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

冯诺依曼代数的分类：与 C* 代数、几何群论和连续模型理论的联系和应用

• 批准号: 2154637
• 负责人: Ionut Chifan
• 金额: $ 24.63万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154637&HistoricalAwards=false
• 关键词: Classification; von; Neumann; Algebras; Connections

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.

冯·诺依曼代数被引入作为研究粒子物理的数学框架。F. Murray和J. von Neumann在1930-1940年已经揭示了冯·诺依曼代数是一种复杂的对象，具有非常丰富的数学结构。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有价值的见解，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·诺依曼代数是高度跨学科的，自然产生于更简单的数学结构，如对称和作用，经常出现在许多数学领域。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中的各种主题保持着密切的联系。本计画探讨冯诺依曼代数与上述领域之间丰富互动的新研究途径。主要目标是通过刚性的透镜来确定对冯·诺依曼代数进行分类的新方法--在刚性的条件下，数学对象的整个结构只能从非常有限的关于该对象的先验已知信息中解开。该项目为研究生的培训和专业发展提供了充足的机会。该项目延续了首席研究员（PI）之前的努力，旨在扩大群/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在形变/刚性理论和几何群论之间的相互作用方面追求新的想法，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von Neumann代数和C*-代数结构中识别出来;特别是，找到满足Connes刚性猜想的性质（T）群的其他例子;（ii）计算不变量，如自同态和性质（T）群因子的基本群和约化群C*-代数;（iii）找到W*-超刚性作用的新的自然例子，这是一个持续感兴趣的问题，因为它统一了轨道等价和冯·诺依曼代数中的两种极端刚性形式;和（iv）揭示新的不变量区分超幂的因素，并深入探讨其应用于连续模型理论。许多这些项目是高度跨学科的性质和结果所产生的这些项目预计将揭示几何群论，遍历理论，C*-代数，模型理论和冯诺依曼代数之间的新桥梁。为了加强PI研究生的职业发展，该奖项支持学生访问计划，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将继续教授高级研究生课程并组织该领域的学习研讨会。PI还将继续通过出版物、系列讲座、在其他研究机构举办的研讨会和座谈会以及在国家和国际会议上的特邀演讲来传播他的研究成果。该奖项反映了NSF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。