Rigidity in von Neumann Algebras and Higher Rank Groups

冯·诺依曼代数和高阶群中的刚性

基本信息

批准号：
1801125
负责人：
Jesse Peterson
金额：
$ 22.34万
依托单位：
Vanderbilt University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801125&HistoricalAwards=false
关键词：
Rigidity von Neumann Algebras Higher

项目摘要

Von Neumann algebras were introduced in the 1930's and 40's in part as a tool for developing a mathematical foundation for quantum physics. Von Neumann algebras have since become a field of independent interest with further applications to areas such as ergodic theory, Voiculescu's free probability theory, Jones' theory of subfactors and planar algebras, knot theory, and many others. The development of von Neumann algebras has also historically been closely connected to the study of measurable dynamics and these connections have recently begun to reemerge in the presence of newly developed rigidity phenomenon. The investigation of this rigidity phenomenon has since led to new connections between von Neumann algebras and other areas of mathematics. Furthering the development of rigidity will in turn lead to new insights and connections among these various fields.Developing alongside the theory of von Neumann algebras has been ergodic theory, and many results in one field has had major applications in the other. A reemergence of this collaboration has occurred in the last ten years with Popa's discovery of deformation/rigidity theory, which juxtaposes deformability properties such as Haagerup's property, free products, or unbounded cocycles, with rigidity properties such as property (T) or spectral gap, allowing one to discover hidden structure in a von Neumann algebra in the case when both types of phenomena occur. This project will investigate more fully these connections, focusing specifically on connections to the deep rigidity theory for ergodic actions of lattices in higher rank groups initiated by Mostow, Margulis, Zimmer, and many others.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.

冯·诺伊曼（Von Neumann）代数是在1930年代和40年代引入的，部分是为量子物理学发展数学基础的工具。冯·诺伊曼（von Neumann）代数从那以后成为一个独立的兴趣领域，并在诸如ergodic理论，Voiculescu的自由概率理论，琼斯的子因子和平面代数，结理论，结理论等领域的领域进一步应用。从历史上看，冯·诺伊曼代数的发展也与可测量动态的研究密切相关，这些连接最近在存在新开发的刚性现象的情况下开始重新出现。此后，对这种僵化现象的调查导致了冯·诺伊曼代数与其他数学领域之间的新联系。进一步发展刚性的发展将导致这些各个领域之间的新见解和联系。与von Neumann代数的理论一起发展是奇特的理论，而在一个领域中，许多结果在另一个领域中都有主要的应用。在过去的十年中，Popa发现了变形/刚度理论，这种合作的重新出现发生在过去的十年中，该理论将诸如Haagerup的特性，免费产品，免费产品或无限制的合作属性等变形性属性并置，并具有属性（T）或光谱差异等刚性属性，从而可以在Von Neumann Algebra中发现隐藏的结构，从而发现了均具有验证剂的情况。该项目将更全面地调查这些联系，专门针对与莫·莫斯托（Mostow），玛格利斯（Margulis），齐默默（Margulis），齐默默（Zimmer）和许多其他许多人在高级组中的深度刚性理论的联系。该奖项反映了NSF的法定任务，并通过使用该基金会的知识优点和广泛的影响来评估NSF的法定任务，并被认为是值得的支持。