Deformation/rigidity theory in von Neumann algebras and ergodic theory

冯诺依曼代数中的变形/刚性理论和遍历理论

基本信息

批准号：
1201565
负责人：
Jesse Peterson
金额：
$ 15.56万
依托单位：
Vanderbilt University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2012
资助国家：
美国
起止时间：
2012-06-01 至 2015-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201565&HistoricalAwards=false
关键词：
Deformation rigidity theory von Neumann

项目摘要

Deformation/rigidity theory, initiated by Sorin Popa in the early 2000's, has been extremely successful over the last decade in answering a number of longstanding problems in von Neumann algebras and ergodic theory. The juxtaposition between deformability properties such as Haagerup's property, free products, or unbounded cocycles, with rigidity properties such as property (T) or spectral gap allows one to discover hidden structure in a von Neumann algebra where both types of phenomena occur. This has led to new insight into the structural properties of these von Neumann algebras, and in turn has found applications to other areas such as measured group theory, or the theory of L2-invariants. The Principal Investigator will continue to investigate and develop these ideas, focusing on their close connection to aspects of ergodic theory. 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 of mathematics 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.

变形/刚度理论由 Sorin Popa 在 2000 年代初期提出，在过去十年中在解决冯·诺依曼代数和遍历理论中的许多长期存在的问题方面取得了极大的成功。哈格鲁普性质、自由积或无界余环等变形性质与性质 (T) 或谱间隙等刚性性质之间的并置允许人们发现冯诺依曼代数中两种现象都发生的隐藏结构。这使得人们对这些冯·诺依曼代数的结构特性有了新的认识，并反过来在其他领域找到了应用，例如可测群论或 L2 不变量理论。首席研究员将继续研究和发展这些想法，重点关注它们与遍历理论各个方面的密切联系。冯·诺依曼代数于 20 世纪 30 年代和 40 年代引入，部分是作为开发量子物理数学基础的工具。冯·诺依曼代数从此成为一个独立的研究领域，并进一步应用于数学领域，如遍历理论、沃伊库列斯库的自由概率论、琼斯的子因子理论和平面代数、纽结理论等。冯·诺依曼代数的发展在历史上也与可测量动力学的研究密切相关，并且这些联系最近在新发展的刚性现象的存在下开始重新出现。对这种刚性现象的研究从此在冯·诺依曼代数和其他数学领域之间建立了新的联系。刚性的进一步发展反过来会带来这些不同领域之间的新见解和联系。