Dynamics and Operator Algebras

该项目涉及两个数学领域。其中之一是研究分析环境中的各种代数，这些代数最初是由冯·诺依曼发明的，用于研究物理学中的量子力学。另一个领域研究系统在演化过程中的长期行为，例如太阳系在10亿年后的状态，以及更普遍的系统对称性。虽然这两个领域看起来非常不同，但在过去的几年里，人们发现了它们之间的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。这些联系导致了这两个领域的应用。主要研究者旨在加深算子代数和动力系统之间的联系。这包括扭转型不变量之间的连接，涉及第一场和熵不变量从第二场，和研究西尔维斯特秩函数的动机，这两个领域之间的连接的早期调查。首席研究员还将研究嵌入问题，在第二个领域使用平均尺寸和伯努利行动。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。新近建立的包括熵和平均维数在内的sofic群作用不变量理论正在迅速发展。这一研究将使我们更好地理解这一理论。算子代数是研究非交换群的代数作用的有力工具，取代了交换代数在研究交换群的代数作用中的地位。关于挠型不变量和熵之间的联系的项目已经应用于第一理论和这种作用的熵理论。该项目下的发现将导致更多的应用，并加强我们对各种数学现象的理解。该奖项反映了NSF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.