Classification of von Neumann Algebras: Connections and Applications to C*-algebras, Geometric Group Theory and Continuous Model Theory

冯·诺伊曼代数作为研究粒子物理的数学框架被引入。f·默里和j·冯·诺伊曼在1930-1940年间的开创性工作已经揭示了冯·诺伊曼代数是一种复杂的对象，它表现出非凡丰富的数学结构。渐渐地，他们的研究演变成一个独立的学科，随着时间的推移，引发了强大的数学理论的发展，并为其他科学带来了宝贵的见解，包括物理学（统计力学）、生物学（DNA分子结构）、工程学（手机网络设计）和计算机科学（纠错码、量子信息理论和量子计算）。冯·诺伊曼代数是高度跨学科的，自然产生于简单的数学结构，如对称和行动，经常出现在数学的许多领域。随着时间的推移，他们的研究与动力系统、测量群论以及最近的连续模型理论和几何群论等各个主题保持着密切的联系。该项目探讨了冯·诺伊曼代数与上述领域之间丰富的相互作用的新研究途径。主要目标是通过刚性透镜确定分类冯·诺伊曼代数的新方法——刚性透镜表明，一个数学对象的整个结构只能从该对象非常有限的先验已知信息中揭示出来。该项目为研究生的培训和专业发展提供了充足的机会。该项目延续了项目负责人（PI）之前的努力，旨在扩展群/测度空间冯·诺伊曼代数分类及其在相关领域的应用的知识边界。具体来说，PI在变形/刚性理论与几何群论之间的相互作用上追求新的想法，以推进几个基本问题：(i)识别新的群和代数群特征，这些群和代数群特征完全可以从冯·诺伊曼代数和C*代数结构中识别出来；特别地，找到满足cones刚性猜想的性质(T)群的其他例子；（ii）计算性质(T)群因子和约简群C*-代数的自同态和基群不变量；（iii）发现W*-超刚性作用的新自然例子，这是一个持续有趣的问题，因为它统一了轨道等价和冯·诺伊曼代数中两种极端形式的刚性；(4)揭示区分因子超能力的新不变量，并深入探讨其在连续模型理论中的应用。其中许多项目在本质上是高度跨学科的，这些项目产生的结果有望揭示几何群论、遍历理论、C*代数、模型理论和冯·诺伊曼代数之间的新桥梁。为了促进PI研究生的职业发展，该奖项支持学生访问同行机构，旨在让学生接触不同的专业知识和研究环境。为了提高该领域的知名度并吸引新的人才，PI将继续教授该领域的高级研究生课程并组织学习研讨会。私家侦探亦会继续透过出版刊物、讲座系列、在其他研究机构举行的研讨会和学术报告会，以及在国内和国际会议上应邀发表演讲，传播他的研究成果。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。

Von Neumann algebras were introduced as a mathematical framework to study particle physics. The pioneering works of F. Murray and J. von Neumann from 1930-1940 already revealed that von Neumann algebras are complex objects that display extraordinary rich mathematical structures. Gradually, their study morphed into a stand-alone discipline which, over time, triggered the development of powerful mathematical theories and brought valuable insight to other sciences, including physics (statistical mechanics), biology (DNA molecule structure), engineering (cell phone network design) and computer science (error-correcting codes, theory of quantum information and quantum computing). Von Neumann algebras are highly interdisciplinary, arising naturally from simpler mathematical structures, such as symmetries and actions, often present in many areas of mathematics. Over time, their study maintained close connections with various topics in dynamical systems, measured group theory, and more recently to continuous model theory and geometric group theory. This project investigates new research avenues at the rich interaction between von Neumann algebras and the aforementioned areas. The main goal is to identify new ways of classifying von Neumann algebras through the lens of rigidity - a condition where it is shown that an entire structure of a mathematical object can be unraveled only from very limited a priori known information on that object. The project provides ample opportunities for the training and professional development of graduate students. Continuing prior efforts of the Principal Investigator (PI), this project is aimed at expanding the boundary of knowledge in the classification of group/measure space von Neumann algebras and their applications to related fields. Specifically, the PI pursues new ideas at the interaction between deformation/rigidity theory and geometric group theory to advance several fundamental problems: (i) identify new groups and algebraic group features that are completely recognizable from the von Neumann algebraic and C*-algebraic structure; in particular, find additional examples of property (T) groups satisfying Connes' rigidity conjecture; (ii) compute invariants like the endomorphisms and the fundamental group of property (T) group factors and reduced group C*-algebras; (iii) find new natural examples of W*-superrigid actions, a problem of continued interest as it unifies two extreme forms of rigidity both in orbit equivalence and von Neumann algebras; and (iv) unveil new invariants distinguishing ultrapowers of factors and explore in depth their applications to continuous model theory. Many of these projects are highly interdisciplinary in nature and results arising from these projects are expected to reveal new bridges between geometric group theory, ergodic theory, C*-algebras, model theory, and von Neumann algebras. To enhance the career development of the PI’s graduate students, the award supports a student visiting program to peer institutions aimed at exposing students to different expertise and research environments. To promote the visibility of and to attract new talent to this field, the PI will continue to teach advanced graduate courses and organize learning seminars in the field. The PI will also continue to disseminate his research findings through publications, lecture series, seminar and colloquium talks at other research institutions, as well as through invited talks at national and international conferences.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.