Approximation, deformation-rigidity and classification in II 1 factor framework

II 1 因子框架中的近似、变形刚度和分类

• 批准号: 1400208
• 负责人: Sorin Popa
• 金额: $ 36万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1400208&HistoricalAwards=false
• 关键词: Approximation; deformation; rigidity; classification; II

"Rigidity" in mathematics occurs when objects in a certain class (functions, function algebras, etc.) can be recognized by knowing very little information about them. Regidity results are usually interdisciplinary and can be relevant to many areas of mathematics, with applications to computer science, complexity theory, design of computer networks, and the theory of error-correcting codes. The principal investigator's work in recent years has to do with the study of rigidity in so-called von Neumann algebras. These are algebras of infinite matrices in which the product of two elements (A times B, say) may be different from the product in reverse order (B times A), a fact that reflects Heisenberg's Uncertainty Principle in the quantum mechanics of particle physics. They are related to group theory and ergodic theory, since actions of groups on spaces give rise to a class of von Neumann algebras, now known as "factors." Rigidity in this context occurs when the group of transformations can be recognized by merely knowing its associated factor. During the period 2001-2010, the principal investigator developed a series of techniques to study such phenomena (namely, deformation-rigidity theory) to which he recently added a powerful approximation technique. In this project, he intends to combine all these tools to study rigidity in factors and to tackle the two most famous problems in the area: the Connes approximate embedding conjecture and the free group factor problem. The Connes conjecture predicts that factors can be "simulated on a computer" and has an interesting reformulation in quantum information theory. The project should contribute to the cross-pollination of several areas of mathematics (e.g., free probability, random matrices, group theory, logic) and lead to progress in each of them. The principal investigator's research in rigidity theory has already had considerable impact on these areas, with a large number of research articles and Ph.D. theses sprouting directly from his work. He expects his techniques to have an even broader impact in the future, leading to new developments and solutions to problems in a variety of subjects, with direct and indirect impact in applied mathematics and the aforememtioned areas of computer science. In this project, the principal investigator intends to further the development of deformation-rigidity theory by incorporating into it a new approximation technology known as incremental patching. Using this array of tools, he will continue to work on the classification of algebras arising from groups and their actions on spaces, as well as on the study of rigidity properties of these objects and the calculation of their symmetries. In particular, the principal investigator will attempt new approaches to two famous problems in two-one factors: the Connes approximate embedding conjecture and the free group factor problem. The problems studied in this project are important to both von Neumann algebra theory and the adjacent areas of group theory, ergodic theory, logic (descriptive set theory), free probability, and subfactor theory. Together with his students and collaborators, the principal investigator will make efforts to broaden the scope of deformation-rigidity theory, strengthening its interactions with all these (and possibly other) areas, an activity that should lead to further surprising results of an interdisciplinary character. Also, the principal investigator intends to continue to disseminate his new techniques through summer programs, mini-courses, textbooks, and expository articles, as well as through conferences that he will conduct and organize.

当某个类中的对象（函数，函数代数等）中，可以通过了解很少的信息来识别数学中的“刚度”。 Endisity的结果通常是跨学科的，并且可能与数学的许多领域有关，并适用于计算机科学，复杂性理论，计算机网络的设计以及错误校正代码的理论。近年来，主要研究者的工作与所谓的冯·诺伊曼代数的刚性研究有关。这些是无限矩阵的代数，其中两个元素的乘积（a times b，se）可能以相反的顺序（b times a）不同，这一事实反映了海森伯格在粒子物理学量子力学中的不确定性原理。它们与群体理论和千古理论有关，因为群体对空间的作用产生了一类von Neumann代数，现在称为“因素”。在这种情况下，当可以通过仅了解其相关因素来识别转换组时，就会发生刚性。在2001 - 2010年期间，首席研究者开发了一系列技术来研究这种现象（即变形 - 独立理论），他最近添加了一种强大的近似技术。 在这个项目中，他打算将所有这些工具结合起来，以研究因素中的刚性并解决该地区的两个最著名的问题：Connes近似嵌入猜想和自由组因子问题。 Connes的猜想预测，可以“在计算机上进行模拟”，并在量子信息理论中具有有趣的重新重新进行。 该项目应有助于数学领域的几个领域（例如自由概率，随机矩阵，群体理论，逻辑）的交叉授粉，并导致每个数学领域的进步。主要研究者在刚性理论方面的研究已经对这些领域产生了很大的影响，并拥有大量的研究文章和博士学位。这是直接从他的工作中发芽的。他期望他的技术在未来会产生更大的影响，从而为各种主题的问题提供新的发展和解决方案，并在应用数学和上述计算机科学领域具有直接和间接的影响。在这个项目中，主要研究者打算通过将新的近似技术纳入其中，称为增量贴片，以进一步发展变形 - 戒烟理论。使用这些工具，他将继续致力于由群体及其对空间的行为以及这些对象的刚性属性的研究以及它们对称性的计算的分类。特别是，首席研究员将尝试解决两种因素的两个著名问题的新方法：康涅斯近似嵌入猜想和自由组因子问题。该项目中研究的问题对于von Neumann代数理论以及群体理论，千古理论，逻辑（描述性集理论），自由概率和亚基因子理论的相邻领域都很重要。首席研究人员将与他的学生和合作者一起，努力扩大变形 - 戒烟理论的范围，增强其与所有这些（甚至可能是其他）领域的互动，这项活动应导致跨学科特征的进一步令人惊讶的结果。此外，首席研究人员打算通过夏季计划，迷你场合，教科书和说明性文章以及他将要进行和组织的会议来继续传播他的新技术。