Mathematical Sciences: Cyclic Cohomology, Symbols, Local Index, Toeplitz-type Operators and Hyponormal Operators

算子理论是现代分析的核心学科。它起源于20世纪初对数学物理和偏微分方程的研究。当时，人们看到平衡理论、振动理论、量子力学等许多物理问题都可以通过模拟现象的积分方程来进行富有成效的研究。目前，算符理论不仅与物理学，而且与工程和许多数学分支都有很强的相互作用。在后者方面值得注意的是k理论、非交换微分几何、指数理论和上同调。夏教授在算子理论的所有这些方面都是专家，他的贡献是深刻而重要的。在来美国定居之前的许多年里，他一直是中国数学界活跃、多产的领导者。他在函数理论、无限维空间上的拟不变测度和拓扑代数方面做了杰出的工作。他在量子场论中研究散射，在谐波分析方面，他与伟大的苏联数学家盖尔芬德合作过。近年来，他主要研究算子理论。这项工作涉及到次反常算子、收缩和k理论之间的深层联系；循环上同调与算子的概李代数算子的几乎李群；差分算子和数学物理。建议的研究集中在三个主要研究领域。第一种方法是确定环上同调以及环、符号和索引之间的关系。这与K理论、非交换微分几何和量子场论有关。第二部分研究了次正规算子和次正规算子的解析模型及其与不变子空间、上同调和代数几何的关系。最后，夏教授将研究Toeplitz算子及其在微分方程和伪微分算子中的应用。

Operator theory is a central discipline in Modern Analysis. Its origins lie in the study of mathematical physics and partial differential equations in the early twentieth century. At that time, it was seen that numerous physical problems in the theory of equilibria, vibration, quantum mechanics, etc. could be studied productively via the integral equations that model the phenomena. Currently, operator theory has a strong interaction not only with physics, but also engineering and many branches of mathematics. Notable in the latter regard are K-theory, non-commutative differential geometry, index theory, and cohomology. Professor Xia is an expert in all of these aspects of operator theory, with deep and significant contributions to his credit. For many years, before taking residency in the United States, he was an active, prolific leader in Chinese mathematics. He has done outstanding work in function theory, quasi-invariant measures on infinite-dimensional spaces, and topological algebras. He has worked on scattering in quantum field theory, and in harmonic analysis he has collaborated with the great Soviet mathematician Gelfand. In recent years, he has concentrated on operator theory. This work has involved deep connections among hyponormal operators, contractions, and K-theory; cyclic cohomology and almost Lie algebras of operators; almost Lie groups of operators; and difference operators and mathematical physics. The proposed research focuses on three major areas of research. The first involves determining cyclic cohomology and relations among cocycles, symbols, and index. This relates to K- theory, non-commutative differential geometry, and quantum field theory. The second is the study of analytic models of hyponormal operators and subnormal operators, with relationship to invariant subspaces, cohomology, and algebraic geometry. Finally, Professor Xia will study Toeplitz operators and applications to differential equations and pseudo-differential operators.