Rigidity in von Neumann Algebras: Connections and Applications to Orbit Equivalence, Geometric Group Theory, and Continuous Model Theory

冯·诺伊曼代数的研究始于理解粒子物理学的努力，但随后成为一门独立的学科，刺激了强大的数学理论的发展，为物理学（统计力学）、生物学（DNA结构）和工程学（手机设计）的其他领域带来了有价值的见解。该项目进一步推进了该领域的研究，并有望揭示新的应用，并为数学的其他领域（遍历论、群论、逻辑和概率论）建立新的桥梁。该项目的活动为参与研究的三名研究生提供了充足的论文主题。本研究项目探索变形/刚性理论的新边界和技术投入，以分类由群产生的冯·诺伊曼代数及其在概率空间上的作用。该项目针对以下具体主题：寻找由冯·诺伊曼代数记忆的群在概率空间上的行动的新的自然例子；构造第一个可以由von Neumann代数重构的性质(T)群的例子；研究由映射类群和自由群的外自同构产生的von Neumann代数的结构；寻找新的不变量对因子超幂进行分类，并深入探索其在连续模型理论中的应用。为了实现这些目标，首席研究员将开发新的方法来实现关于群及其行为的代数、上同调、几何和动态信息到冯·诺伊曼代数的分析框架中。此外，该项目还寻求新的分析工具来计算超功率因子的对称群。该项目产生的结果有望揭示在表示理论、几何群论、轨道等价和逻辑方面的重要新应用和联系。

The study of von Neumann algebras began in efforts to understand particle physics but subsequently became an independent discipline that has stimulated the development of powerful mathematical theories, bringing valuable insight to other areas of physics (statistical mechanics), biology (DNA structure), and engineering (cell-phone design). This project further advances research in this field and is expected to reveal new applications and establish new bridges to other areas in mathematics (ergodic theory, group theory, logic, and probability). The activities of this project provide ample dissertation subject matter for three graduate students who are involved in the research.This research project explores new boundaries and technical inputs in deformation/rigidity theory towards the classification of von Neumann algebras arising from groups and their actions on probability spaces. The project targets the following specific themes: find new natural examples of actions of groups on probability spaces that are remembered by their von Neumann algebras; construct the first examples of property (T) groups that can be reconstructed from their von Neumann algebras; investigate the structure of von Neumann algebras arising from mapping class groups and outer automorphisms of free groups; find new invariants to classify ultrapowers of factors and explore in depth their applications to continuous model theory. To achieve these goals the principal investigator will develop new methods to implement algebraic, cohomological, geometric, and dynamical information about a group and its actions into the analytic framework of von Neumann algebras. In addition, the project pursues new analytic tools towards the calculation of the symmetry groups of ultrapower factors. The results arising from this project are expected to reveal significant new applications and connections to representation theory, geometric group theory, orbit equivalence, and logic.