Derivations, quantum Dirichlet forms, and deformation/rigidity theory in von Neumann algebras

冯诺依曼代数中的导数、量子狄利克雷形式和变形/刚性理论

• 批准号: 0901510
• 负责人: Jesse Peterson
• 金额: $ 12.43万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901510&HistoricalAwards=false
• 关键词: Derivations; quantum; Dirichlet; forms; deformation

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The theory of von Neumann algebras is a non-commutative integration theory which was first developed by F. Murray and J. von Neumann as a tool for understanding representation theory for groups as well as providing a mathematical framework for quantum mechanics. Recently the appearance of new rigidity phenomena in von Neumann algebras has led to deeper understanding and to the solutions to many longstanding problems both in von Neumann algebras and in other areas such as orbit equivalence ergodic theory and measurable equivalence relations. There has also emerged recently a connection between deformation/rigidity theory, free probability, group cohomology, and derivations/quantum Dirichlet forms on von Neumann algebras. It is the intent of this author to investigate these new connections in order to gain more understanding of the relationship which is developing, and to use this understanding in order to approach some deep problems from new perspectives.

该奖项根据 2009 年《美国复苏和再投资法案》（公法 111-5）提供资金。冯·诺依曼代数理论是一种非交换积分理论，最初由 F. Murray 和 J. von Neumann 提出，作为理解群表示论的工具以及为量子力学提供数学框架。最近，冯·诺依曼代数中新的刚性现象的出现导致了对冯·诺依曼代数和其他领域（例如轨道等效遍历理论和可测量等价关系）中许多长期存在的问题的更深入的理解和解决。最近还出现了变形/刚性理论、自由概率、群上同调和冯·诺依曼代数上的导数/量子狄利克雷形式之间的联系。作者的目的是研究这些新的联系，以便更好地理解正在发展的关系，并利用这种理解从新的角度解决一些深层问题。