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Soficity, Dynamics, and Operator Algebras

Soficity、动力学和算子代数

基本信息

批准号：
1600717
负责人：
HanFeng Li
金额：
$ 14.4万
依托单位：
SUNY at Buffalo
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600717&HistoricalAwards=false
关键词：
Soficity Dynamics Operator Algebras

项目摘要

This project concerns two mathematical fields. One of them studies the shape of geometric objects such as spheres using a tool that was originally invented by John von Neumann to study quantum mechanics in physics. The other field studies the long-time behavior of systems under evolution (e.g., the state of the solar system one billion years in the future) and, more generally, symmetries of systems. Although these two fields might appear to be quite different, in the last several years some surprising connections between them have been discovered, especially about those "infinite" symmetries that can be approximated by finite symmetries. These connections have led to applications in both fields. The principal investigator plans to deepen and broaden such connections. The principal investigator seeks to deepen the recently-found connection between the theory of invariants for manifolds via group von Neumann algebras and the theory of dynamical systems by introducing various relative invariants. This includes the connection between von Neumann-Lueck rank in the former theory and mean topological dimension in the latter and the connection between torsion-type invariant in the first theory and entropy in the second. The project will shed new light on the geometry and dynamics of various spaces and algebras. The newly-created theory of invariants for sofic group actions, including both entropy and mean dimension, is developing rapidly. The project will provides a better understanding of this theory. It will also deepen the interplay between different fields of mathematics. Operator algebras have become a powerful tool in the study of algebraic actions of nonabelian groups, replacing the commutative algebra tool that is used in the study of algebraic actions of abelian groups. The parts of the project on the connection between mean dimension and von Neumann-Lueck rank and on the connection between entropy and torsion-type invariants have already found applications to both the mean dimension and entropy theory of such group actions. Discoveries made in this project will lead to more applications and will strengthen one's understanding of a variety of mathematical phenomena.

这个项目涉及两个数学领域。其中一个研究几何物体的形状，如球体，使用的工具最初是由约翰·冯·诺依曼发明的，用于研究物理学中的量子力学。另一个领域研究进化下系统的长期行为（例如，未来十亿年太阳系的状态），更普遍的是，系统的对称性。虽然这两个领域可能看起来完全不同，但在过去几年中，人们发现了它们之间的一些令人惊讶的联系，特别是关于那些可以用有限对称近似的“无限”对称。这些联系导致了在这两个领域的应用。首席研究员计划加深和扩大这种联系。主要研究者试图通过引入各种相对不变量来加深最近发现的流形不变量理论与动力系统理论之间的联系。这包括前者的von Neumann-Lueck秩与后者的平均拓扑维数之间的联系，以及前者的挠型不变量与后者的熵之间的联系。该项目将为各种空间和代数的几何和动力学提供新的见解。近年来，新兴的包括熵和平均维数在内的sofic群作用不变量理论得到了迅速发展。该项目将提供一个更好的理解这一理论。它还将加深不同数学领域之间的相互作用。算子代数已经成为研究非交换群代数作用的有力工具，取代了研究交换群代数作用的工具。关于平均维数和冯·诺依曼-吕克秩之间的联系以及熵和挠型不变量之间的联系的项目部分已经应用于这种群作用的平均维数和熵理论。在这个项目中的发现将导致更多的应用，并将加强一个人对各种数学现象的理解。