Multivariable Operator Theory and Several Complex Variables

建议：DMS-9970518首席研究员：Norberto Salinas摘要：作为该项目的主要活动，Salinas计划继续他在多变量算子理论的特定领域的研究。其中包括：解析上Bergman空间的（Hilbert和Frechet）子模和商模n维复空间中（光滑有界）伪凸域的谱图和C ~*-代数性质（如本质正规性和类型单一性）的n元组诱导的模的这种类型的有界全纯映射的这种域是光滑的到边界;光滑的相应的诱导扩张;具有非交换横对称的伪凸域（如广义Reinhardt域）上的Toeplitz C*-代数，这类C*-代数的K-理论的计算，以及光滑KK-理论与光滑相对拟对角性之间的关系.他还打算考虑一个新的几何现象;即，在给定算子T的“数值范围”中的无限曲率点，这些点在T的广义表示谱中产生点。萨利纳斯还计划调查光滑的某些运营商的二阶光滑有界的pseudoclave域的伯格曼投影不是全球regular.The这个项目的目标是探索和利用重要的相互作用之间的两个丰富和历史悠久的领域的数学，算子理论和理论的函数的几个复杂的变量。一个新出现的数学跨学科活动的焦点，这一领域充满了兴奋。从纯数学的角度来看，促进异花授粉自然是有利的，因为它将增加算子理论家和复杂分析师在各自的分支学科中解决基本问题的工具数量，但婚姻的成功超出了所有人的预期。从更实际和具体Vantage来看，这两个领域的结合是重要的，因为它为理论和实验物理以及工程应用提供了更大的潜力。例如，算子理论和多复变量理论已经成功地应用于控制理论，作为航天飞机自动着陆软件开发的关键部分。Salinas的活动旨在为盲人教师创造替代教学方法，他在发展科学的8点盲文方面继续发挥作用，为人力资源开发领域的项目增加了价值。

Proposal: DMS-9970518Principal Investigator: Norberto SalinasAbstract: As the main activity of this project, Salinas plans to continue his research in specific areas of multivariable operator theory. These include: analytic (Hilbert and Frechet) submodules and quotient modules of the Bergman space over a (smoothly bounded) pseudoconvex domain in n-dimensional complex space; the spectral picture and C*-algebraic properties (such as essential normality and type oneness) of the n-tuples induced on modules of this type by bounded holomorphic mappings of such domains that are smooth up to the boundary; the smoothness of the corresponding induced extensions; Toeplitz C*-algebras over pseudoconvex domains with nonabelian transverse symmetries (such as generalized Reinhardt domains), the calculation of the K-theory of such C*-algebras, and the relationship between the smooth KK-theory and smooth relative quasidiagonality. He also intends to consider a new geometric phenomenon; namely, points of infinite curvature in the "numerical range" of a given operator T that produce points in the generalized representation spectrum of T. Salinas also plans to investigate the smoothness of certain operators of second order on smoothly bounded pseudoconvex domains where the Bergman projection is not globally regular.The goal of this project is to explore and to exploit the important interplay between two rich and time-honored areas of mathematics, operator theory and the theory of functions of several complex variables. A newly emerging focal point of mathematical interdisciplinary activity, this area is brimming with excitement. From a purely mathematical point of view, it was naturally advantageous to foster the cross-pollination because it would increase the number of tools that operator theorists and complex analysts alike have to attack fundamental problems in their respective subdisciplines, but the success of the marriage has exceeded all expectations. From a more practical and concrete vantage point, the combination of the two fields is significant because of the expanded potential it offers for applications to both theoretical and experimental physics, as well as to engineering. For example, operator theory and the theory of several complex variables have already been used successfully in control theory as a crucial part of the development of the software that automatically lands the space shuttle. The activities of Salinas aimed at creating alternative teaching methods for use by blind instructors and his continuing role in the evolution of scientific 8-dot Braille add value to the project in the area of human resource development.