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Dynamics and Operator Algebras

动力学和算子代数

基本信息

批准号：
1900746
负责人：
HanFeng Li
金额：
$ 22.5万
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：
2019
资助国家：
美国
起止时间：
2019-06-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900746&HistoricalAwards=false
关键词：
Dynamics Operator Algebras

项目摘要

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.

这个项目涉及两个数学领域。其中之一是研究解析环境下的各种代数，它最初是由冯·诺伊曼发明的，目的是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期行为，如太阳系在10亿年后的状态，以及更一般的系统对称性。尽管这两个领域看起来非常不同，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，特别是关于可以由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。主要研究人员致力于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型不变量和来自第二场的熵不变量之间的联系，以及早期对这两个场之间联系的研究所推动的西尔维斯特秩函数的研究。主要研究人员还将使用平均维度和Bernoulli作用量研究第二域中的嵌入问题。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。这项研究将使我们对这一理论有更好的理解。算子代数是研究非阿贝尔群的代数作用的有力工具，在研究阿贝尔群的代数作用时取代了交换代数工具。关于扭型不变量和熵之间联系的项目已经应用于第一种理论和这类行为的熵理论。在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智力优势和更广泛的影响审查标准进行评估，被认为值得支持。