Subfactors, Tensor Categories, and Higher Dimensional Algebra

对称是物体或系统向自身的运动。例如，洗牌或围绕原子核旋转原子。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学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引人注目的应用是伽罗瓦利用数系的对称性来解释为什么用三次或四次幂来解方程可以推广我们熟悉的二次公式，而用五次幂来解方程却没有这样的公式。伽罗瓦的数字系统与用于研究量子力学的冯·诺伊曼因子是平行的，而我们所知的量子群或量子对称性的最初几个引人注目的应用出现在琼斯对子因子的研究中。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。在这个项目中使用的主要技术涉及高维代数。这里的想法是，我们通常在做计算时将数学符号写在一条线上，但在许多情况下，最好使用整个平面来绘图，或者在“纸”上以更有趣的形状（如圆圈或球体）进行计算。有时几何图形允许深入了解数学，例如，您可以“旋转”发生在平面上的计算。这个项目的目标是使用高维代数来研究量子对称中的问题。该项目还将通过培养博士和高中生，为美国劳动力的发展做出贡献。在更专业的语言中，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冯·诺依曼子因子和张量范畴的问题。这些问题包括非常范例驱动的问题，如构建和研究特殊的小索引子因子，以及更多的理论问题，如理解融合类别的3类结构（其中子因子表现为1-态射）。更具体地说，该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的作者以前的工作对小指数子因子的分类。其次，我们将更系统地使用模块类别来处理有关从标准不变量重建子因子的分析问题。第三，我们将使用束理论方法和琼斯平面代数来研究张量范畴，包括非半简单张量范畴。最后，我们利用基于Lurie-Baez-Dolan协同假设的高范畴和局部拓扑量子场论研究了张量范畴和编织张量范畴。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.