Rigidity, Cohomology, and Approximate Embeddings in von Neumann Algebra Factors

近年来，数学及其应用中许多令人兴奋的发展背后的驱动力是结构（有序、刚性等）和随机性（缺乏结构、无序、粗糙）之间的二分法。这个项目涉及一种强大的技术，称为变形-刚性理论，在冯·诺伊曼代数的框架内研究这些相反的现象，即无限矩阵的代数，其中两个元素的乘积a乘以B可能不同于逆顺序的乘积B乘以a，这一事实反映了粒子物理学中的量子力学定律（海森堡的测不准原理）。冯·诺伊曼代数也与群论和遍历理论有关，因为空间变换会产生一类特殊的代数，称为II1因子。在这种情况下，当仅通过了解其相关的II1因子（这是一个非常粗糙的对象）就可以识别转换组时，就会出现刚性。因子刚性的研究是一项跨学科的努力，可以与数学的许多领域有关。它还应用于计算机科学、复杂性理论、设计计算机网络和纠错码理论。本研究项目旨在加深对这一重要领域的理解。首席研究员最近为这项研究增加了几个新工具，即近似/模拟技术，表示理论和上同源。在这个项目中，他打算结合所有这些工具来进一步研究II1因子中的刚性与随机性现象，并解决该领域的几个著名问题：cones近似嵌入（CAE）猜想、sofic群问题、自由群因子问题、铺装问题和类群对象的不变量计算。值得注意的是，CAE猜想预测i1因子可以在计算机上“模拟”，这将对量子信息理论产生惊人的影响。

The driving force behind many exciting developments in mathematics and its applications in recent years has been the dichotomy between structure (order, rigidity, etc.) and randomness (lack of structure, disorder, coarseness). This project concerns a powerful technique, called deformation-rigidity theory, to study these opposing phenomena in the framework of von Neumann algebras, i.e., algebras of infinite matrices, where the product of two elements, A times B, may be different from the product in reverse order, B times A, a fact that reflects the laws of quantum mechanics in particle physics (Heisenberg's Uncertainty Principle). Von Neumann algebras are also related to group theory and ergodic theory, since transformations of spaces give rise to a special class of such algebras called II1 factors. Rigidity in this context occurs when the group of transformations can be recognized by merely knowing its associated II1 factor (which is a very coarse object). The study of rigidity of factors is an interdisciplinary endeavor and can be relevant to many areas of mathematics. It also has applications to computer science, complexity theory, design computer networks, and the theory of error-correcting codes. This research project aims to deepen understanding in this important area.The principal investigator recently added several new tools to this study, namely an approximation/simulation technique, a representation theory, and a cohomology. In this project, he intends to combine all these tools to further study rigidity-versus-randomness phenomena in II1 factors and to tackle several famous problems in this area: the Connes Approximate Embedding (CAE) conjecture, the sofic group problem, the free group factor problem, the paving problem, and calculations of invariants for group-like objects. It should be noted that the CAE conjecture, which predicts that II1 factors can be "simulated" on a computer, would have striking consequences in quantum information theory.