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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·诺依曼代数在1930年的S和40年的S被引入，部分原因是作为一种工具，为量子物理奠定了数学基础。自那以后，von Neumann代数已经成为一个独立的研究领域，并进一步应用于遍历理论、Voulescu的自由概率理论、Jones的子因子和平面代数理论、纽结理论等许多领域。Von Neumann代数的发展历史上也一直与可测动力学的研究密切相关，最近随着新发展的刚性现象的出现，这些联系开始重新出现。从那时起，对这种刚性现象的研究导致了冯·诺依曼代数和其他数学领域之间的新联系。刚性的进一步发展将反过来导致这些不同领域之间的新的见解和联系。与von Neumann代数理论一起发展的一直是遍历理论，一个领域的许多结果在另一个领域具有重要的应用。在过去的十年里，随着Popa的变形/刚性理论的发现，这种合作再次出现，该理论将可变形性质(如Haagerup性质、自由积或无界上循环)与刚性性质(如性质(T)或谱间隙)并列在一起，使人们能够在这两种现象都发生的情况下发现冯·诺依曼代数中的隐藏结构。这个项目将更全面地调查这些联系，特别关注Mostow，Marguis，Zimmer和许多其他人发起的较高等级群中晶格遍历行为的深度刚性理论的联系。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。