Mathematical Sciences: Spectral Duality of Differential Operators Affiliated to von Neumann Algebras

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、量子理论等许多物理问题可以通过对现象建模的积分方程进行有效的研究。在随后的几年里，算符理论的主题在这些研究和核心数学中已经发展到一个中心位置。现代分析的核心是算子代数的相关学科，在这个学科中，人们同时研究算子的集合。作为最好地将概率论、测度论、拓扑学和几何的概念转移到非交换环境的数学结构，算子代数涉及到从数学本身到物理学和微生物学的丰富而重要的应用阵列。夏夏教授是算子理论领域的年轻专家。虽然刚刚获得博士学位，但他已经建立了令人印象深刻的出版记录。他的研究成果包括周期椭圆算子谱的连续性结果，例如薛定谔算子。这项工作利用了（可交换的）n维向量群对适当算子的作用。他现在提议将他的谱对偶理论推广到紧空间上非交换群的作用，以及与此作用相关的椭圆微分算子的谱。这是一个困难的项目，需要算子理论，算子代数，几何，调和分析和微分方程。然而，沿着所提出的路线的重要结果将对无序系统的数学物理有重要的应用，通过随机微分算子进行连接。

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 theory, etc. could be studied productively via the integral equations that model the phenomena. In the ensuing years, the subject of operator theory has grown to a central position in such investigations, and in core mathematics as well. Also central to Modern Analysis is the related discipline of operator algebras in which one studies collections of operators simultaneously. As the mathematical construct which best transfers the concepts of probability, measure theory, topology, and geometry to noncommutative contexts, operator algebras relate to a rich and important array of applications, from within mathematics itself to physics and microbiology. Professor Xia ia a young specialist in the area of operator theory. Although a recent Ph.D., he has already established an impressive publication record. His research successes include continuity results for the spectrum of periodic elliptic operators, for example Schrodinger operators. This work utilizes the action of the (commutative) n-dimensional vector group on the appropriate operators. He proposes now to extend his spectral duality theory to the action of noncommutative groups on compact spaces, and to the spectrum of the elliptic differential operators associated with the action. This is a difficult project that requires operator theory, operator algebras, geometry, harmonic analysis, and differential equations. However, significant results along the proposed lines would have important applications to the mathematical physics of disordered systems, the connection being through random differential operators.