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.

算子代数领域出现于海森堡所创立的量子力学的矩阵力学公式，并由冯·诺依曼随后发展。 粒子的物理性质被解释为随时间演化的无限矩阵，并且可以被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，算子代数可能具有重要的对称性，这些对称性对应于在变换群下保持的内在性质。主要研究人员将研究离散有限维近似，捕获这些无限维结构的拓扑性质及其稳定性。在一个不同的方向，主要研究人员将研究拓扑不变量所产生的束理论的算子代数。 该项目的一个教育部分致力于在算子代数领域培训学生，该领域与计算机科学中的群稳定性和可测试性问题以及固态物理学中的拓扑绝缘体理论有直接联系。 三个项目有关的分析和拓扑方面的算子代数将进行调查。 第一个项目的目的是研究离散群的稳定性，相对于运营商规范和拓扑障碍，在各种情况下的群体稳定性。第二个项目是致力于有限维近似性质的非顺从离散群，重点是准对角作为一种工具，在建设几乎平坦的向量丛和群准表示进行拓扑信息。 第三个项目是关于C*-代数连续域的不变量及其在C*-动力系统和高阶扭K-理论中的应用。其目的是获得一个完整的计算的上同调群，分类的连续领域的稳定强自吸收C*-代数的广义Dixmier-Douady理论的一部分，主要研究者已开发与Pennig。这个奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的智力价值和更广泛的影响审查标准。