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，群作用的冯·诺依曼代数已成为研究各种刚性现象的中心舞台和各个数学领域相互作用的游乐场：算子代数、群论（测量、几何、算术等）、遍历理论、轨道等效关系和描述性集合论等等。 2001-2010年期间，主要研究者在此框架下开发了一系列研究刚性的技术，现在称为变形/刚性理论。这导致了冯·诺依曼代数和轨道遍历理论设置中的大量惊人的刚性结果，并解决了许多长期存在的问题。主要研究者的技术和结果自然会带来所有这些领域中一些令人兴奋的新研究方向和问题。它们还提供了解决冯诺依曼代数中一些经典（迄今为止“棘手”）问题的新工具，例如：康内斯刚性猜想；自由群体因素的结构和分类；由群及其作用产生的代数的超刚性性质。在这个项目中，首席研究员将与他的学生和合作者一起系统地研究这些方向。他打算利用算子代数技术加深与群论、遍历理论和描述性集合论领域的互动。这项活动应该会带来更多令人惊讶的结果并解决所有这些领域的问题。首席研究员预计因素框架将继续在数学不同领域之间的相互作用中发挥至关重要的作用。当某一类对象（函数、函数代数等）可以在没有太多初始信息的情况下被识别时，数学中的“刚性”就会出现。此类结果通常是跨学科的，并且可能与数学的许多领域相关。它们还可以在计算机科学、复杂性理论、计算机网络设计和纠错码理论中产生有趣的应用。首席研究员近年来的工作重点是冯·诺依曼代数类对象的刚性研究。这些是无限矩阵的代数，其中两个元素 A 和 B 相乘的结果可能会有所不同，具体取决于乘积的顺序（即 AB 可能与 BA 不同）。这些代数由冯·诺依曼在 20 年代引入，旨在为粒子物理学中的量子力学提供严格的方法。他一开始就将代数与群论和遍历理论等数学领域联系起来，注意到群在所谓的概率测度空间上的作用产生了一类值得注意的冯·诺依曼代数。当仅通过了解相关的冯·诺依曼代数就可以识别群体行为时，就会出现这种情况下的刚性。首席研究员最近开发了一套全新的技术来研究此类现象，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且本质上美丽的结果，这些结果架起了从冯·诺依曼代数到其他数学领域的刚性的桥梁，并导致了深入的跨学科活动。首席研究员打算在未来三年内解决的问题越来越雄心勃勃，涉及有关“刚性群体”和“自由群体”产生的因素分类的著名未解决问题。这些项目对于冯·诺依曼代数理论以及群论、遍历理论、逻辑（描述性集合论）、自由概率和子因子理论等邻近数学领域都很重要。该提案应进一步促进这些领域的交叉授粉，并推动每个领域取得实质性进展。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，发表了大量的研究论文和博士学位。论文直接从中萌芽。他预计他的技术将在未来产生更广泛的影响，从而为各种学科的问题带来新的发展和解决方案。他还预计这项研究将对应用数学和上述计算机科学领域产生直接和间接的影响。