Free Information Theory Techniques in von Neumann Algebras

冯诺依曼代数中的自由信息理论技术

• 批准号: 2348633
• 负责人: Dimitri Shlyakhtenko
• 金额: $ 42.1万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348633&HistoricalAwards=false
• 关键词: Free; Information; Theory; Techniques; von

Von Neumann algebras arose in the 1930s as a mathematical framework for quantum mechanics. In classical mechanics it is possible to simultaneously observe and measure various properties of a physical system — for example, the locations and velocities of all of its components. Such properties are often called observables. Observables be viewed as functions of the underlying system and form an algebra — they can be added and multiplied. In quantum mechanics, simultaneous measurements are no longer possible. Mathematically this is reflected by the non-commutativity of the algebra of observables for quantum systems. Nonetheless, many of the operations that can be done with ordinary functions have quantum analogs. The current proposal studies such non-commutative algebras of observables from the angle of Voiculescu’s free probability theory, which treats observables as random variables. This results in an extremely rich theory that leads to free probability generalizations of classical objects such as partial differential equations and Brownian motion, amenable to analysis by techniques inspired by classical information theory. This project will promote human resource development through graduate and undergraduate research opportunities and will support students under the auspices of the UCLA Olga Radko Endowed Math Circle. The proposed research deals with several questions in von Neumann algebras which are approached by free probability and free information methods, including free entropy theory. This includes further developing PDE based methods in the non-commutative context and strengthening the connection between free probability and random matrix theory. Among the research directions is a notion of dimension that is based on the behavior of optimal transportation distance, as well as applications of free information theory techniques to von Neumann algebra theory. The project includes a mixture of problems, some coming from existing research directions and some exploring new lines of inquiry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

冯诺依曼代数作为量子力学的数学框架出现于 20 世纪 30 年代，在经典力学中，可以同时观察和测量物理系统的各种属性，例如，其所有组件的位置和速度。可观测量被视为基础系统的函数并形成代数——在量子力学中，同时测量不再可能。然而，量子系统可观测量代数的非交换性，许多可以用普通函数完成的运算都具有量子类似物，当前的提案从 Voiculescu 自由概率论的角度研究了此类可观测量代数。它将可观测值视为随机变量，这产生了极其丰富的理论，可以对偏微分方程和布朗运动等经典对象进行自由概率推广，并且可以通过技术进行分析。受经典信息论的启发，该项目将通过研究生和本科生的研究机会促进人力资源开发，并将在加州大学洛杉矶分校奥尔加·拉德科捐赠数学圈的支持下支持学生。拟议的研究涉及冯·诺依曼代数中的几个问题。自由概率和自由信息方法，包括自由熵理论，这包括在非交换背景下进一步发展基于偏微分方程的方法，并加强自由概率和随机矩阵理论之间的联系。最佳运输距离的行为，以及自由信息论技术在冯诺依曼代数理论中的应用。该项目包括一系列问题，一些来自现有的研究方向，一些探索新的研究方向。该奖项反映了 NSF 的法定使命。通过使用基金会的智力价值和更广泛的影响审查标准进行评估，并被认为值得支持。