Matrix Approximations, Stability of Groups and Cohomology Invariants

矩阵近似、群稳定性和上同调不变量

• 批准号: 2247334
• 负责人: Marius Dadarlat
• 金额: $ 26.95万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247334&HistoricalAwards=false
• 关键词: Matrix; Approximations; Stability; Groups; Cohomology

The field of operator algebras emerged from the matrix mechanics formulation of quantum mechanics created by Heisenberg and developed subsequently by von Neumann. Physical properties of particles are interpreted as infinite matrices which evolve in time and can be organized as algebraic structures of linear operators acting on Hilbert spaces. Just like geometric spaces, operator algebras may feature important symmetries corresponding to intrinsic properties that are preserved under groups of transformations. The principal investigator will study discrete finite-dimensional approximations that capture topological properties of these infinite-dimensional structures and their stability properties. In a different direction the principal investigator will study topological invariants arising in the bundle theory of operator algebras. An educational component of the project is devoted to the training of students in an area of operator algebras that has direct connections to group stability and testability problems in computer science and the theory of topological insulators from solid state physics. Three projects concerned with analytical and topological aspects of operator algebras will be investigated. The purpose of the first project is to study the stability of discrete groups with respect to the operator norm and topological obstructions to group stability in various contexts. The second project is devoted to finite-dimensional approximation properties of non-amenable discrete groups, with a focus on quasidiagonality as a tool in the construction of almost flat vector bundles and group quasi-representations that carry topological information. The third project is concerned with invariants of continuous fields of C*-algebras and their applications to C*-dynamical systems and higher twisted K-theory. The aim is to obtain a complete calculation of the cohomology groups that classify the continuous fields of stable strongly self-absorbing C*-algebras as part of the generalized Dixmier-Douady theory that the principal investigator has developed with Pennig.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.

算子代数的领域来自海森伯格创建的量子力学的基质力学公式，随后由冯·诺伊曼（Von Neumann）开发。 颗粒的物理特性被解释为无限矩阵，它们会随着时间的流逝而进化，可以组织为作用于希尔伯特空间的线性算子的代数结构。就像几何空间一样，操作员代数可能具有与内在特性相对应的重要对称性，这些属性保留在转换组下。主要研究者将研究离散的有限维近似值，以捕获这些无限维结构的拓扑特性及其稳定性。在不同的方向上，主要研究者将研究在操作员代数束理论中产生的拓扑不变性。 该项目的一个教育组成部分致力于在操作员代数方面对学生进行培训，该领域与计算机科学中的组稳定性和可检验性问题以及固态物理学的拓扑绝缘体理论有直接联系。 将研究与操作员代数的分析和拓扑方面有关的三个项目。 第一个项目的目的是研究离散基团在操作员规范方面的稳定性以及在各种情况下对群体稳定性的拓扑障碍。第二个项目致力于非无分离心组的有限维近似特性，重点是准双节，作为构建几乎平坦的矢量捆绑包和构建群体准代理的工具，这些工具携带拓扑信息。 第三个项目涉及C* - 代数连续领域的不变式及其应用于C*-Dyanical Systems和更高扭曲的K理论。目的是获得对共同体的完整计算，这些群体将稳定的强烈自我吸收的c* - 代数分类为普遍的dixmier-douady理论的一部分，该理论是，该奖项与Pennig一起开发的普遍性dixmier-douady理论。该奖项颁发了该奖项，反映了NSF的法定任务，并通过评估了Infestia crigia critia corit and Founlitia croits and Founditia infortial and Founditial的支持者。