Approximation properties of groups and operator algebras

这个项目解决了与有限结构的群、图和算子代数逼近有关的几个开放问题。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适应性的技术。cones嵌入问题（CEP）涉及von Neumann代数上迹态的一个基本近似性质。首席研究员计划开发一种代数方法来研究CEP，这是她在最近的几篇论文中开始的。这种方法是基于酉变量上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有限群对离散群的逼近，特别是所谓的sofic逼近。有几个长期存在的猜想被认为是正确的。其中包括群von Neumann代数的CEP、Gottschalk猜想、行列式猜想和Kaplansky的直接有限猜想。此外，目前还没有已知的非sofic群体的例子。该项目的目的之一是研究几类群体的数学近似的性质，以期对这种情况有更大的了解。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，群论的方法是代数许多部分的基础。“可服从群”的概念是由约翰·冯·诺伊曼于1929年提出的，近年来，许多著名的猜想被证明是适用于可服从群的。拟议项目的核心部分是发展首席研究员的最新成果，以确定某些类别的群体是可服从的，或证明他们表现出这种属性的某些较弱形式。在这个过程中，该项目有望加强几个数学领域之间的联系。最后，首席研究员将设法吸引年轻研究人员到该领域，并组织有关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.