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Subfactors, Tensor Categories, and Higher Dimensional Algebra

子因子、张量类别和高维代数

基本信息

批准号：
2000093
负责人：
Noah Snyder
金额：
$ 25.85万
依托单位：
Indiana University
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：
2020-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000093&HistoricalAwards=false
关键词：
Subfactors Tensor Categories Higher Dimensional

项目摘要

Symmetries are movements of an object or system back onto itself. For example, shuffling a deck of cards or rotating an atom around its nucleus. Over the last 200 years, Group Theory has been spectacularly successful in using symmetry to study a wide variety of problems in mathematics, physics, chemistry, and engineering. More recently, starting in the 1980s, mathematicians and mathematical physicists have developed a theory of Quantum Groups and Quantum Symmetry which adapt ideas from Group Theory to more general settings appearing in quantum physics and related mathematics. The first spectacular application of what we now call Group Theory was Galois' use of symmetries of number systems to explain why there's a generalization of the familiar quadratic formula to solve equations using third or fourth powers, but no such formula for solving equations with fifth powers. Galois's number systems are parallel to von Neumann factors developed to study quantum mechanics, and several of the first spectacular applications of what we would know call Quantum Groups or Quantum Symmetries appeared in Jones's study of subfactors. More recently Quantum Symmetry has become central to certain approaches to constructing quantum computers using so-called topological phases of matter. The main techniques used in this project involve higher dimensional algebra. Here the idea is that we typically write mathematical symbols on a line when doing calculations, but in many settings it is better to use the whole plane to draw pictures, or to do the calculation on "paper" in more interesting shapes like circles or spheres. Sometimes the geometry allows for insight into the mathematics, for example you can "rotate" a calculation taking place in the plane. The goal of this project is to use higher dimensional algebra to study questions in quantum symmetry. The project will also contribute to the development of the US workforce through the training of Ph.D. and high school students.In more technical language, we use planar algebras, topological quantum field theory, and higher categories to study questions coming from von Neumann subfactors and tensor categories. These include very example-driven questions like constructing and studying exceptional small index subfactors, as well as more theoretical questions like understanding the structure of the 3-category of fusion categories (where subfactors appear as 1-morphisms). More specifically, this project focuses on four closely related research programs. First, we will develop a purely algebraic analogue of the author's previous work on the classification of small index subfactors. Second we will use module categories more systematically to approach analytic questions about reconstructing subfactors from their standard invariants. Third, we will use skein theoretic approaches and Jones's planar algebras to study tensor categories, including non-semisimple tensor categories. Finally, we study tensor categories and braided tensor categories using higher-categories and local topological quantum field theories building on the Lurie-Baez-Dolan cobordism hypothesis.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.

对称性是物体或系统返回自身的运动。例如，洗一副牌或围绕原子核旋转一个原子。在过去的200年里，群论在利用对称性研究数学、物理、化学和工程中的各种问题方面取得了惊人的成功。最近，从20世纪80年代开始，数学家和数学物理学家发展了量子群和量子对称理论，将群论的思想应用于量子物理和相关数学中出现的更一般的设置。我们现在所说的群论的第一个壮观的应用是伽罗瓦利用数系的对称性来解释为什么我们熟悉的二次公式可以推广到使用三次或四次幂来求解方程，但没有这样的公式来求解五次幂的方程。伽罗瓦的数系与为研究量子力学而发展起来的冯·诺依曼因子是平行的，我们所知的量子群或量子对称性的几个最早的壮观应用出现在琼斯的子因子研究中。最近，量子对称性已经成为使用所谓的物质拓扑相构建量子计算机的某些方法的核心。这个项目中使用的主要技术涉及高维代数。这里的想法是，我们通常在计算时在一条线上写数学符号，但在许多情况下，更好的做法是使用整个平面来绘制图形，或者在更有趣的形状(如圆形或球体)的“纸”上进行计算。有时几何学允许洞察数学，例如，你可以“旋转”在平面上进行的计算。这个项目的目标是使用高维代数来研究量子对称性的问题。该项目还将通过对博士和高中生的培训来促进美国劳动力的发展。用更专业的语言，我们使用平面代数、拓扑量子场论和更高的范畴来研究来自von Neumann子因子和张量范畴的问题。这些问题包括非常以实例为导向的问题，如构造和研究异常小的指标子因素，以及更多的理论问题，如理解融合范畴的3-范畴的结构(其中，子因素表现为1-态射)。更具体地说，这个项目侧重于四个密切相关的研究项目。首先，我们将开发一个纯代数模拟作者以前关于小指标子因子分类的工作。其次，我们将更系统地使用模块范畴来处理有关从其标准不变量重构子因子的分析问题。第三，我们将使用Skein理论方法和Jones平面代数来研究张量范畴，包括非半单张量范畴。最后，我们使用建立在Lurie-Baez-Dolan Cobordism假设基础上的更高范畴和局部拓扑量子场理论来研究张量范畴和辫子张量范畴。这一奖项反映了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。