Operator Algebras, Groups, and Topological Invariants

将研究与算子代数的分析和拓扑方面有关的三个项目。算子代数是一个数学领域，源于海森堡发现的量子力学的矩阵力学公式，并由冯·诺依曼进一步发展。该理论中使用的新（量子）变量是满足适当正则性的（可能是无限的）矩阵的集合。它们被组织成各种代数结构，称为算子代数，可以实现为作用于希尔伯特空间的线性算子。从函数到矩阵的转变通常涉及量化/变形的过程，其中几何结构被离散化，从而重要的拓扑空间关系以数值不变量的形式被提炼出来。首席研究员将在更大的算子代数背景下探索这种变形的理论。目标是展示和研究新类别的代数和几何结构，这些结构允许在有限矩阵中进行特殊变形，从而捕获微妙的拓扑特性。要研究的不变量与新材料物理学中出现的不变量密切相关，例如具有晶体对称性的拓扑绝缘体。用更专业的术语来说，第一个项目的目的是探索连接 C* 代数类。连接性是同伦不变的属性，具有显着的持久性。它对于缺乏投影、有限维近似（例如准对角性）和 K-同调的几何实现具有重要的影响。第二个项目致力于 AF 代数的嵌入性。主要研究者将研究与准表示相关的 K 理论不变量和准对角性的拓扑障碍。对于第三个项目，首席研究员将在与 Ulrich Pennig 共同开发的广义 Dixmier-Douady 理论框架内探索强自吸 C* 代数连续域的特征类。

Three projects concerned with analytical and topological aspects of operator algebras are to be investigated. Operator algebras is an area of mathematics that emerged from the matrix mechanics formulation of quantum mechanics discovered by Heisenberg and further developed by von Neumann. The new (quantum) variables used in this theory are ensembles of (possibly infinite) matrices that satisfy suitable regularity properties. They are organized in various algebraic structures, called operator algebras, that can realized as linear operators acting on Hilbert spaces. The passage from functions to matrices often involves a process of quantization/deformation in which geometric structures are discretized in such a way that important topological spatial relations are distilled in the form of numerical invariants. The principal investigator will explore the theory of such deformations in the larger context of operator algebras. The goal is to exhibit and study new classes of algebraic and geometric structures which admit special deformations into finite matrices that capture subtle topological properties. The invariants that are to be investigated are intimately related to those that arise in the physics of novel materials such as topological insulators with crystalline symmetry.In more technical terms, the purpose of the first project is to explore the class of connective C*-algebras. Connectivity is a homotopy invariant property with significant permanence properties. It has important consequences pertaining to absence of projections, finite dimensional approximations such as quasidiagonality, and geometric realizations of K-homology. The second project is devoted to embeddability into AF-algebras. The principal investigator will investigate K-theory invariants associated to quasi-representations and topological obstructions to quasidiagonality. For the third project, the principal investigator will explore characteristic classes of continuous fields of strongly self-absorbing C*-algebras in the framework of the generalized Dixmier-Douady theory that were developed with Ulrich Pennig.