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

数学的主线之一是研究物体的对称性。另一个是对群体的研究，这在整个数学和科学中都非常重要。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，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。这些同样出现在科学(例如物质的相)和数学(例如纽结理论)中。群的最早出现之一是在伽罗瓦理论的研究中，其中一个人研究的是不移动子系统的数系统的对称性，平行地，量子群最早的出现之一是在冯·诺伊曼因子的类似的伽罗瓦理论中。这个项目的研究部分集中在量子群和子因子的研究上。用来研究这种量子群的一种重要技术被称为“高维代数”。该项目的教育和外展部分延伸了调查员十年来在暑期数学课程中参与高中教育，以及最近在高中数学教师教育中的参与。他将在加拿大/美国数学营为高中生开发新的课程和研究项目基于他的研究，他将启动一个项目，为有非凡教育和推广记录的研究生颁发论文写作奖学金，特别强调代表不足的群体，他将为数学教育专业的学生建立一个数学俱乐部，他将监督本科生的研究项目。本研究项目运用高等范畴论和拓扑学中的技巧来更好地理解张量范畴理论和从子因子理论中得到的具体例子。这将通过两个主要的研究项目来完成。第一个是研究研究者在以前的工作中分类的小指标子因子的结构，并证明与双模和中间子因子有关的其他分类结果。第二个程序是利用贝兹-多兰-霍普金斯-卢里共边假设的洞察力，理解具有价值的局部拓扑场理论，即张量范畴的3-范畴和相关的更高范畴。

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.