Regularity Properties and K-Theory of Crossed Product Operator Algebras

叉积算子代数的正则性质与K理论

基本信息

批准号：
2055736
负责人：
Sherry Gong
金额：
$ 11.22万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055736&HistoricalAwards=false
关键词：
Regularity Properties Theory Crossed Product

项目摘要

Algebra, analysis, geometry, and dynamics are some of the major branches of modern mathematics. Algebra studies the rules to work with mathematical symbols in ways that generalize addition and multiplication. Analysis deals with approximations of numbers, functions and other mathematical objects. Geometry concerns itself with notions such as shape, distance, angle, etc. Dynamics studies motions and in particular their long-term behaviors. This research project lies in the area of operator algebras and noncommutative geometry, which uses advanced techniques from algebra and analysis to study mathematical problems that are often geometrically and dynamically motivated. Outreach activities to foster communication and collaboration, especially among early-stage mathematicians will also be carried out.This project focuses on regularity properties and K-theory of operator algebras of dynamical nature, in particular, crossed product C*-algebras. It is motivated by applications in several areas of mathematical research. On the one hand, to further advance the (already spectacular) applications of noncommutative geometry to the celebrated Novikov conjecture in differential topology, the PI aims to study the K-theory of group C*-algebras of groups of diffeomorphisms on smooth manifolds. On the other hand, the classification program for simple separable nuclear C*-algebras has spawned and highlighted a number of regularity properties of C*-algebras, and the PI is working on transporting these properties to the dynamical setting, so as to aid future applications to the classification and structure theory of topological dynamical systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

代数、分析、几何和动力学是现代数学的一些主要分支。代数研究以推广加法和乘法的方式处理数学符号的规则。分析涉及数字、函数和其他数学对象的近似值。几何学关注形状、距离、角度等概念。动力学研究运动，特别是它们的长期行为。该研究项目属于算子代数和非交换几何领域，它使用代数和分析的先进技术来研究通常由几何和动态驱动的数学问题。还将开展促进交流与合作的外展活动，特别是在早期数学家之间。该项目重点关注动态算子代数的正则性性质和 K 理论，特别是交叉积 C* 代数。它是由数学研究多个领域的应用推动的。一方面，为了进一步推进非交换几何在微分拓扑中著名的诺维科夫猜想中的应用（已经很引人注目），PI 旨在研究光滑流形上微分同胚群的 C* 群代数的 K 理论。另一方面，简单可分离核C*代数的分类程序已经产生并突出了C*代数的许多正则性性质，PI正在致力于将这些性质转移到动态环境中，以帮助未来该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。