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

算子理论是现代分析中的一个中心学科。 它的起源在于研究数学物理和部分 微分方程在世纪早期。 在那个 当时，人们发现，理论中的许多物理问题 平衡，振动，量子理论等都可以研究 有效地通过模拟现象的积分方程。 在随后的几年里，算子理论的主题已经发展到 在这些调查中处于中心地位， 数学也是。 现代分析的另一个核心是 算子代数的相关学科，其中一个 同时研究操作员的集合。 为 数学结构，最好地转移概念， 概率、测度论、拓扑学和几何学， 非交换的情况下，算子代数涉及到丰富的， 一系列重要的应用，从数学本身 物理学和微生物学。 夏教授是运算符领域的年轻专家 理论 尽管刚拿到博士学位，他已经建立了一个 令人印象深刻的出版记录。 他的研究成果包括 周期椭圆型谱的连续性结果 例如，Schrodinger算子。这项工作利用 （交换）n维向量群对 合适的运营商。 他现在打算扩展他的光谱 紧空间上非交换群作用的对偶理论 空间，以及椭圆微分的谱 与操作相关的操作符。 这是一 需要算子理论的困难项目，算子 代数、几何、调和分析和微分 方程 然而，沿着所提议的路线， 在数学物理学中有重要的应用 无序的系统，连接是通过随机 微分算子

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.