Entropy and Boundary Methods in von Neumann Algebras

冯·诺伊曼代数理论起源于20世纪30年代，是量子物理学的数学基础，后来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·诺伊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各个领域，如动力系统、遍历理论、解析和几何群论、连续模型论、拓扑学和结论。他们还继续密切参与科学和工程的各种领域，包括量子物理、量子计算、密码学和算法复杂性。PI将通过结合熵（定量）和边界（定性）方法，并将其应用于各种基本开放问题，致力于为冯·诺依曼代数的结构特性研究开辟新的视野。该项目还将通过多样性倡议和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。在本项目中，PI将在有限冯诺依曼代数的分类理论中开辟两个新的研究方向：Voiculescu的自由熵理论在冯诺依曼代数的自由积和超幂结构中的应用；由相对适当的近端群产生的冯诺依曼代数的无穷小紧化和结构。这将涉及通过三种不同的成功方法之间的创新相互作用，对结构，刚度和不可分解性特性进行细致的研究：Voiculescu的自由熵理论，Popa的变形刚度理论，Ozawa的无限小边界理论和可调节作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.