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

该项目旨在开发经典分析中常用的数学工具（例如，正则性的概念，来自偏微分方程理论的思想）在非对易的背景下，在所谓的自由概率论中出现。 基础数学非常丰富，与量子物理学有许多联系。 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”的非对易态的结构。这些物体出现在Voiculescu的自由概率理论中，作为经典概率吉布斯分布的类似物。 除了自由概率，它们的性质在冯·诺依曼代数理论、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输运。 由A. Guoinnet和首席研究员，这个工具已经被证明是有用的回答问题，如同构的某些冯诺依曼代数和计算标准不变量的某些子因子。 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交换技术，使人们能够构建运输图，以及研究关键现象。 沿着的方式，主要研究人员将调查的问题，正规性的非交换地图。

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.