影长度以及某些C*-代数的酉群和Grassmann空间的同伦群。他还将继续利用伴随C*-代数的K-理论及其扭子群来研究乘子代数的内部结构。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。

Zhang will continue his research on various invariants of C*- algebras, including the real rand, the topological stable rand, the exponential rand and length, the C*-projective length of the Grassmann space, and the homotopy groups of the unitary group and the Grassmann space of certain C*-algebras. He will also continue his investigation on the internal structure of the multiplier algebras in terms of the K-theory and its torsion subgroup of the associated C*-algebra. 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.