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.

该项目旨在开发经典分析中常用的数学工具（例如，正则性概念，偏微分方程理论的思想），在所谓的自由概率论中出现的非交换背景下。其基础数学极其丰富，与量子物理学有许多联系。这些工具的发展进展有望对冯·诺依曼代数理论和随机矩阵理论以及子因子理论产生重大影响。该项目的主要目的是了解一类称为“自由吉布斯态”的非交换态的结构。这些对象在Voiculescu的自由概率论中作为经典概率吉布斯分布的类似物出现。除了自由概率外，它们的性质在冯·诺伊曼代数理论、随机矩阵理论和子因子理论中也很重要。对理解这些状态有用的主要新工具是自由运输。由A. Guoinnet和首席研究员开发的这个工具已经被证明对回答诸如某些冯·诺伊曼代数的同构和某些子因子的标准不变量的计算等问题很有用。需要更多的发展和理解。该项目将研究和发展偏微分方程中的非交换技术，使人们能够构建运输图，以及研究关键现象。在此过程中，首席研究员将研究非交换映射的正则性问题。