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

冯诺依曼代数：刚性、可测量动力学的应用和模型理论

基本信息

批准号：
1600857
负责人：
Thomas Sinclair
金额：
$ 18万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600857&HistoricalAwards=false
关键词：
Von Neumann Algebras Rigidity Applications

项目摘要

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.

冯·诺依曼代数理论最初是由弗朗西斯·J·默里和约翰·冯·诺依曼在20世纪30年代和40年代发展起来的，作为矩阵代数的一个广泛推广，通过编码量子力学系统的对称性，以大致相同的方式编码经典力学和广义相对论的对称性，来捕捉量子力学的数学复杂性。在随后的几十年里，人们已经意识到冯·诺依曼代数代表了物理和生物科学中许多研究对象的对称性，导致了统计力学理论，DNA分子结构，纠错码理论，量子信息理论和量子计算理论等不同领域的应用。这些对称性的组成部分被称为“因子”。冯·诺依曼代数理论的主要目标是提供一个有效的因素分类，以了解对称性如何在这些不同的设置中出现。一种方法是通过所谓的刚性，其中表明，一个小的，已知的对称性组实际上描述了整个，一个因素的巨大的对称性集。另一个方向是通过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。由于Sorin Popa在21世纪初提出的变形刚性理论的发展，过去十年对因子结构的理解迅速增长。这些结果和方法还产生了显着的影响领域的可测动力学的团体，描述集理论的波兰组行动，概率论，以及对几何群论和粗几何。致力于发展和扩展这一理论，首席研究员的研究重点是算子代数，几何和可测群论，李群和遍历理论之间的联系，特别是通过使用几何变形，首先由首席研究员和Ionut Chifan在工作中进行了研究。主要研究者也对逻辑和模型理论技术在算子代数中的应用感兴趣。该项目旨在实现以下广泛目标：（1）继续发展一种观点，从这种观点出发，可以使用李群和几何群论的工具对群和测度-测度空间冯诺依曼代数的结构进行分类，并展示刚性现象的新例子;（2）找到这些技巧在可测群理论、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，并将其应用于Connes嵌入问题。首席研究员将继续教授高级课程，并组织算子代数研讨会，与研究生密切合作，培养和培养人才和发现。他还将继续通过出版物、研讨会和学术讨论会以及国家和国际一级的会议和讲习班，广泛和积极地传播他的研究成果。