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

在过去的十年里，群作用的von Neumann代数已经成为研究各种刚性现象的中心舞台和不同数学领域相互作用的游乐场：算子代数、群论(测量的、几何的、算术的等)、遍历理论、轨道等价关系和描述集论，仅举几例。在2001-2010年间，首席研究员发展了一系列在这一框架内研究刚性的技术，现在被称为变形/刚性理论。这导致了在冯·诺依曼代数和轨道遍历理论设置下的大量显著的刚性结果，并解决了许多长期存在的问题。首席研究人员的技术和结果自然会带来一些令人兴奋的新研究方向和所有这些领域的问题。它们还为解决von Neumann代数中的一些经典问题提供了新的工具，例如：Connes刚性猜想；自由群因子的结构和分类；由群及其作用产生的代数的超刚性性质。在这个项目中，首席研究员与他的学生和合作者将系统地研究这些方向。他打算利用算符代数技术，深化他与群论、遍历论和描述集合论领域的互动。这项活动应该会进一步产生令人惊讶的结果，并解决所有这些领域的问题。首席研究人员预计，因素框架将继续在数学的不同领域之间的相互作用中发挥关键作用。数学中的“刚性”是指某一类对象(函数、函数代数等)可以在没有太多关于它们的初始信息的情况下被识别。这种类型的结果通常是跨学科的，可能与许多数学领域相关。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有有趣的应用。这位首席研究员近年来的工作主要集中在研究一类被称为von Neumann代数的对象的刚性。这些是无限矩阵的代数，其中两个元素A和B的乘法结果可能根据乘积中的顺序而不同(即AB可能不同于BA)。这些代数是由冯·诺伊曼在20世纪20年代提出的，他努力为粒子物理中的量子力学提供一种严格的方法。他首先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度量空间上的作用产生了一类非凡的冯·诺依曼代数。在这种情况下，当仅通过知道相关的冯·诺伊曼代数就可以识别群体行动时，就发生了刚性。首席研究人员最近开发了一套全新的技术来研究这种现象，创建了一个现在被称为变形/刚性理论的框架。他获得了许多令人惊讶的、本质上美丽的结果，这些结果建立了从冯·诺伊曼代数到数学其他领域的刚性的桥梁，并导致了深度的跨学科活动。首席研究人员打算在未来三年内致力于的问题越来越雄心勃勃，与著名的悬而未决的问题有关，这些问题涉及到“刚性群”和“自由群”产生的因素的分类。这些项目对冯·诺依曼代数理论和邻近的群论、遍历论、逻辑(描述集合论)、自由概率和子因子论等数学领域都很重要。该提案应进一步促进这些地区的异花授粉，并在每个地区取得实质性进展。这位主要研究人员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，大量的研究文章和博士论文直接由此而生。他预计他的技术将在未来产生更广泛的影响，导致各种主题的问题得到新的发展和解决方案。他还预计这项研究将对应用数学和前述计算机科学领域产生直接和间接的影响。

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.