Deformation and Rigidity for Groups, Actions, and von Neumann Algebras

群、作用和冯诺依曼代数的变形和刚度

• 批准号: 1101718
• 负责人: Sorin Popa
• 金额: $ 45万
• 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: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101718&HistoricalAwards=false
• 关键词: Deformation; Rigidity; Groups; Actions; von

During the last decade, von Neumann algebras of group actions have become a center stage for studying a variety of rigidity phenomena and a playground for various areas of mathematics to interact: operator algebras, group theory (measured, geometric, arithmetic, etc.), ergodic theory, orbit equivalence relations, and descriptive set theory, to name a few. During the period 2001-2010, the principal investigator has developed a series of techniques for studying rigidity in this framework, which is now called deformation/rigidity theory. This led to a large number of striking rigidity results in both the von Neumann algebra and orbit ergodic theory settings, and to the solution of many long-standing problems. The principal investigator's techniques and results naturally entail some exciting new directions of research and problems in all these areas. They also provide new tools for approaching some of the classical (hitherto "intractable") problems in von Neumann algebras, such as: the Connes rigidity conjecture; the structure and classification of free group factors; superrigidity properties of algebras arising from groups and their actions. In this project the principal investigator, with his students and collaborators, will systematically investigate these directions. He intends to deepen his interaction with the areas of group theory, ergodic theory, and descriptive set theory, using operator algebra techniques. This activity should lead to further surprising results and solutions to problems in all these areas. The principal investigator expects the framework of factors to continue to play a crucial role in this interplay between diverse areas of mathematics. "Rigidity" in mathematics occurs when objects in a certain class (functions, function algebras, etc.) can be recognized without having very much initial information about them. Results of this type are usually interdisciplinary and can be relevant to many areas of mathematics. They can also have interesting 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 focused on the study of rigidity in the class of objects known as von Neumann algebras. These are algebras of infinite matrices, wherein the outcome of the multiplication of two elements A and B may be different depending on the order in the product (i.e., AB may be different from BA). Theses algebras where introduced by von Neumann in the 1920s in his effort to provide a rigorous approach to quantum mechanics in particle physics. He related the algebras, at the outset, with such areas of mathematics as group theory and ergodic theory by noticing that actions of groups on so-called probability measure spaces give rise to a remarkable class of von Neumann algebras. Rigidity in this context occurs when the group action can be recognized by merely knowing the associated von Neumann algebra. The principal investigator has recently developed a completely new set of techniques for studying such phenomena, creating a framework that is now called deformation/rigidity theory. He has obtained a number of surprising and intrinsically beautiful results that create a bridge from von Neumann algebras to rigidity in other areas of mathematics and lead to deep interdisciplinary activity. The problems that the principal investigator intends to work on over the next three years are increasingly ambitious, having to do with famous unsolved problems about the classification of factors arising from "rigid groups" and "free groups." The projects are important to both von Neumann algebra theory and to the adjacent mathematical areas of group theory, ergodic theory, logic (descriptive set theory), free probability, and subfactor theory. The proposal should further contribute to the cross-pollination of these areas and to substantial progress in each of them. The principal investigator's work in rigidity theory has already had considerable impact in many areas, with a large number of research articles and Ph.D. theses sprouting directly from it. 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. He also expects this research to have direct and indirect impact in applied mathematics and in the aforementioned areas of computer science.

在过去的十年中，团体行动的冯·诺伊曼代数已成为研究各种刚性现象的中心舞台，并且是各种数学领域相互作用的游乐场：操作员代数，群体理论（测量，几何，几何，算术等），奇特的理论，Orgodic理论，Orbit Greita，Orbit等效性关系和参见性关系和得分设置理论，以下几个。在2001 - 2010年期间，首席研究员开发了一系列用于研究该框架刚性的技术，现在称为变形/刚性理论。这导致了大量惊人的刚性导致冯·诺伊曼代数和轨道千古理论设置，以及解决许多长期存在的问题的解决方案。首席研究者的技术和结果自然需要在所有这些领域的研究和问题方面进行一些令人兴奋的新方向。他们还提供了新的工具，以解决冯·诺伊曼代数中的一些经典（迄今“棘手”）问题，例如：connes僵化猜想；自由小组因素的结构和分类；群体及其作用产生的代数的超级汇率特性。在这个项目中，主要研究人员及其学生和合作者将系统地研究这些指示。他打算使用操作员代数技术来加深与群体理论，千古理论和描述性理论领域的互动。这项活动应为所有这些领域的问题带来进一步的令人惊讶的结果和解决方案。首席研究人员预计，因素的框架将继续在数学领域之间的这种相互作用中发挥关键作用。数学中的“刚度”发生在某个类中的对象（函数，功能代数等）中，而无需大量有关它们的初始信息。这种类型的结果通常是跨学科的，并且可能与数学的许多领域有关。他们还可以在计算机科学，复杂性理论，计算机网络设计以及错误校正代码理论方面具有有趣的应用。近年来，主要研究者的工作集中在被称为von Neumann代数的对象类别中的刚性研究。这些是无限矩阵的代数，其中两个元素A和B的繁殖结果可能会有所不同，具体取决于产品中的顺序（即，AB可能与BA不同）。冯·诺伊曼（Von Neumann）在1920年代引入的代数为代数提供了一种严格的粒子物理学方法。他一开始就将代数与诸如群体理论和千古理论之类的数学领域相关联，该领域通过注意到群体对所谓概率度量空间的作用会引起一类显着的von Neumann代数。在这种情况下，在仅知道相关的von Neumann代数来识别小组行动时，就会发生刚性。主要研究者最近开发了一套全新的技术来研究这种现象，创建了一个现在称为变形/刚性理论的框架。他获得了许多令人惊讶和内在的美丽结果，这些结果从冯·诺伊曼代数到其他数学领域的刚性创造了一座桥，并导致了深层的跨学科活动。首席研究人员打算在未来三年中进行的问题越来越雄心勃勃，这与著名的未解决问题有关“僵化群体”和“自由群体”的因素的分类。这些项目对von Neumann代数理论以及群体理论，千古理论，逻辑（描述性集理论），自由概率和子因素理论的相邻数学领域都很重要。该提案应进一步有助于这些领域的交叉授粉，并在每个地区的实质性进展中取得实质性进展。主要研究者在刚性理论方面的工作已经在许多领域产生了相当大的影响，并拥有大量的研究文章和博士学位。这些直接从中发芽。他希望他的技术在将来会产生更大的影响，从而为各种主题中的问题提供新的发展和解决方案。他还希望这项研究对应用数学和上述计算机科学领域有直接和间接的影响。