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

数学中的“刚性”是指某一类对象(函数、函数代数等)可以通过对它们知之甚少的信息来识别。在计算机科学、复杂性理论、计算机网络设计和纠错码理论的应用中，算术运算的结果通常是跨学科的，并且可以与许多数学领域相关。这位主要研究者近年来的工作是研究所谓的von Neumann代数中的刚性。这些是无限矩阵的代数，其中两个元素的乘积(比方说A乘B)可能与逆序乘积(B乘A)不同，这一事实反映了粒子物理量子力学中的海森堡测不准原理。它们与群论和遍历理论有关，因为群在空间上的作用产生了一类von Neumann代数，现在被称为“因子”。在这种情况下，当仅通过知道其相关因素就可以识别该组变换时，就会出现刚性。在2001-2010年间，首席研究人员发展了一系列技术来研究这种现象(即变形-刚性理论)，最近他又增加了一种强大的近似技术。在这个项目中，他打算结合所有这些工具来研究因子中的刚性，并解决该领域最著名的两个问题：Connes近似嵌入猜想和自由群因子问题。康奈斯猜想预测，因子可以“在计算机上模拟”，并在量子信息论中有一个有趣的重新表述。该项目应促进数学的几个领域(例如，自由概率、随机矩阵、群论、逻辑)的交叉授粉，并在每一个领域取得进展。这位首席研究员在刚性理论方面的研究已经在这些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从他的工作中萌生出来。他预计他的技术将在未来产生更广泛的影响，导致各种学科的问题的新发展和解决方案，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。在这个项目中，首席研究人员打算通过将一种新的近似技术--增量拼接技术引入变形-刚度理论中来进一步发展该理论。使用这一系列工具，他将继续致力于群产生的代数的分类及其在空间上的作用，以及研究这些对象的刚性性质和它们的对称性的计算。特别是，首席研究者将尝试新的方法来解决两个著名的二一因素问题：Connes近似嵌入猜想和自由群因素问题。本课题所研究的问题对冯·诺依曼代数理论以及群论、遍历论、逻辑(描述集合论)、自由概率和子因子论等相关领域都具有重要意义。与他的学生和合作者一起，首席研究员将努力扩大变形-刚性理论的范围，加强其与所有这些(可能还有其他)领域的互动，这一活动应该会带来更多令人惊讶的跨学科性质的结果。此外，首席研究人员打算继续通过暑期计划、迷你课程、教科书和说明性文章以及他将主持和组织的会议来传播他的新技术。

"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.