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·诺依曼代数已经成为研究各种刚性现象的中心舞台和各种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、轨道等价关系和描述集合论，仅举几例。在2001-2010年期间，主要研究者开发了一系列技术，用于研究该框架中的刚度，现在称为变形/刚度理论。这导致了大量惊人的刚性结果在冯诺依曼代数和轨道遍历理论的设置，并解决了许多长期存在的问题。首席研究员的技术和结果自然会在所有这些领域带来一些令人兴奋的新研究方向和问题。它们还提供了新的工具， （迄今为止“棘手”）的问题，冯诺依曼代数，如：康纳斯刚性猜想;结构和分类的自由群因子; superrigidity性质的代数所产生的群体和他们的行动。在这个项目中，首席研究员，与他的学生和合作者，将系统地调查这些方向。他打算加深他的互动与领域的群论，遍历理论和描述集理论，使用算子代数技术。这一活动应导致进一步取得令人惊讶的成果，并解决所有这些领域的问题。首席研究员预计，因素框架将继续在不同数学领域之间的相互作用中发挥至关重要的作用。数学中的“刚性”发生在某个类中的对象（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出来。这种类型的结果通常是跨学科的，可以与数学的许多领域。它们也可以在计算机科学、复杂性理论、计算机网络设计和纠错码理论中有有趣的应用。主要研究者的工作，近年来一直集中在研究刚性的一类对象被称为冯诺依曼代数。这些是无限矩阵的代数，其中两个元素A和B相乘的结果可以根据乘积中的阶而不同（即，AB可以不同于BA）。论文代数介绍了冯诺依曼在20世纪20年代在他的努力，以提供一个严格的办法，量子力学的粒子物理。他有关的代数，在一开始，这些领域的数学作为群论和遍历理论注意到，行动的群体对所谓的概率措施空间产生了显着的一类冯诺依曼代数。在这种情况下，当只需要知道相关的冯·诺依曼代数就可以识别群作用时，刚性就出现了。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。他已经获得了一些令人惊讶的和本质上美丽的结果，创造了一个桥梁，从冯诺依曼代数刚性在其他领域的数学，并导致深入的跨学科活动。主要研究者打算在未来三年内研究的问题越来越雄心勃勃，必须与著名的未解决的问题有关，即“刚性群体”和“自由群体”的因素分类。“该项目是重要的冯诺依曼代数理论和相邻的数学领域的群论，遍历理论，逻辑（描述集理论），自由概率，和子因子理论。该提案应进一步促进这些领域的相互借鉴，并在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，拥有大量的研究文章和博士学位。他希望他的技术在未来产生更广泛的影响，导致各种学科的新发展和问题的解决方案。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。