Approximation properties of groups and operator algebras

这个项目解决了几个与有限结构的群、图和算子代数的近似有关的公开问题。该项目旨在推进用于证明康托极小系统的全拓扑群对其他类的顺从性的技术。Connes嵌入问题（CEP）涉及vonNeumann代数上迹态的一个基本逼近性质.首席研究员计划开发一种代数方法CEP，她开始在最近的几篇论文。这种方法是基于酉变量上的非交换多项式的某些代数性质。群论中发展迅速的领域之一是研究离散群的有限群近似，特别是所谓的sofic近似。众所周知，对于sofic群体来说，几个长期存在的猜测是正确的。其中包括CEP群冯诺依曼代数，戈特沙尔克猜想，行列式猜想，Kaplansky的直接有限性猜想。此外，目前还没有已知的非sofic基团的例子。该项目的目的之一是研究几类群体的sofic近似的性质，希望能更好地阐明这种状况。群体和算子代数自然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，群论的方法是代数许多部分的基础。“顺从群”的概念是由约翰·冯·诺依曼在1929年提出的，近年来许多著名的定理被证明对顺从群类是正确的。拟议项目的核心部分是发展主要研究者的最新结果，以确定某些类别的群体是顺从的，或者证明它们表现出这种性质的某些较弱形式。在这个过程中，该项目预计将加强数学的几个领域之间的联系。最后，首席研究员将设法吸引年轻研究人员到该领域，并组织关于该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.