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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。代数学研究的是以归纳加法和乘法的方式使用数学符号的规则。分析处理数字，函数和其他数学对象的近似值。几何学关注的是形状、距离、角度等概念。动力学研究运动，特别是它们的长期行为。该研究项目位于算子代数和非交换几何领域，使用代数和分析的先进技术来研究通常是几何和动态动机的数学问题。此外，亦会举办外展活动，以促进沟通和合作，特别是在早期数学家之间。本计划集中于动力性质的算子代数，特别是交叉积C*-代数的正则性和K-理论。它的动机是在数学研究的几个领域的应用。一方面，为了进一步推进非交换几何在微分拓扑学中著名的诺维科夫猜想中的应用（已经很壮观了），PI的目标是研究光滑流形上的群C*-代数的K-理论。另一方面，简单可分核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正在努力将这些性质转移到动态设置中，从而有助于拓扑动力系统的分类和结构理论的未来应用。该奖项反映了NSF的法定使命，并通过评估被认为值得支持使用基金会的知识价值和更广泛的影响审查标准。