Von Neumann Algebras: Rigidity, Applications to Measurable Dynamics, and Model Theory

冯·诺伊曼代数理论最初是由弗朗西斯·J·默里和约翰·冯·诺伊曼在20世纪30年代和40年代发展起来的，最初是对矩阵代数的广泛推广，通过用矩阵代数编码经典力学和广义相对论的对称性的大致相同的方式来编码量子力学系统的对称性，从而捕捉量子力学的数学复杂性。在接下来的几十年里，人们已经意识到von Neumann代数代表了物理和生物科学中许多研究对象的对称性，导致在统计力学理论、DNA分子结构、纠错码理论、量子信息论和量子计算理论等不同领域中得到应用。这些对称性的基础被称为“因子”。冯·诺依曼代数理论的主要目的是提供一种有效的因素分类，以了解对称性是如何在这些不同的环境中产生的。一种方法是通过所谓的刚性，它表明一小群已知的对称性实际上描述了一个因子的整个、巨大的对称性集。另一个方向是通过用矩阵代数在计算机中模拟因子的能力来解决因子对称性的可计算性，这被称为Connes嵌入问题。过去十年来，由于索林·波帕在21世纪初开创的形变-刚性理论的发展，人们对因子结构的理解有了快速的增长。这些结果和方法还对群的可测动力学、波兰群行动的描述集合论、概率论以及几何群论和粗几何等领域产生了显著的影响。为了发展和扩展这一理论，主要研究人员的研究集中在算子代数、几何和可测群论、李群和遍历理论之间的联系，特别是通过使用主要研究人员和Ionut Chifan在工作中首次研究的几何变形。主要研究人员还对逻辑和模型论技术在算子代数中的应用感兴趣。该项目的目标是实现以下广泛的目标：(1)继续发展一种观点，从中可以使用李群和几何群论的工具来分类群和测度空间von Neumann代数中的刚性现象的新实例；(2)寻找这些技术在可测群论、概率论和C*-代数分类中的应用；(3)进一步探索和发展连续模型理论与von Neumann代数、C*-代数和算子系统理论之间的联系以及在Connes嵌入问题中的应用。首席研究员将继续教授算子代数的高级课程和组织研讨会，与研究生密切合作，培养和培养人才和发现能力。他还将继续在国家和国际两级通过出版物、研讨会和座谈以及会议和讲习班广泛和积极地传播他的研究成果。

The theory of von Neumann algebras was initially developed in the 1930s and 1940s by Francis J. Murray and John von Neumann as a vast generalization of matrix algebra necessary to capture the mathematical complexities of quantum mechanics by encoding the symmetries of the quantum mechanical system in roughly the same way that matrix algebra encodes the symmetries of classical mechanics and general relativity. Over the intervening decades it has been realized that von Neumann algebras represent the symmetries of many objects studied in the physical and biological sciences, leading to applications in diverse fields such as the theory of statistical mechanics, the structure of DNA molecules, the theory of error correcting codes, and the theory of quantum information theory and quantum computing. The building blocks of these symmetries are called "factors." The main goal of the theory of von Neumann algebras is to provide an effective classification of factors to understand how symmetries arise in these various settings. One approach is through so-called rigidity, where it is shown that a small, known group of symmetries actually describes the entire, vast set of symmetries of a factor. Another direction addresses the computability of the symmetries of a factor via the ability to simulate the factor in a computer by matrix algebras, what is known as the Connes embedding problem.The last decade has seen a rapid growth in the understanding of the structure of factors, due to the development of deformation-rigidity theory that was initiated by Sorin Popa in the early 2000s. These results and approaches have additionally had a remarkable impact on the fields of measurable dynamics of groups, descriptive set theory of Polish group actions, probability theory, as well as on geometric group theory and coarse geometry. Working to develop and expand this theory, the principal investigator's research focuses on the connections between operator algebras, geometric and measurable group theory, Lie groups, and ergodic theory, specifically through the use of geometric deformations that were first investigated in work by the principal investigator and Ionut Chifan. The principal investigator is also interested in the application of logic and model-theoretic techniques to operator algebras. The project aims to achieve the following broad objectives: (1) to continue developing a perspective from which tools from Lie groups and geometric group theory can be used to classify the structure of and to demonstrate new examples of rigidity phenomena in the context of group and measure-measure space von Neumann algebras; (2) to find applications of these techniques to measurable group theory, probability theory, and the classification of C*-algebras; (3) to further explore and develop connections between continuous model theory and von Neumann algebras, C*-algebras, and the theory of operator systems with applications to the Connes embedding problem. The principal investigator will continue to teach advanced courses and organize seminars in operator algebras, working closely with graduate students to develop and foster talent and discovery. He will also continue to broadly and actively disseminate his research through publications, seminar and colloquium talks, and conferences and workshops, at both the national and international level.