Operator Algebra, Dynamics and Geometry

主要研究人员将在算子代数的框架内研究动力学和几何问题。样本课题包括：1.研究可数群的主代数作用的熵与Fuglede-Kadison行列式之间的关系；2.建立SOFIC群作用的熵的算子代数方法；3.研究可测动力学和拓扑动力学中的组合独立性；4.建立非对易Choket边界的凸分析方法；5.研究紧量子度量空间的量子等距.为了获得研究量子力学的新的数学工具，冯·诺伊曼引入了算符代数。算符代数理论已经发展成为现代数学中一个令人兴奋的巨大领域，并与物理和数学的许多其他领域有关。遍历理论和动力学源于对复杂过程的长期行为的研究。这个项目将深化和拓宽几何、动力学和算子代数之间的联系，并在物理学中有应用。对代数作用和SOFIC群作用的熵的研究将加深我们对复杂对称性的理解。对组合学和动力学之间新的相互关系的研究应用于Banach空间的局部理论。非对易Choquite边界的发展在算子理论中有一定的应用。对度规几何的研究将为理论高能物理文献中的某些陈述提供具体的数学基础。

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.