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

夏教授将研究几乎可换的希尔伯特空间算子的n元组理论中的一些主题。他将研究几乎可交换的n元组的循环上同调和迹公式。他将研究几乎未受扰动的薛定谔算子的n元组理论，并试图为次正规的算子组找到一组完备的酉不变量。本研究涉及到希尔伯特空间算子的理论。这些运算符可以被认为是复数的无限矩阵。它们几乎在纯数学和应用数学的每一个分支中都有应用。夏教授的研究对象是对这些算子的重要家族进行分类和理解。

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.