Rigidity, Cohomology, and Approximate Embeddings in von Neumann Algebra Factors

冯诺依曼代数因子中的刚性、上同调和近似嵌入

• 批准号: 1700344
• 负责人: Sorin Popa
• 金额: $ 18.6万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700344&HistoricalAwards=false
• 关键词: Rigidity; Cohomology; Approximate; Embeddings; von; Rigidity; Cohomology; Approximate; Embeddings; von

The driving force behind many exciting developments in mathematics and its applications in recent years has been the dichotomy between structure (order, rigidity, etc.) and randomness (lack of structure, disorder, coarseness). This project concerns a powerful technique, called deformation-rigidity theory, to study these opposing phenomena in the framework of von Neumann algebras, i.e., algebras of infinite matrices, where the product of two elements, A times B, may be different from the product in reverse order, B times A, a fact that reflects the laws of quantum mechanics in particle physics (Heisenberg's Uncertainty Principle). Von Neumann algebras are also related to group theory and ergodic theory, since transformations of spaces give rise to a special class of such algebras called II1 factors. Rigidity in this context occurs when the group of transformations can be recognized by merely knowing its associated II1 factor (which is a very coarse object). The study of rigidity of factors is an interdisciplinary endeavor and can be relevant to many areas of mathematics. It also has applications to computer science, complexity theory, design computer networks, and the theory of error-correcting codes. This research project aims to deepen understanding in this important area.The principal investigator recently added several new tools to this study, namely an approximation/simulation technique, a representation theory, and a cohomology. In this project, he intends to combine all these tools to further study rigidity-versus-randomness phenomena in II1 factors and to tackle several famous problems in this area: the Connes Approximate Embedding (CAE) conjecture, the sofic group problem, the free group factor problem, the paving problem, and calculations of invariants for group-like objects. It should be noted that the CAE conjecture, which predicts that II1 factors can be "simulated" on a computer, would have striking consequences in quantum information theory.

近年来，许多令人兴奋的数学发展及其应用的驱动力是结构（顺序，刚性等）和随机性（缺乏结构，混乱，粗糙）之间的二分法。 This project concerns a powerful technique, called deformation-rigidity theory, to study these opposing phenomena in the framework of von Neumann algebras, i.e., algebras of infinite matrices, where the product of two elements, A times B, may be different from the product in reverse order, B times A, a fact that reflects the laws of quantum mechanics in particle physics (Heisenberg's Uncertainty Principle).冯·诺伊曼（Von Neumann）代数也与群体理论和千古理论有关，因为空间的变换引起了一类称为II1因子的代数类别。在这种情况下，刚性在这种情况下仅通过知道其相关的II1因子（这是一个非常粗糙的对象）来识别的转换群。对因素的刚性研究是一项跨学科的努力，并且可能与许多数学领域有关。它还对计算机科学，复杂性理论，设计计算机网络以及错误校正代码理论的应用。该研究项目旨在加深对这一重要领域的理解。首席研究者最近在这项研究中添加了几种新工具，即一种近似/仿真技术，代表理论和共同体学。在这个项目中，他打算将所有这些工具结合起来，以进一步研究II1因素中的刚性与随机性现象，并解决该领域的几个著名问题：Connes近似嵌入（CAE）猜想，SOFIC组问题，自由小组因素问题，自由组因子问题，自由群体问题，对小组式对象的不变对象的计算。应当指出的是，CAE猜想预测可以在计算机上“模拟” II1因素，这将在量子信息理论中产生巨大的后果。