Rigidity in von Neumann Algebras and Higher Rank Groups

冯·诺依曼代数是在20世纪30年代和40年代引入的，部分原因是为了发展量子物理学的数学基础。 冯诺依曼代数已成为一个领域的独立利益与进一步的应用领域，如遍历理论，Voiculescu的自由概率理论，琼斯理论的子因子和平面代数，结理论，和许多其他。冯·诺依曼代数的发展在历史上也与可测动力学的研究密切相关，这些联系最近开始在新发展的刚性现象中重新出现。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。进一步发展刚性反过来又会导致新的见解和这些不同领域之间的联系。发展与冯诺依曼代数的理论一直是遍历理论，并在一个领域的许多结果有重大的应用在其他领域。在过去的十年里，波帕发现了变形/刚性理论，并将变形性（如Haagerup的性质）、自由产物或无界上循环与刚性性质（如性质（T）或谱隙）并列，从而发现了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连接，特别是对连接到深层刚性理论的遍历行动的晶格在更高的秩组发起的Mostow，马古利斯，齐默，和许多其他人。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.