FRG: Collaborative Research: von Neumann Algebras Associated to Groups Acting on Hyperbolic Spaces

冯诺依曼代数的研究是由F.Murray和J.von Neumann在20世纪30年代和40年代作为理解粒子物理的数学工具而发起的。随后，它成为一门独立的学科，刺激了强大的数学理论的发展，并为物理学(统计力学)、生物学(DNA结构)和工程学(手机网络设计)带来了宝贵的见解。冯·诺依曼代数在本质上是高度跨学科的，因为它们典型地起源于更简单的数学结构，例如经常出现在许多数学领域的对称和作用量。随着时间的推移，他们的研究仍然与动力系统、测量群论和最近的几何群论中的各种主题密切相关。本研究受算子代数与上述领域的丰富相互作用的启发，研究了几个主要的公开问题，探索了群von Neumann代数分类的新视野。这个项目的第一个目标是提出Connes的刚性猜想，这是一个重大的公开问题，它预言ICC性质(T)群可以从它们的von Neumann代数(W*-超刚性)中完全识别出来。PI基于几何群论和形变/刚性理论的新发展，提出了W*-超刚性(T)群的几种自然构造。该项目的第二个目标围绕着对素数II1因子的研究。主要的焦点是理解群论中负曲率的各种表现形式与相应的群因子的素性方面之间的关系。这个项目的结果有望揭示几何群论、遍历理论、随机游动、C*-代数和von Neumann代数之间显著的交叉授粉。私人投资机构打算组织一系列讲习班，旨在促进这些领域的专家之间的合作。为了促进研究生的职业发展，该提案还涉及参与机构之间的学生交流计划，旨在让学生接触不同的专业知识和研究环境。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。

The study of von Neumann algebras was initiated in the 1930s and 1940s by F. Murray and J. von Neumann as a mathematical tool to understand particle physics. Subsequently, it became an independent discipline that has stimulated the development of powerful mathematical theories and bringing valuable insight to physics (statistical mechanics), biology (DNA structure), and engineering (cell phone network design). Von Neumann algebras are highly interdisciplinary in nature as they arise canonically from simpler mathematical structures, such as symmetries and actions, often present in many areas of mathematics. Over time their study remained closely connected with various topics in dynamical systems, measured group theory, and more recently geometric group theory. This project investigates several major open problems inspired by the rich interaction between operator algebras and the aforementioned fields.This research project explores new horizons in the classification of group von Neumann algebras. The first objective of the project is to advance Connes' rigidity conjecture, a major wide-open problem predicting that ICC property (T) groups are completely recognizable from their von Neumann algebras (W*-superrigid). The PIs proposed several natural constructions of W*-superrigid property (T) groups based on new developments in geometric group theory and deformation/rigidity theory. The second objective of the project revolves around the study of prime II1 factors. The main focus is to understand the relationship between various manifestations of negative curvature in group theory and primeness aspects of the corresponding group factor. The results arising from this project are expected to reveal significant cross-pollination between, geometric group theory, ergodic theory, random walks, C*-algebras, and von Neumann algebras. The PIs intend to organize a series of workshops aimed at stimulating collaboration between experts in these fields. To promote the career development of graduate students the proposal also involves a student exchange program between the participating institutions aimed at exposing students to different expertise and research environments.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.