Quantifying Rigidity in von Neumann Algebras

冯·诺依曼代数理论是由F.J.默里和约翰·冯·诺伊曼在20世纪30年代和40年代提出的，作为量子力学的数学框架。随着最近在量子计算方面的突破，冯·诺伊曼代数的研究有望深入了解量子计算理论中的深层次问题，这些问题必须被克服，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von Neumann代数中的工具来提供对所谓的“量子扩展器”的见解，这些扩展器应用于量子纠错和量子密码学。这是该项目研究von Neumann代数的量化方面的更广泛目标的一部分。其他潜在的应用在于随机矩阵理论，它在从量子物理到生物学和大数据的许多领域都有不同的应用。该项目将通过在本科生和研究生一级提供研究、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在几个方向上取得进展，以探索von Neumann代数中的刚性现象。一个目标是在Popa的变形/刚性理论中进一步发展上同调刚性技术的理论和应用，这些技术是由PI与合作者共同开发的关于变形的极大刚性子代数的存在和唯一性的技术。这可能导致解决该领域的两个悬而未决的猜想的进展，即Peterson-Thom猜想和具有非平凡第一上同调且系数在左正则表示中的系数的von Neumann代数的Cartan子代数的缺失。来自连续模型理论的技术也将被探索为这些猜想的潜在途径，试图找到与安德森和凯斯勒的随机微分方程的模型理论方法类似的非对易类似方法。第二个目标是开发伽马性质的实验和定量方法，部分基于PI在他与Mulcahy的工作中发现的异常矩阵。PI将使用遍历理论、随机矩阵理论、可计算性理论和von Neumann代数的混合技术来处理这些问题。这一方向的结果可能会在冯·诺伊曼代数和量子计算的界面上带来新的见解。第三个目标是探索一致2-范数在核C*-代数分类理论中的应用，该分类理论是基于PI与戈德布林和哈特在对应的连续模型理论上的工作。这一奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.