Basic course description
Math 201a is a topics course in the basic theory of monoidal and tensor categories. The formal prerequisite is Math 200C or equivalent experience. The most important prerequisite is a basic grounding in ring and module theory, including tensor products.
Goal
The goal is to give a taste of an important area of algebra, the language and results of which have become increasingly influential. My research has touched on this subject, but I wouldn't call myself an expert on tensor categories and it is not the core of what I do. Still, tensor categories are definitely a "noncommutative" subject, and so the material of the course will give you an idea of the general feel of research in noncommutative algebra. I hope it will be of interest also to those working in other related fields.
Syllabus
Lectures Lectures are MWF 1pm-1:50pm in B402A AP&M. Note that this room changed last minute from the originally scheduled 2402 AP&M.
Accessing Lectures I will record lectures using EVT, an lecture capture and notetaking system we have installed in B402A. Here are the access instructions: Signup: https://learn.evt.ai/n/signup Login: https://learn.evt.ai/n/login Term: Fall 2023 MATH 201A PIN #: 658219 Please use the Support button on the EVT Platform or email hello@evt.ai for any support related questions or feedback.
Exercises I may state some exercises during lecture, and may post some occasional separate sets of exercises. You are not required to hand in written solutions, but I encourage you to look these over and attempt some of them. I also personally find that I get more out of topics courses if I review the lecture notes to think about the points I didn't understand in real time, so I encourage you to do this. You can also read about the topics in the other references below for another point of view.
Grading The course is not graded strictly, as is usual for a graduate topics course. Your main responsibility is interest in the subject as well as some degree of attendance and participation. If you are not sure you will engage with the course beyond attending the first few lectures, it would be better to audit. If you decide midway through the quarter that the remainder of the course would not be helpful to you, just keep me informed.
Office hours I will not schedule regular office hours, but I will happily meet any student in person or on Zoom by appointment, to talk about the course material or anything else. I will be in my office most MWF late morning and all afternoon.
References
I will not precisely follow any source. The most comprehensive source is the book "Tensor Categories" by Etingof, Gelaki, Nikshych, and Ostrik (EGNO). Etingof maintains a free pdf of the book online: EGNO: Tensor Categories It is a great reference for the subject, but parts are tough going for the novice (and for me) as a lot of details are left to the reader; and some of the exercises are actually major results. Most of what we cover will be in chapters 1, 2, 4, 5, 7, and 8 of the that book, but I plan my lectures to have a self-contained presentation.
My friend Chelsea Walton is writing a book that treats some of this material from a more user-friendly viewpoint. The current draft version of Chapters 1-3 of her book can be found here: Chelsea Walton: Symmetries of Algebras
Chapter 1 of her book reviews background on rings, algebras and modules, much of which I expect our audience to have seen in Math 200b or the equivalent; though I will review the necessary facts about noncommutative ring theory as we need them. Chapter 2 of her book reviews the basic theory of categories, which we will go over very quickly, so her notes are a good place to see more details. Chapter 3 of the book discusses the basics of monoidal categories, which will also be the material for a good part of our course. This basic material also roughly corresponds to parts of Chapters 1, 2, 4 of EGNO.