Dynamics and Operator Algebras

这个项目涉及两个数学领域。其中之一是在解析环境中研究各种代数，这些代数最初是由冯·诺伊曼发明的，用于研究物理学中的量子力学。另一个领域研究演化中的系统的长时间行为，例如10亿年后太阳系的状态，以及更一般的系统对称性。虽然这两个领域看起来非常不同，但在过去的几年里，它们之间已经发现了一些令人惊讶的联系，特别是关于可以用有限对称近似的无限对称。这些联系导致了两个领域的应用。主要研究者的目标是加深算子代数和动力系统之间的联系。这包括涉及第一场的扭转型不变量与来自第二场的熵不变量之间的联系，以及由这两个场之间的联系的早期研究激发的Sylvester秩函数的研究。首席研究员还将研究一个嵌入问题在第二个领域使用平均维数和伯努利动作。这个项目将帮助我们理解各种空间和代数的几何和动力学。新建立的sofic群作用不变量理论，包括熵和平均维，正在迅速发展。这项研究将使我们更好地理解这一理论。算子代数已成为研究非阿贝尔群代数作用的有力工具，取代了交换代数研究阿贝尔群代数作用的工具。关于扭转型不变量与熵之间关系的研究已经应用于此类作用的第一理论和熵理论。在这个项目下的发现将带来更多的应用，并加强我们对各种数学现象的理解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is concerned with two mathematical fields. One of these studies various algebras in the analytic setting, which were originally invented by von Neumann to study quantum mechanics in physics. The other field studies the long time behavior of systems under evolution, such as the status of the solar system after one billion years and, more generally, symmetries of systems. Although these two fields look very different, in the last several years some surprising connections between them have been discovered, especially about infinite symmetries that can be approximated by finite symmetries. These connections have led to applications to both fields. The principal investigator aims to deepen the connections between operator algebras and dynamical systems. This includes the connection between the torsion-type invariant involving the first field and entropy invariant from the second field, and the study of Sylvester rank functions motivated by earlier investigations of connections between these two fields. The principal investigator will also study an embedding problem in the second field using mean dimension and Bernoulli actions. This project will help us understand the geometry and dynamics of various spaces and algebras. The newly-founded theory of invariants for sofic group actions, including both entropy and mean dimension, is developing rapidly. This research will give us a better understanding of this theory. Operator algebras have been found to be a powerful tool in the study of algebraic actions of non-abelian groups, replacing the commutative algebra tool in the study of algebraic actions of abelian groups. The project on the connection between the torsion-type invariant and entropy already has applications to the first theory and the entropy theory of such actions. Discoveries made under this project would lead to more applications and strengthen our understanding of a variety of mathematical phenomena.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.