Soficity, Dynamics, and Operator Algebras

这个项目涉及两个数学领域。其中一种是使用约翰·冯·诺伊曼最初发明的一种工具来研究球体等几何物体的形状，该工具最初是为了研究物理学中的量子力学而发明的。另一个领域研究演化中的系统的长期行为(例如，未来10亿年太阳系的状态)，以及更一般的系统对称性。尽管这两个领域可能看起来完全不同，但在过去的几年里，人们已经发现了它们之间一些令人惊讶的联系，特别是关于那些可以用有限对称来近似的“无限”对称。这些联系导致了这两个领域的应用。首席调查员计划加深和扩大这种联系。主要研究人员试图通过引入各种相对不变量来深化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之间的关系，以及前者的扭型不变量与后者的熵之间的关系。该项目将为各种空间和代数的几何和动力学提供新的线索。新创立的SOFIC群作用不变量理论，包括熵和平均维，正在迅速发展。该项目将提供对这一理论的更好理解。它还将加深不同数学领域之间的相互作用。算子代数已成为研究非交换群代数作用的有力工具，取代了用于研究交换群代数作用的交换代数工具。该项目中关于平均维度和von Neumann-Lueck等级之间的联系以及关于熵和扭型不变量之间的联系的部分已经在这类群作用的平均维度和熵理论中找到了应用。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。

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.