Operator Algebra, Dynamics and Geometry

算子代数、动力学和几何

• 批准号: 1001625
• 负责人: HanFeng Li
• 金额: $ 11.98万
• 依托单位: SUNY at Buffalo
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001625&HistoricalAwards=false
• 关键词: Operator; Algebra; Dynamics; Geometry

The principal investigator will investigate problems in dynamics and geometry in the framework of operator algebras. Sample topics include: 1. Study the relation between entropy of principal algebraic actions of countable groups and Fuglede-Kadison determinant; 2. Develop an operator algebraic approach to entropy of actions of sofic groups; 3. Investigate combinatorial independence in measurable and topological dynamics; 4. Develop a convex analysis approach to the study of noncommutative Choquet boundary; 5. Study quantum isometry of compact quantum metric spaces. In order to get a new mathematical tool to study quantum mechanics, von Neumann introduced operator algebras. The theory of operator algebras has grown into a huge exciting area of modern mathematics, and is related to many other areas of physics and mathematics. Ergodic theory and dynamics arose from studying the long-term behavior of complicated processes. This project will deepen and broaden connections between geometry, dynamics, and operator algebras, and has application in physics. The proposed study of entropy of algebraic actions and actions of sofic groups will enhance our understanding of complicated symmetries. The study of the new interrelation between combinatorics and dynamics has application to the local theory of Banach spaces. The development of noncommutative Choquet boundary has application in operator theory. The study of metric geometry will provide a concrete mathematical foundation for certain statements in the theoretical high-energy physics literature.

主要研究者将在算子代数的框架下研究动力学和几何学问题。 示例主题包括：1.研究可数群主代数作用的熵与Fuglede-Kadison行列式的关系; 2.发展了一种求解sofic群作用熵的算子代数方法; 3.研究可测动力学和拓扑动力学中的组合独立性; 4.发展了一种凸分析方法来研究非对易Choquet边界; 5.研究紧致量子度量空间的量子等距性。 为了给量子力学的研究提供一种新的数学工具，冯·诺依曼引入了算子代数。算子代数理论已经发展成为现代数学中一个巨大的令人兴奋的领域，并且与物理学和数学的许多其他领域有关。遍历理论和动力学起源于研究复杂过程的长期行为。这个项目将加深和扩大几何，动力学和算子代数之间的联系，并在物理学中的应用。对代数作用和sofic群作用的熵的研究，将有助于加深我们对复杂对称性的理解。 组合学和动力学之间新的相互关系的研究已应用到局部理论的Banach空间。非对易Choquet边界的发展在算子理论中有应用。 度量几何的研究将为高能物理学理论文献中的某些陈述提供具体的数学基础。