K-Theory of Operator Algebras and Its Applications

算子代数的K理论及其应用

• 批准号: 1700021
• 负责人: Guoliang Yu
• 金额: $ 18.6万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700021&HistoricalAwards=false
• 关键词: Theory; Operator; Algebras; Its; Applications

In classical geometry, one studies geometric objects whose coordinates commute. Noncommutative geometry is a mathematical theory specifically designed to study "geometric objects" whose coordinates do not commute but which do occur naturally in mathematics and physics. In the last decade or so, with the help of new ideas from noncommutative geometry, great advances have been made towards the solutions of long-standing problems in classical geometry and topology. K-theory, higher index theory, and secondary index theory serve as bridges between noncommutative geometry and classical geometry and topology. The principal investigator and his students plan to apply noncommutative geometry methods to study problems in differential geometry and topology.The K-groups of certain operator algebras are receptacles of higher indices and secondary index invariants of elliptic differential operators and have important applications to problems in differential geometry and in the topology of manifolds. Examples of such applications include estimation of the size of the moduli space of all Riemannian metrics with positive scalar curvature and questions concerning the rigidity or nonrigidity of a manifold. The principal investigator and his students plan to apply the techniques of quantitative operator K-theory and dynamic complexity to study K-theory of operator algebras, higher index theory, and secondary index theory. The principal investigator and his students also intend to apply quantitative techniques to study the isomorphism conjectures in algebraic K-theory.

在经典几何中，人们研究坐标可交换的几何对象。非对易几何（英语：Noncommutative geometry）是一种数学理论，专门用来研究“几何对象”，其坐标不对易，但在数学和物理学中自然发生。在过去十年左右的时间里，在非对易几何新思想的帮助下，经典几何和拓扑学中长期存在的问题的解决取得了很大的进展。K-理论、高阶指数理论和次级指数理论是非交换几何与经典几何和拓扑之间的桥梁。主要研究者和他的学生计划应用非交换几何方法来研究微分几何和拓扑学中的问题。某些算子代数的K-群是椭圆微分算子的高指数和次指数不变量的容器，在微分几何和流形拓扑学中有重要的应用。这样的应用的例子包括估计的大小的模空间的所有黎曼度量与正标量曲率和问题的刚性或nonrigidity的一个流形。主要研究者和他的学生计划应用定量算子K理论和动态复杂性的技术来研究算子代数的K理论，高级指标理论和二级指标理论。主要研究者和他的学生也打算应用定量技术来研究代数K-理论中的同构代数。

项目成果

Additivity of Higher Rho Invariants and Nonrigidity of Topological Manifolds

• DOI: 10.1002/cpa.21962
• 发表时间: 2016-08
• 期刊: Communications on Pure and Applied Mathematics
• 影响因子: 3
• 作者: S. Weinberger;Zhizhang Xie;Guoliang Yu

Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and $$C^*$$ C ∗ -algebras

动态渐近维数：与动力学、拓扑、粗略几何和 $$C^*$$ C â -代数的关系

• DOI: 10.1007/s00208-016-1395-0
• 发表时间: 2017
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Guentner, Erik;Willett, Rufus;Yu, Guoliang
• 通讯作者: Yu, Guoliang

Persistence approximation property and controlled operator K-theory

• 发表时间: 2014-03
• 期刊: arXiv: Operator Algebras
• 作者: H. Oyono-Oyono-H.-Oyono-Oyono-1402854792;Guoliang Yu

A new index theorem for monomial ideals by resolutions

• DOI: 10.1016/j.jfa.2018.04.003
• 发表时间: 2017-08
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: R. Douglas;M. Jabbari;Xiang Tang;Guoliang Yu

The Novikov conjecture for algebraic K-theory of the group algebra over the ring of Schatten class operators

• DOI: 10.1016/j.aim.2016.11.027
• 发表时间: 2011-06
• 期刊: arXiv: K-Theory and Homology
• 作者: Guoliang Yu