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

算子论是现代分析的中心学科。它起源于二十世纪初对数学物理和偏微分方程式的研究。当时可以看到，平衡论、振动、量子力学等理论中的许多物理问题都可以通过模拟这些现象的积分方程组进行富有成效的研究。目前，算子理论不仅与物理学有很强的相互作用，而且与工程学和数学的许多分支都有很强的相互作用。在后一方面值得注意的是K-理论、非对易微分几何、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，对他的功劳有深刻而重大的贡献。多年来，在到美国定居之前，他是中国数学领域一位活跃、多产的领导者。他在函数论、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，并在调和分析方面与伟大的苏联数学家盖尔方合作。近年来，他一直专注于算子理论。这项工作涉及到亚正规算子、压缩和K-理论、循环上同调和算子的几乎李代数、几乎李算子群以及差算子和数学物理之间的深层联系。拟议的研究集中在三个主要研究领域。第一个涉及确定循环上同调以及上循环、符号和指数之间的关系。这涉及到K理论、非对易微分几何和量子场论。第二是研究次正规算子和次正规算子的分析模型，以及与不变子空间、上同调和代数几何的关系。最后，夏教授将学习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.