Operator Algebras, Groups, and Topological Invariants

算子代数、群和拓扑不变量

基本信息

批准号：
1700086
负责人：
Marius Dadarlat
金额：
$ 18万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-15 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700086&HistoricalAwards=false
关键词：
Operator Algebras Groups Topological Invariants

项目摘要

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.

将研究有关操作员代数的分析和拓扑方面的三个项目。操作员代数是一个数学领域，它来自海森伯格发现的量子力学的基质力学制定，并由冯·诺伊曼（Von Neumann）进一步开发。该理论中使用的新（量子）变量是（可能是无限）矩阵的集合，可满足适当的规律性特性。它们是在各种代数结构中组织的，称为操作员代数，这些结构可以实现为作用于希尔伯特空间的线性操作员。从函数到矩阵的段落通常涉及一个量化/变形的过程，在这种过程中，几何结构被离散化，以使重要的拓扑空间关系以数值不变的形式提炼。主要研究者将探索在较大的操作员代数背景下这种变形的理论。目的是展示和研究新的代数和几何结构，这些结构将特殊变形置于有限的矩阵中，以捕获微妙的拓扑特性。要研究的不变剂与新型材料的物理学（例如具有晶体对称性的拓扑绝缘子）中出现的不变性密切相关。在更多的技术术语中，第一个项目的目的是探索连接c* - 代理的类别。连接性是具有显着永久特性的同质性不变属性。它具有与缺乏预测，有限维近似值（如准经相结合性）和k-总体几何实现等有限维度近似有关的重要后果。第二个项目致力于嵌入到AF-Elgebras中。首席研究者将调查与准代表和准式拓扑相关的K理论不变性。对于第三个项目，主要研究人员将探索在用Ulrich Pennig开发的广义dixmier-douady理论框架中，强烈自我吸收C*代数的连续领域的特征类别。