Matrix Approximations, Stability of Groups and Cohomology Invariants

算符代数领域起源于海森堡创立的量子力学的矩阵力学公式，随后由冯·诺伊曼发展。粒子的物理性质被解释为随时间演化的无限矩阵，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，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留的内在性质。首席研究员将研究离散有限维近似，以捕获这些无限维结构的拓扑特性及其稳定性。在另一个方向上，首席研究员将研究算子代数束理论中出现的拓扑不变量。该项目的教育部分致力于培养学生在算子代数领域的能力，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中的拓扑绝缘体理论有直接联系。三个项目涉及算子代数的分析和拓扑方面将被调查。第一个项目的目的是研究离散群在不同情况下相对于算子范数和拓扑障碍的稳定性。第二个项目致力于研究不可服从离散群的有限维逼近性质，重点研究拟对角性作为构造几乎平坦的向量束和携带拓扑信息的群准表示的工具。研究C*-代数连续域的不变量及其在C*-动力系统和高扭曲k理论中的应用。目的是获得一个完整的计算，分类稳定的强自吸收C*-代数的连续场的上同群，作为主要研究者与Pennig共同开发的广义Dixmier-Douady理论的一部分。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.