Approximation properties of groups and operator algebras

这个项目解决了与有限结构逼近群、图和算子代数有关的几个公开问题。该项目旨在将证明Cantor极小系统的完全拓扑群的可修饰性的技术推广到其他类。Connes嵌入问题(CEP)涉及von Neumann代数上迹态的一个基本逼近性质。这位首席研究员计划开发一种代数方法来研究CEP，她在最近的几篇论文中开始了这一方法。这种方法是基于么正变量上非对易多项式的某些代数性质。群论中一个迅速发展的领域是研究离散群与有限群的逼近，特别是所谓的SOFIC逼近。一些长期存在的猜测被认为对SOFIC集团是正确的。其中包括群von Neumann代数的CEP、Gottschalk猜想、行列式猜想和Kaplansky的直接有限性猜想。此外，目前还没有已知的非软性团体的例子。该项目的目的之一是研究几类群的SOFIC逼近的性质，以期更好地揭示这种情况。群和算子代数自然地出现在数学的所有领域，也出现在物理和化学的某些部分。特别是，群论的方法植根于代数的许多部分。“顺从群”的概念是由约翰·冯·诺伊曼在1929年提出的，近年来，许多著名的猜想被证明对顺从群的类是成立的。拟议项目的核心部分是发展首席调查员的最新结果，以确定某些类别的集团是服从的，或证明它们表现出这种性质的某些较弱的形式。在这一过程中，预计该项目将加强数学几个领域之间的联系。最后，首席调查员将设法吸引年轻研究人员到外地工作，并组织关于这一主题的会议和研讨会。

This project addresses several open problems related to approximations of groups, graphs, and operator algebras by finite structures. The project aims to advance the technique that is used to prove amenability of the full topological group of a Cantor minimal system to other classes. The Connes embedding problem (CEP) concerns a fundamental approximation property of tracial states on von Neumann algebras. The principal investigator plans to develop an algebraic approach to CEP that she started in several recent papers. This approach is based on certain algebraic properties of noncommutative polynomials over unitary variables. One of the rapidly developing areas in group theory is a study of approximations of discrete groups by finite groups, in particular, so-called sofic approximations. Several long-standing conjectures are known to be true for sofic groups. Among them are CEP for group von Neumann algebras, Gottschalk's conjecture, the determinant conjecture, and Kaplansky's direct finiteness conjecture. Moreover, there are currently no known examples of nonsofic groups. One of the aims of the project is to study properties of sofic approximations for several classes of groups in the hope of shedding greater light on this state of affairs.Groups and operator algebras arise naturally in all areas of mathematics and also in certain parts of physics and chemistry. In particular, the methods of group theory lie at the roots of many parts of algebra. The notion of an "amenable group" was introduced by John von Neumann in 1929, and in recent years many famous conjectures have been shown to be true for the class of amenable groups. The core part of the proposed project is to develop recent results of the principal investigator in order to establish that certain classes of groups are amenable or to prove that they exhibit certain weaker forms of this property. In the process, the project is expected to strengthen the connections between several areas of mathematics. Finally, the principal investigator will seek to attract young researchers to the field and organize conferences and seminars on the subject.