Operator Algebras, Groups, and Topological Invariants

关于算子代数的分析和拓扑方面的三个项目将被研究。算符代数是由海森伯格发现并由冯·诺伊曼进一步发展起来的量子力学的矩阵力学公式的一个数学领域。这个理论中使用的新的(量子)变量是满足适当的正则性的(可能是无限的)矩阵的集合。它们被组织在各种代数结构中，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。从函数到矩阵的传递通常涉及量化/变形过程，在该过程中，几何结构被离散化，重要的拓扑空间关系以数值不变量的形式被提取。主要研究人员将在算子代数的更大背景下探索这种变形的理论。其目的是展示和研究新的代数和几何结构，这些结构允许到有限矩阵的特殊变形，从而捕捉到微妙的拓扑属性。要研究的不变量与新材料物理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通C*-代数类。连通性是一种具有显著持久性的同伦不变性质。它具有与无投影、有限维近似(如拟对角性)和K-同调的几何实现有关的重要结果。第二个项目致力于研究AF-代数的可嵌入性。主要研究人员将研究与拟表示有关的K-理论不变量和准对角性的拓扑障碍。对于第三个项目，主要研究者将在Ulrich Pennig发展的广义DixmierDouady理论的框架下，探索强自吸收C*-代数的连续场的特征类。

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.