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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.

冯·诺依曼代数的理论是由F. J. Murray和John von Neumann在20世纪30年代和40年代作为量子力学的数学框架提出的。随着量子计算的最新突破，对冯·诺依曼代数的研究有望深入了解量子计算理论中的深层问题，这些问题必须克服，才能使量子计算和量子密码学成为实用、高效的技术。该项目的一个目标是使用冯诺依曼代数的工具来提供对所谓的“量子扩展器”的见解，这些扩展器在量子纠错和量子密码学中有应用。这是该项目更广泛目标的一部分，以研究冯诺依曼代数的定量方面。其他潜在的应用在于随机矩阵理论，该理论用于从量子物理到生物学和大数据的许多领域的各种应用。该项目将通过提供本科生和研究生水平的研究培训和指导机会，促进劳动力发展。该项目旨在围绕量化和开发新的不变量，以探索冯诺依曼代数中的刚性现象，在几个方向上取得进展。一个目标是进一步发展的理论和使用上同调刚度技术在波帕的变形/刚度理论的基础上开发的技术PI共同与合作者的存在性和唯一性的最大刚性子代数的变形。这可能会导致解决该领域两个突出问题的进展，彼得森-托姆猜想和缺乏Cartan子代数的冯诺依曼代数群具有非平凡的第一上同调系数在左正则表示。从连续模型理论的技术也将被探索作为潜在的途径，这些acquitures试图找到非交换类似物安德森和凯斯勒的模型理论方法随机微分方程。第二个目标是开发实验和定量的方法来属性伽玛，部分基于PI的发现异常矩阵在他的工作与Mulcahy。PI将使用遍历理论、随机矩阵理论、可计算性理论和冯诺伊曼代数等技术来解决这些问题。这一方向的结果可能会在冯诺依曼代数和量子计算的界面上产生新的见解。第三个目标是探索应用程序的统一2-规范的分类理论的核C*-代数的基础上PI的工作与Goldbring和哈特的连续模型理论的correspondings.This奖反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。