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·诺伊曼（Von Neumann）代数在1930年代作为量子力学的数学框架出现。在经典力学中，可以同时观察和测量物理系统的各种特性，例如，其所有组件的位置和速度。这样的属性通常称为可观察物。可观察到可观察到基础系统的函数并形成代数 - 可以添加并乘以它们。在量子力学中，简单的测量已不再可能。从数学上讲，这是由量子系统观察力的代数的非共同性反映的。尽管如此，许多可以使用普通函数进行的操作都具有量子类似物。当前的建议研究从语音的自由概率理论的角度研究了可观察物的非共同代数，该理论将观察物视为随机变量。这导致了非常丰富的理论，它导致了经典对象（例如部分微分方程和布朗运动）的自由概率概括，可以通过受经典信息理论启发的技术进行分析。该项目将通过研究生和本科研究机会来促进人力资源的发展，并将在UCLA Olga Radko endowed Math Circle的主持下为学生提供支持。拟议的研究涉及冯·诺伊曼代数中的几个问题，这些问题是通过自由概率和免费信息方法（包括自由熵理论）来接近的。这包括进一步开发基于PDE的方法在非共同环境中，并加强自由概率与随机矩阵理论之间的联系。在研究方向中，有一个尺寸的概念，该概念基于最佳传输距离的行为，以及免费信息理论技术对von Neumann代数理论的应用。问题的混合，一些来自现有的研究方向以及一些探索新的询问线的混合。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响标准来评估，被视为珍贵的支持。