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

数学中的“刚性”发生在某个类中的对象（函数，函数代数等）可以通过很少的信息来识别它们。Regidity的结果通常是跨学科的，可以与许多数学领域相关，应用于计算机科学，复杂性理论，计算机网络设计和纠错码理论。主要研究者的工作在最近几年做的研究刚性在所谓的冯诺依曼代数。这些是无限矩阵的代数，其中两个元素的乘积（比如A乘B）可能不同于逆序的乘积（B乘A），这一事实反映了粒子物理学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冯·诺依曼代数，现在称为“因子”。“在这种情况下，刚性发生在变换群可以通过仅仅知道其相关因子而被识别的时候。在2001-2010年期间，首席研究员开发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，他打算联合收割机所有这些工具来研究刚性的因素，并解决两个最著名的问题在该地区：康纳斯近似嵌入猜想和自由群因子问题。康纳斯猜想预言，因素可以“在计算机上模拟”，并在量子信息理论中有一个有趣的重新表述。 该项目应有助于数学的几个领域的交叉授粉（例如，自由概率，随机矩阵，群论，逻辑），并导致在每一个进步。主要研究者在刚性理论方面的研究已经对这些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。论文直接从他的工作中萌芽。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，直接和间接地影响应用数学和计算机科学的上述领域。在这个项目中，首席研究员打算进一步发展的变形刚度理论，将其纳入一个新的近似技术称为增量修补。使用这一系列的工具，他将继续工作的分类代数所产生的群体和他们的行动对空间，以及对研究的刚性性质的这些对象和计算其对称性。特别是，首席研究员将尝试新的方法来解决两个著名的问题，在两个因素：康纳斯近似嵌入猜想和自由群因子问题。在这个项目中研究的问题是重要的冯诺依曼代数理论和相邻领域的群论，遍历理论，逻辑（描述集理论），自由概率，和子因子理论。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）领域的相互作用，这一活动将导致跨学科性质的进一步令人惊讶的结果。此外，首席研究员打算继续通过暑期课程，迷你课程，教科书和临时文章以及他将举办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.