Rigidity in von Neumann Algebras; Connections and Applications to Orbit Equivalence and Geometric Group Theory

在Popa的变形/刚性理论的一般背景下，并以首席研究员之前的研究结果为基础，本项目的目标是寻找与概率空间上群作用相关的冯·诺伊曼代数的新刚性结果。该项目围绕以下几个主要问题展开：(1)寻找更多“奇异”群在概率空间上的作用的新例子，这些例子可以从它们的冯·诺伊曼代数完全重构；(2)找到导致具有唯一Cartan子代数的von Neumann代数的群作用的其他例子；(3)获得遍历理论（特别是等价关系的轨道等价和结构性质）和概率论（图上的渗流）的新应用；(4)深化与几何群论和表征理论的联系。首席研究员计划通过进一步完善他以前的技术，并继续扩展他和他的合作者带来的上同调和几何群论视角来研究冯·诺伊曼代数的刚性来实现这些目标。他希望这些技术能够揭示理论的新方面，从而使这些领域之间的相互作用更加富有成效。冯·诺伊曼代数的研究是在20世纪30年代由默里和冯·诺伊曼发起的，作为研究量子力学的工具，它逐渐演变成一门独立的学科。它还为强大的数学理论的发展奠定了基础，这些理论最终为物理学（统计力学）、生物学（DNA结构）和工程学（手机设计）等领域带来了有价值的见解。该项目继续研究冯·诺依曼代数中的刚性现象，预计将产生新的应用，并为其他活跃的数学研究领域（概率论、遍历理论和群论中的几何方面）建立新的桥梁。在更广泛的范围内，首席研究员预计刚性现象在数学谱之外的应用将继续揭示自己，特别是在工程和计算机科学（例如，纠错码）。该项目将为刚刚进入这个非常活跃的研究领域的研究生或年轻研究人员提供许多发展机会。虽然项目的主题主要集中在先进趋势上，但它也包含了大量的研究方向和开放问题，可以成为博士论文的优秀主题。首席研究员目前正在教授这一主题的两个学期的入门课程，目的是吸引新的年轻人才到该地区。他将继续通过出版物、系列讲座、研讨会和学术讨论来传播他的研究成果，并促进与数学等相关领域的联系。

Within the general context of Popa's deformation/rigidity theory and building on principal investigator's prior research results, the goal of this project is to find new rigidity results for von Neumann algebras associated with group actions on probability spaces. The project revolves around the following main problems: (1) find new examples of actions of more "exotic" groups on probability spaces that can be completely reconstructed from their von Neumann algebras; (2) find additional examples of group actions that lead to von Neumann algebras with unique Cartan subalgebras; (3) obtain new applications to ergodic theory (particularly orbit equivalence and structural properties for equivalence relations) and probability theory (percolation on graphs); and (4) deepen the connections with geometric group theory and representation theory. The principal investigator plans to achieve these goals by further refining his previous techniques and continuing to expand the cohomological and geometric group theory perspective that he and his coauthors have brought to the study of rigidity in von Neumann algebras. He expects these techniques to reveal new aspects of the theory that will enable an even more fruitful interplay between these fields.The study of von Neumann algebras was initiated in the 1930s by Murray and von Neumann as a tool to study quantum mechanics, and it has progressively morphed into a stand-alone discipline. It also created a basis for the development of powerful mathematical theories that ultimately brought valuable insight to areas of physics (statistical mechanics), biology (DNA structure), and engineering (cell-phone design). This project, which continues the study of rigidity phenomena in von Neumann algebras, is expected to generate novel applications and to establish new bridges to other active research areas in mathematics (probability, ergodic theory, and geometric aspects in group theory). On a broader scale, the principal investigator anticipates that applications of rigidity phenomena outside the mathematical spectrum will continue to unveil themselves, particularly in engineering and computer science (e.g., error-correcting codes). The project will offer numerous development opportunities for graduate students or young researchers just entering this very active research field. While the topic of the project is focused heavily on advanced trends, it also contains substantial research directions and open problems that can be excellent subjects for Ph.D. theses. The principal investigator is currently teaching a two-semester introductory course on this subject with the purpose of attracting new young talent to the area. He will continue to disseminate his findings from the project through publications, lecture series, and seminar and colloquium talks, and to promote the connections with related areas of mathematics and beyond.