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

冯·诺依曼代数中的刚性：与轨道等效、几何群论和连续模型理论的联系和应用

• 批准号: 1600688
• 负责人: Ionut Chifan
• 金额: $ 17.45万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600688&HistoricalAwards=false
• 关键词: Rigidity; von; Neumann; Algebras; Connections

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.

冯·诺依曼代数的研究开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这一领域的研究，并有望揭示新的应用，并建立新的桥梁，以其他领域的数学（遍历理论，群论，逻辑和概率）。这个项目的活动提供了充足的论文主题为三个研究生谁是参与research.This研究项目探讨新的边界和技术投入变形/刚性理论对冯诺依曼代数的分类所产生的群体和他们的行动概率空间。该项目的目标是以下具体主题：发现群在概率空间上的作用的新的自然例子，这些例子被它们的冯·诺依曼代数记住;构造第一个可以从它们的冯·诺依曼代数重构的性质（T）群的例子;研究由映射类群和自由群的外自同构产生的冯·诺依曼代数的结构;寻找新的不变量来分类因子的超幂，并深入探索它们在连续模型理论中的应用。为了实现这些目标，主要研究人员将开发新的方法来实现代数，上同调，几何和动力学信息的一组及其行动到冯诺依曼代数的分析框架。此外，该项目还寻求新的分析工具来计算超幂因子的对称群。从这个项目产生的结果预计将揭示显着的新的应用和连接表示理论，几何群论，轨道等价和逻辑。