Operator Algebras, Groups, and Topological Invariants

三个项目有关的分析和拓扑方面的算子代数进行调查。算子代数是数学的一个领域，从海森堡发现的量子力学的矩阵力学公式中出现，并由冯·诺依曼进一步发展。在这个理论中使用的新的（量子）变量是满足适当的正则性质的（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/变形的过程，其中几何结构以这样的方式离散化，即重要的拓扑空间关系以数值不变量的形式提取。主要研究者将探索这种变形的理论在更大的背景下的算子代数。其目标是展示和研究新的代数和几何结构，允许特殊变形到有限矩阵，捕捉微妙的拓扑性质。研究的不变量与新材料物理学中出现的不变量密切相关，比如晶体对称的拓扑绝缘体。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。连通性是一个同伦不变的性质，具有重要的持久性。它有重要的后果有关的情况下，投影，有限维近似，如拟对角，和几何实现的K-同调。第二个项目是致力于嵌入到AF-代数。主要研究者将研究与准表示和拓扑障碍准对角性相关的K-理论不变量。对于第三个项目，主要研究人员将在与Ulrich Pennig开发的广义Dixmier-Douady理论的框架内探索强自吸收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.