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.

对von Neumann代数的研究始于理解粒子物理学的努力，但随后成为一项独立的学科，刺激了强大的数学理论的发展，为其他物理学（统计力学），生物学（DNA结构）（DNA结构）和工程（工程手机设计）带来了宝贵的见解。该项目进一步促进了该领域的研究，并有望揭示新的应用，并为数学其他领域建立新的桥梁（Ergodic理论，群体理论，逻辑和概率）。该项目的活动为参与研究的三名研究生提供了充分的论文主题。该研究项目探讨了变形/刚性理论中的新边界和技术投入，以分类由群组及其对概率空间的行为及其行动产生的von Neumann代数分类。该项目针对以下特定主题：找到群体对冯·诺伊曼代数记住的概率空间的新自然示例；构建可以从其von Neumann代数重建的财产（t）组的第一个示例；研究由自由群体的绘制阶级组和外部自动形态绘制的von Neumann代数的结构；找到新的不变性来对因素的超能力进行分类，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，主要研究者将开发新的方法来实施有关群体及其在von Neumann代数分析框架中的组合，几何和动态信息。此外，该项目还为计算超能因素的对称群体提供了新的分析工具。该项目产生的结果有望揭示与表示理论，几何群体理论，轨道等效性和逻辑的重要新应用和联系。