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

遍历嵌入、双模分解和 II1 型因子的结构

• 批准号: 1955812
• 负责人: Sorin Popa
• 金额: $ 40.34万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1955812&HistoricalAwards=false
• 关键词: Ergodic; Embeddings; Bimodule; Decomposition; Structure

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.

数学中的刚性发生在某些几何对象（例如一组转换）可以通过仅知道某些部分信息（例如其因素）来识别时发生刚性。 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该领域的几个长期问题。 刚性结果可能与数学及其应用的许多领域有关，包括计算机科学，复杂性理论，量子信息理论，计算机网络的设计以及误差校正代码的理论。该项目通过培训与项目研究相关的主题的研究生培训为劳动力发展做出了贡献。II1因子框架的惊人特征是其具有刚性和随机性现象的能力。较早利用这些相反范式之间紧张关系的工作导致了惊人的发现以及II1因素与其他领域的研究之间的富有成效的相互作用，例如C* - 代数，自由概率，千古理论，群体理论，群体理论（测量，几何，算法，算术等），量子组，随机矩阵，随机矩阵和分布性集理论。这项工作开发了几种重要的技术来研究II1因素：亚比例理论中的有限维近似和重建方法，变形刚度理论，双模型相互交织以及增量贴合。该项目的目的是利用迭代性沿牙嵌入的技术，并结合了以前的技术来控制双模型结构，以解决几个重要问题：（1）自由组因素及其“贵族”的非同态性，其“粗糙”（其“粗糙”），深层结构性的结构性概述了许多先前的结果并猜测了这些因素； （2）Connes的嵌入猜想（尤其是针对组）和SOFIC组问题； （3）III1因子的Conner's conne concontralizer猜想。该项目旨在扩大开发技术的范围，并加深操作员代数与其他数学领域的相互作用。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准通过评估来获得支持的。