Entropy and Boundary Methods in von Neumann Algebras

冯·诺依曼代数理论起源于20世纪30年代的S，是量子物理的数学基础，后来发展成为现代泛函分析的一个非常丰富的子领域。研究von Neumann代数的精确结构是有意义的，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、拓扑学和纽结理论。他们还继续密切参与科学和工程领域的各个领域，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，并应用于各种基本的开放问题，为研究von Neumann代数的结构性质开辟新的天地。该项目还将通过多样性倡议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。在这个项目中，PI将在有限的von Neumann代数的分类理论中发展两个新的研究方向：Voular escu的自由熵理论在von Neumann代数的自由积结构和超幂结构中的应用；von Neumann代数的无穷小紧化和von Neumann代数的结构产生于相对真的近邻群。这将涉及对结构、刚性和不可分解属性的精细研究，通过三种不同的成功方法之间的创新相互作用：沃库列斯库的自由熵理论、波帕的变形刚性理论、小泽的无限边界小理论和服从行动。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.