Regularity Properties and K-Theory of Crossed Product Operator Algebras

代数、分析、几何和动力学是现代数学的一些主要分支。代数研究以加法和乘法为基础的数学符号的使用规则。分析处理数字、函数和其他数学对象的近似。几何学本身涉及形状、距离、角度等概念。动力学研究运动，特别是它们的长期行为。这项研究项目位于算子代数和非对易几何领域，它利用代数和分析的先进技术来研究通常是几何和动态激励的数学问题。还将开展推广活动，以促进交流和合作，特别是早期数学家之间的交流和合作。本项目侧重于动态性质的算子代数的正则性和K-理论，特别是交叉积C*-代数。它的动机是在数学研究的几个领域中的应用。一方面，为了进一步将非对易几何应用于微分拓扑学中著名的Novikov猜想，PI旨在研究光滑流形上微分同胚群的群C*-代数的K-理论。另一方面，简单可分核C*-代数的分类程序产生并突出了C*-代数的一些正则性，PI正致力于将这些性质转移到动力学环境中，以帮助未来应用于拓扑动力系统的分类和结构理论。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.