Quantifying Rigidity in von Neumann Algebras

量化冯·诺依曼代数中的刚性

基本信息

批准号：
2055155
负责人：
Thomas Sinclair
金额：
$ 30万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-06-01 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055155&HistoricalAwards=false
关键词：
Quantifying Rigidity von Neumann Algebras

项目摘要

The theory of von Neumann algebras was initiated in the 1930s and 40s by F.J. Murray and John von Neumann as a mathematical framework for quantum mechanics. With recent breakthroughs in quantum computing, the study of von Neumann algebras is poised to yield insights into deep problems in the theory of quantum computation which must be overcome to make quantum computing and quantum cryptography practical, efficient technologies. One goal of this project is to use tools from von Neumann algebras to provide insights into so-called “quantum expanders” which have applications to quantum error correction and quantum cryptography. This is part of the broader goal of the project to investigate quantitative aspects of von Neumann algebras. Other potential applications lie in the theory of random matrices, which are used in diverse applications in many fields from quantum physics to biology and big data. This project will contribute to workforce development by providing research training and mentoring opportunities at the undergraduate and graduate level. The project aims to make progress in several directions around quantifying and developing new invariants for exploring the phenomenon of rigidity in von Neumann algebras. One objective is to further develop the theory and use of cohomological rigidity techniques in Popa’s deformation/rigidity theory based on techniques developed by the PI jointly with collaborators on the existence and uniqueness of maximal rigid subalgebras of deformations. This could lead to progress towards settling two outstanding conjectures in the field, the Peterson-Thom conjecture and absence of Cartan subalgebras for von Neumann algebras of groups having nontrivial first cohomology with coefficients in the left-regular representation. Techniques from continuous model theory will also be explored as potential avenues to these conjectures by attempting to find noncommutative analogs to Anderson and Keisler’s model theoretic approach to stochastic differential equations. A second objective is to develop experimental and quantitative approaches to property Gamma, in part based on the PI’s discovery of malnormal matrices in his work with Mulcahy. The PI will approach these problems using a mix of techniques from ergodic theory, random matrix theory, computability theory, and von Neumann algebras. Results in this direction could lead to new insights at the interface of von Neumann algebras and quantum computing. A third objective is to explore the applications of uniform 2-norms to the classification theory of nuclear C*-algebras based on the PI’s work with Goldbring and Hart on the continuous model theory of correspondences.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.

F.J. Murray和John von Neumann在1930年代和40年代启动了冯·诺伊曼代数理论，作为量子力学的数学框架。随着量子计算的最新突破，对von Neumann代数的研究被毒死，以产生对量子计算理论中深层问题的见解，量子计算理论必须克服，以进行量子计算和量子加密实践，有效的技术。该项目的目标之一是使用von Neumann代数的工具来提供对所谓的“量子扩张器”的见解，这些“量子扩张器”具有用于量子误差校正和量子密码学的应用。这是该项目研究冯·诺伊曼代数的定量方面的更广泛目标的一部分。其他潜在的应用在于随机物质理论，这些理论用于从量子物理学到生物学和大数据的许多领域的潜水员应用中使用。该项目将通过在基础和研究生层面提供研究培训和心理机会来为劳动力发展做出贡献。该项目旨在围绕量化和开发新的不变性来探索von Neumann代数的刚性现象的多个方向。一个目的是进一步发展POPA变形/刚性理论中的共同体学刚性技术的理论和使用，该技术基于PI与合作者共同开发的技术和最大刚性刚性下层的独特性开发的技术。这可能会导致在该领域设定两个杰出概念的进展，彼得森 - 学者的概念和cartan subergebras von neumann代数的cartan subergebras的群体具有非平凡的第一共同体学，并在左侧表示中具有系数。连续模型理论的技术还将通过试图找到与安德森和凯斯勒的非交通类似物来探讨这些概念的潜在途径。第二个目标是开发实验性和定量方法，部分基于PI在与Mulcahy的合作中发现疟疾物质的发现。 PI将使用来自Ergodic理论，随机矩阵理论，计算理论和von Neumann代数的技术混合来解决这些问题。结果朝这个方向发展可能导致对von Neumann代数和量子计算的界面的新见解。第三个目标是根据PI与Goldbring和Hart的工作探索统一2-Norms对核C*代数分类理论的应用。该奖项反映了NSF的法定任务，并通过评估了该基金会的知识分子优先级和广泛的影响来评估NSF的法定任务，并被认为是珍贵的支持。