Mathematical Sciences: Trace Formulas, Cyclic Cohomology, Complete Unitary Invariants and Tuples of Operators

夏教授将研究Hilbert空间算子的几乎可交换n元理论中的一些课题。他将研究几乎可交换n元组的循环上同调和迹公式。他将研究几乎无摄动薛定谔算子的n元组理论，他将试图找到算子的次正规元组的完整的酉不变量集。本研究涉及到希尔伯特空间算子理论。这些运算符可以被认为是无限的复数矩阵。它们几乎应用于纯数学和应用数学的每一个分支。夏教授的研究目的是对这些算子的重要族进行分类和认识。

Professor Xia will study a number of topics in the theory of almost commuting n-tuples of Hilbert space operators. He will investigate cyclic cohomology and trace formulas for almost commuting n-tuples. He will study the theory of n-tuples of almost unperturbed Schrodinger operators and he will try to find a complete set of unitary invariants for subnormal tuples of operators. This research involves the theory of Hilbert space operators. These operators can be thought of as infinite matrices of complex numbers. They find application in almost every branch of pure and applied mathematics. The object of Professor Xia's research is to classify and understand important families of these operators.