Ergodic Embeddings, Bimodule Decomposition, and the Structure of Type II1 Factors

当某些几何对象，如一组变换，仅通过知道某些部分信息，如它们的因子，就可以识别时，数学中的刚性就发生了。因子是由冯·诺伊曼在20世纪30年代引入研究量子力学的，它是无限矩阵的不可约代数，其中两个元素的乘积A乘B通常不同于逆序乘积B×A。该项目旨在结合包括变形刚性理论、遍历嵌入、近似和模拟技术以及重建方法在内的工具来研究因子中的刚性，并解决该领域中的几个长期存在的问题。刚性结果可能与数学及其应用的许多领域相关，包括计算机科学、复杂性理论、量子信息理论、计算机网络设计和纠错码理论。这个项目通过培训研究生与项目研究相关的主题来促进劳动力的发展。II1因素框架的一个显著特点是它能够同时容纳刚性和随机性现象。早期利用这些对立范式之间的紧张关系的工作导致了惊人的发现，并在II1因子的研究和其他领域之间进行了卓有成效的互动，如C*-代数、自由概率、遍历理论、群论(测量的、几何的、算术的等)、量子群、随机矩阵和描述集合论。这项工作发展了几种研究II1因子的重要技术：子因子理论中的有限维近似和重建方法、变形刚性理论、双模缠绕和增量修补。这个项目的目的是利用控制双模结构的迭代遍历嵌入技术与以前的技术相结合来解决几个重要问题：(1)自由群因子的非同构及其“粗糙性”，这是一个深层次的结构性质，推广了许多关于这些因子的先前结果和猜想；(2)Connes的嵌入猜想(特别是对于群)和SOFIC群问题；以及(3)Connes的关于III1因子的双集中化猜想。该项目旨在扩大正在开发的技术的范围，并深化算子代数与其他数学领域的互动。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Rigidity in mathematics occurs when certain geometric objects, such as a group of transformations, can be recognized by merely knowing some partial information, such as their factors. Introduced by von Neumann in the 1930s to study quantum mechanics, factors are irreducible algebras of infinite matrices, where the product of two elements, A times B, is in general different from the product in reverse order, B times A. This project aims to combine tools including deformation-rigidity theory, ergodic embeddings, approximation and simulation techniques, and reconstruction methods to study rigidity in factors and to tackle several longstanding questions in this area. Rigidity results can be relevant to many areas of mathematics and its applications, including computer science, complexity theory, quantum information theory, the design of computer networks, and the theory of error-correcting codes. This project contributes to workforce development through the training of graduate students in topics related to the project research.A striking feature of the II1 factor framework is its ability to host both rigidity and randomness phenomena. Earlier work exploiting the tension between these opposing paradigms led to striking discoveries and to fruitful interaction between study of II1 factors and other areas, such as C*-algebras, free probability, ergodic theory, group theory (measured, geometric, arithmetic, etc.), quantum groups, random matrices, and descriptive set theory. This work developed several important techniques to study II1 factors: finite dimensional approximation and reconstruction methods in subfactor theory, deformation rigidity theory, intertwining by bimodules, and incremental patching. This project aims to employ the technique of iterative ergodic embeddings with control of bimodule structure in combination with previous techniques to tackle several important questions: (1) the non-isomorphism of the free group factors and their "coarseness," a deep structural property that generalizes many prior results and conjectures about these factors; (2) Connes' embedding conjecture (notably for groups) and the sofic group problem; and (3) Connes' bicentralizer conjecture for III1 factors. The project aims to broaden the scope of the techniques under development and to deepen the interaction of operator algebra with other areas of mathematics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.