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

II 1 因子框架中的近似、变形刚度和分类

• 批准号: 1400208
• 负责人: Sorin Popa
• 金额: $ 36万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1400208&HistoricalAwards=false
• 关键词: Approximation; deformation; rigidity; classification; II

"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.

当某一类对象（函数、函数代数等）可以通过了解很少的信息来识别时，数学中的“刚性”就会出现。刚性结果通常是跨学科的，可以与数学的许多领域相关，包括计算机科学、复杂性理论、计算机网络设计和纠错码理论的应用。首席研究员近年来的工作与所谓冯·诺依曼代数中的刚性研究有关。这些是无限矩阵的代数，其中两个元素的乘积（例如 A 乘以 B）可能不同于逆序的乘积（B 乘以 A），这一事实反映了粒子物理量子力学中海森堡的不确定性原理。它们与群论和遍历理论相关，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”。当仅通过知道其相关因素就可以识别一组变换时，就会出现这种情况下的刚性。 2001年至2010年期间，首席研究员开发了一系列技术来研究此类现象（即变形-刚性理论），最近他在其中添加了强大的近似技术。 在这个项目中，他打算结合所有这些工具来研究因子的刚性，并解决该领域两个最著名的问题：Connes 近似嵌入猜想和自由群因子问题。康尼斯猜想预测因素可以“在计算机上模拟”，并且在量子信息论中进行了有趣的重新表述。 该项目应有助于数学多个领域（例如自由概率、随机矩阵、群论、逻辑）的交叉授粉，并推动每个领域的进步。主要研究者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，发表了大量的研究论文和博士学位。论文直接源于他的工作。他预计他的技术将在未来产生更广泛的影响，带来各种学科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。在该项目中，主要研究人员打算通过将一种称为增量修补的新近似技术纳入其中，进一步发展变形刚度理论。使用这一系列工具，他将继续致力于对群产生的代数及其在空间上的作用进行分类，以及研究这些物体的刚性特性及其对称性的计算。特别是，首席研究员将尝试新的方法来解决二一因子中的两个著名问题：Connes近似嵌入猜想和自由群因子问题。该项目研究的问题对于冯·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的邻近领域都很重要。首席研究员将与他的学生和合作者一起努力扩大变形刚度理论的范围，加强其与所有这些（可能还有其他）领域的相互作用，这项活动应该会带来跨学科特征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项目、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传播他的新技术。