Mathematical Sciences: Problems in Operator Algebra

Muhly将研究算子代数中的各种问题，这些问题可以分为三组，标题包括：一般算子代数、群类群和Toeplitz及相关算子。在第一个标题下，他将致力于理解非自伴代数结构的5个战略问题。重点将放在与算子代数相关的Hilbert模的范畴上。灵感来源于有限维纯环理论，但关键是要理解代数和所涉及的度量之间的相互作用。在Muhly的工作中，群类群为表示算子代数提供了具体的坐标。这个标题下的项目是他早期努力的自然延续。其中最重要的问题可能是为群类群建立一个好的拓扑上同理论。他提出要研究的Toeplitz算子的问题自然也来源于他早期的工作，这些问题的进展将有助于揭示经典函数理论/调和分析与算子理论之间重要的新联系。虽然在许多方面，Muhly将研究的问题的细节方面受到核心数学问题的启发，但它们对应用数学和“现实世界”的意义是实质性的。Muhly的大部分工作都为控制理论，特别是h∞控制理论的数学基础做出了贡献，而h∞控制理论又是工程师设计飞机、汽车、机器人和其他需要响应外部刺激并在指定指导方针内进行控制的设备的理论框架。他在类群和Toeplitz算子方面的工作与随机控制问题密切相关，即涉及随机不确定性的控制问题，人们试图引导一条平均最好的路线。他在算子代数方面的工作，一般来说，对多变量控制理论和非平稳控制问题做出了贡献，即控制响应是输入时间的函数的问题。这些问题在各种设备的设计中变得越来越重要，从声学滤波器到数字处理器，再到神经网络。

Abstract Muhly Muhly will investigate a variety of problems in operator algebra that may be divided into three groups covered by the headings: General Operator Algebras, Groupoids, and Toeplitz and Related Operators. Under the first heading, he will pursue 5 strategic problems dedicated to understanding the structure of non-self-adjoint algebras. The focus will be on the categories of Hilbert modules associated with operator algebras. Inspiration for this derives from finite dimensional pure ring theory, but the key is to understand the interplay between the algebra and the metrics involved. In Muhly's work, groupoids provide concrete coordinates for representing operator algebras. The projects under this rubric are natural continuations of his earlier efforts. Most important among them, perhaps, is the problem of developing a good, topological cohomology theory for groupoids. The problems on Toeplitz operators that he proposes to investigate also derive naturally from his earlier work and progress on them should help shed light on important new connections between classical function theory/harmonic analysis and operator theory. Although in many respects, the detailed aspects of the problems Muhly will investigate are inspired by questions from core mathematics, their significance to applied mathematics and the "real world" is substantial. Much of Muhly's work contributes to the mathematical underpinning of control theory, H-infinity control theory, in particular, which, in turn, is the theoretical framework on which engineers base the design of aircraft, automobiles, robots, and other devices that need to respond to external stimuli and to be controlled to perform within specified guidelines. His work on groupoids and some of his work on Toeplitz operators is closely connected to issues of stochastic control, i.e., control problems where there are random uncertainties involved and one tries to steer a course that is best on the average. His work on operator algebras, generally, c ontributes to the field of multivariable control theory and to non-stationary control problems, i.e., problems where the control response is a function of the time of the input. Such problems are of increasing importance in the design of all types of devices, from acoustical filters, to digital processors, and neural networks.