CAREER: Subfactors, Tensor Categories, and Local Topological Field Theory

数学的主线之一是研究对象的对称性。另一个是对群的研究，这在整个数学和科学中非常重要。一个物体的所有经典对称性的集合形成一个群，在量子环境中，一个被称为量子群或张量范畴的群的广义版本扮演着类似的角色。这些又出现在科学（如物质相）和数学（如纽结理论）中。群的最早出现之一是在伽罗瓦理论的研究中，其中一个研究的是不移动子系统的数系统的对称性，而量子群的最早出现之一是在冯诺依曼因子的类似伽罗瓦理论中。该项目的研究部分侧重于量子群和子因子的研究。用来研究这种量子群的一种重要技术叫做“高维代数”。“这个项目的教育和推广部分扩展了研究人员在夏季数学项目中长达十年的高中教育参与，以及最近在高中数学教师教育中的参与。他将开发新的课程和研究项目的高中生在加拿大/美国数学夏令营的基础上，他的研究，他将开始授予学位论文写作奖学金的研究生与非凡的教育和推广的既定记录，特别强调代表性不足的群体，他将启动数学俱乐部的数学教育专业，他将监督本科生的研究项目。本研究项目使用高级范畴理论和拓扑学的技术来更好地理解张量范畴理论和来自子因子理论的具体例子。这将通过两个主要的研究项目来实现。首先是研究小指标子因子的结构，研究者在以前的工作中分类，并证明其他分类结果与双模和中间子因子。第二个程序是理解局部拓扑场论的值的3-范畴的张量范畴和相关的更高的范畴使用的洞察力所给出的贝兹-多兰-霍普金斯-卢里协边假设。

One of the main threads in mathematics is the study of the symmetries of an object. Another is the study of groups, which are very important throughout mathematics and science. The collection of all classical symmetries of an object forms a group, and in quantum settings, a similar role is played by a generalized version of groups called quantum groups or tensor categories. These again appear both in science (e.g. phases of matter) and mathematics (e.g. knot theory). One of the earliest appearances of groups was in the study of Galois theory, where one looks at symmetries of a number system that do not move a subsystem, and in parallel one of the earliest appearances of quantum groups was in a similar Galois theory for von Neumann factors. The research portion of this project focuses on the study of quantum groups and subfactors. One important technique used to study such quantum groups is called "higher dimensional algebra." The education and outreach portion of this project extends the investigator's decade-long involvement in high school education at summer math programs and more recently in education of high school math teachers. He will develop new courses and research projects for high school students at the Canada/USA Mathcamp based on his research, he will start a program for awarding dissertation writing fellowships for graduate students with an established record of extraordinary education and outreach, with a particular emphasis on underrepresented groups, he will start a math club for mathematics education majors, and he will supervise undergraduate research projects. This research project uses techniques from higher category theory and topology to better understand the theory of tensor categories and specific examples coming from the theory of subfactors. This will be done through two major research programs. The first is to study the structure of small index subfactors that the investigator classified in prior work and to prove other classification results related to bimodules and intermediate subfactors. The second program is to understand local topological field theories with values the 3-category of tensor categories and related higher categories using the insight given by the Baez-Dolan-Hopkins-Lurie cobordism hypothesis.