Free Gibbs States: Von Neumann Algebras, Random Matrices, and Subfactors

自由吉布斯态：冯诺依曼代数、随机矩阵和子因子

• 批准号: 1500035
• 负责人: Dimitri Shlyakhtenko
• 金额: $ 42万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500035&HistoricalAwards=false
• 关键词: Free; Gibbs; States; Von; Neumann

This project intends to develop mathematical tools that are in common use in classical analysis (e.g., notions of regularity, ideas from the theory of partial differential equations) in the noncommutative context that arises in so-called free probability theory. The underlying mathematics is extremely rich and has many connections with quantum physics. Progress in development of these tools is expected to have a significant impact on von Neumann algebra theory and random matrix theory, as well as on subfactor theory. The principal aim fo the project is to understand the structure of a class of noncommutative states called "free Gibbs states." These objects appear in Voiculescu's free probability theory as analogs of classical-probability Gibbs distributions. Besides free probability, their properties turn out to be important in von Neumann algebra theory, random matrix theory, and subfactor theory. The main new tool useful for understanding these states is that of free transport. Developed by A. Guoinnet and the principal investigator, this tool has already been proved to be useful for answering questions such as isomorphism of certain von Neumann algebras and computations of standard invariants of certain subfactors. More development and understanding is needed. The project will study and develop noncommutative techniques in partial differential equations that would enable one to construct transport maps, as well as to study critical phenomena. Along the way, the principal investigator will investigate questions of regularity of noncommutative maps.

这个项目的目的是开发在经典分析中常用的数学工具(例如，正则性概念，偏微分方程式理论中的思想)，在所谓的自由概率理论中出现的非对易背景下。基本的数学知识极其丰富，与量子物理有许多联系。这些工具的开发进展预计将对冯·诺依曼代数理论和随机矩阵理论以及子因素理论产生重大影响。该项目的主要目的是了解一类被称为“自由吉布斯态”的非对易状态的结构。这些天体作为经典概率吉布斯分布的类似物出现在沃库列斯库的自由概率理论中。除了自由概率外，它们的性质在von Neumann代数理论、随机矩阵理论和子因子理论中被证明是重要的。有助于理解这些状态的主要新工具是自由运输。这个工具是由A.Guoinnet和首席研究员开发的，已经被证明对回答某些von Neumann代数的同构和某些子因子的标准不变量的计算等问题是有用的。需要更多的发展和理解。该项目将研究和开发偏微分方程中的非对易技术，使人们能够构建交通地图，以及研究临界现象。在此过程中，主要研究者将研究非对易映射的正则性问题。