C*-algebras, Groups, and Topological Invariants

对物理定律的研究导致了一种精细的数学框架的发展，其中数值函数的代数被一种基于无限矩阵的理论所包含。无限矩阵结构，如算符代数，能够描述基本粒子的相互作用和量子物理学背后的对称性。本项目是将分析和几何的基本思想和技术扩展到算子(矩阵)代数的非交换背景下的共同努力的一部分。该项目将调查算子代数的有限维近似的存在，该算子代数的有限维近似足够丰富，以捕获初始数据的关键特征。传递到有限维模型很重要，因为它提供了访问具体的数值不变量的途径。不可避免的是，有限维模型的结构将比无限维模型的结构不那么对称。精确对称性的丧失是近似有限元模型的一个基本特征。它反映了主要研究人员致力于以数字形式量化的微妙的拓扑属性。由此产生的不变量与支持新材料物理的数学中出现的不变量有关，例如具有晶体对称性的拓扑绝缘体。研究涉及具有分析和拓扑方面的算子代数中的两个项目。第一个项目致力于群、C*-代数及其有限维矩阵模型的逼近。它将研究离散群和群C*-代数到矩阵代数的变形的存在，由这些变形产生的不变量，以及在这个过程中遇到的潜在的拓扑障碍。这项研究的一个基本目标是更好地理解拟对角性的拓扑性质，拟对角性是一种有限维逼近性质，在C*-代数的结构理论中起着核心作用。第二个项目涉及C*-代数的连续场理论及其在一般空间上的推广。虽然C*-代数的本原谱是一个基本不变量，但它有一个重要的限制。它只对理想是如何粘合在一起的进行了较低层次的描述。该项目的第一个目标是开发可计算的不变量，以捕捉理想和局部商之间的K理论相互作用。第二个目标是由Ulrich Pennig和主要研究者进一步发展强自吸收C*-代数的连续场的广义DixmierDouady理论。

The study of physical laws has led to the development of a refined mathematical framework where the algebra of numerical functions is subsumed by a theory based on infinite matrices. Infinite matriceal structures such as operator algebras are able to depict and model the interactions of elementary particles and the symmetries underlying quantum physics. The present project is part of a concerted effort to extend fundamental ideas and techniques of analysis and geometry to the noncommutative context of operator (matrix) algebras. The project will investigate the existence of finite-dimensional approximations of operator algebras that are sufficiently rich to capture key features of the initial data. Passing to finite-dimensional models is important since it gives access to concrete numerical invariants. Inevitably, the structure of the finite-dimensional models will be somewhat less symmetric than their infinite-dimensional counterparts. The loss of exact symmetries is an essential feature of the approximant finite models. It reflects subtle topological properties that the principal investigator aims to quantify in numerical form. The resulting invariants are related to those arising in the mathematics underpinning the physics of novel materials, such as topological insulators with crystalline symmetry.The research concerns two projects in operators algebras that have analytical and topological aspects. The first project is devoted to groups, C*-algebras, and their approximations by finite-dimensional matrix models. It will examine the existence of deformations of discrete groups and group C*-algebras into matrix algebras, the invariants that arise from these deformations, and potential topological obstructions encountered in the process. One underlying goal of the investigation is to develop a better understanding of the topological nature of quasidiagonality, a finite-dimensional approximation property that plays a central role in the structure theory of C*-algebras. The second project concerns the theory of continuous fields of C*-algebras and their generalizations to C*-algebras over general spaces. While the primitive spectrum of a C*-algebra is a fundamental invariant, it has one important limitation. It gives only a low-level description of how the ideals are glued together. A first goal of the project is to develop computable invariants that capture K-theoretical interactions between ideals and local quotients. A second goal is to further develop the generalized Dixmier-Douady theory of continuous fields of strongly self-absorbing C*-algebras due to Ulrich Pennig and the principal investigator.