Rigidity in von Neumann Algebras and Higher Rank Groups

冯·诺伊曼代数在20世纪30年代和40年代被引入，部分原因是作为发展量子物理学数学基础的工具。冯·诺伊曼代数从此成为一个独立的兴趣领域，并进一步应用于遍历理论、Voiculescu的自由概率论、Jones的子因子和平面代数理论、结理论等领域。冯·诺伊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，这些联系最近在新发展的刚性现象中开始重新出现。对这种刚性现象的研究导致了冯·诺伊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见解和联系。与冯·诺伊曼代数理论一起发展的是遍历理论，一个领域的许多结果在另一个领域有重要的应用。在过去的十年中，随着Popa的变形/刚性理论的发现，这种合作的重新出现，该理论将哈格鲁普性质、自由积或无界环等可变形性特性与性质(T)或谱间隙等刚性特性并列，允许人们在两种现象发生的情况下发现冯·诺伊曼代数中的隐藏结构。本项目将更全面地研究这些联系，特别关注与Mostow、Margulis、Zimmer等人提出的高阶群中晶格遍历作用的深度刚性理论的联系。该奖项反映了美国国家科学基金会的法定使命，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。

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.