Operator Algebras, Groups, and Topological Invariants

三个项目涉及算子代数的分析和拓扑方面的研究。算符代数是一个数学领域，从海森堡发现的量子力学的矩阵力学公式中出现，并由冯·诺伊曼进一步发展。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。从函数到矩阵的过渡通常涉及到一个量化/变形的过程，在这个过程中，几何结构被离散化，重要的拓扑空间关系被提炼成数值不变量的形式。首席研究员将在算子代数的大背景下探索这种变形的理论。目标是展示和研究新的代数和几何结构类别，这些结构允许特殊变形为有限矩阵，从而捕获微妙的拓扑特性。所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一类连接C*-代数。连通性是一个具有显著持久性的同伦不变性质。它在缺乏投影、有限维近似（如拟对角性）和k -同调的几何实现方面具有重要的意义。第二个项目致力于嵌入af -代数。主要研究者将研究与拟表示和拟对角的拓扑障碍相关的k理论不变量。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.