Entropy and Boundary Methods in von Neumann Algebras

冯诺依曼代数中的熵和边界方法

• 批准号: 2350049
• 负责人: Srivatsav Kunnawalkam Elayavalli
• 金额: $ 14.51万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing 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=2350049&HistoricalAwards=false
• 关键词: Entropy; Boundary; Methods; von; Neumann

The theory of von Neumann algebras, originating in the 1930's as a mathematical foundation for quantum physics, has since evolved into a beautifully rich subfield of modern functional analysis. Studying the precise structure of von Neumann algebras is rewarding for many reasons, as they appear naturally in diverse areas of modern mathematics such as dynamical systems, ergodic theory, analytic and geometric group theory, continuous model theory, topology, and knot theory. They also continue to be intimately involved in a variety of fields across science and engineering, including quantum physics, quantum computation, cryptography, and algorithmic complexity. The PI will focus on developing a new horizon for research on structural properties of von Neumann algebras, by combining entropy (quantitative) and boundary (qualitative) methods, with applications to various fundamental open questions. This project will also contribute to US workforce development through diversity initiatives and mentoring of graduate students and early career researchers. In this project, the PI will develop two new research directions in the classification theory of finite von Neumann algebras: applications of Voiculescu's free entropy theory to the structure of free products and of ultrapowers of von Neumann algebras; the small at infinity compactification and structure of von Neumann algebras arising from relatively properly proximal groups. This will involve a delicate study of structure, rigidity and indecomposability properties via innovative interplays between three distinct successful approaches: Voiculescu's free entropy theory, Popa's deformation rigidity theory, Ozawa's theory of small at infinity boundaries and amenable actions.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·诺依曼代数理论起源于 1930 年代，作为量子物理学的数学基础，现已发展成为现代泛函分析的一个美丽而丰富的子领域。研究冯·诺依曼代数的精确结构有很多好处，因为它们自然地出现在现代数学的不同领域，如动力系统、遍历理论、解析和几何群论、连续模型理论、拓扑和纽结理论。 他们还继续密切参与科学和工程的各个领域，包括量子物理学、量子计算、密码学和算法复杂性。 PI 将致力于通过结合熵（定量）和边界（定性）方法，以及对各种基本开放问题的应用，为冯诺依曼代数的结构特性研究开辟新视野。该项目还将通过多元化举措以及对研究生和早期职业研究人员的指导，为美国劳动力发展做出贡献。在该项目中，PI将在有限冯·诺依曼代数分类理论中发展两个新的研究方向：Voiculescu的自由熵理论在自由积结构和冯·诺依曼代数超幂结构中的应用；由相对适当的近端群产生的冯·诺依曼代数的无穷远小紧化和结构。这将涉及通过三种不同成功方法之间的创新相互作用对结构、刚性和不可分解性特性进行细致的研究：Voiculescu 的自由熵理论、Popa 的变形刚性理论、Ozawa 的无限边界小理论和顺从的作用。该奖项反映了 NSF 的法定使命和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，该项目被认为值得支持。