CO 739 -- combinatorial Hopf algebras with a focus on renormalization, Winter 2020

Information

We will be looking at combinatorial Hopf algebras. We will study a number of classical and interesting Hopf algebras that appear in combinatorics. There will be some emphasis on how Hopf algebras appear in renormalization in quantum field theory, and those examples which lead there, but we will also discuss many other examples.

You do not need to know any physics to take this course. You should know something about at least one of enumerative combinatorics, abstract algebra, or renormalization in quantum field theory.

I hope that we will have people with a variety of backgrounds in the course. How many different departments do you think we can get represented? How many different research areas from within departments? Undergrads can be signed in.

Instructor: Karen Yeats
Office:MC 5126
Email: kayeats at uwaterloo.ca
Office Hours: Tuesdays 1-3
Lectures: Tuesdays and Thursdays 11:30-12:50, MC 5417
Syllabus.

Announcements

• The room is now MC 5417 rather than a room in engineering.
• On the first day we will vote on details of the course, like how you will be evaluated and when my office hours will be, so please come and contribute your view and your vote.
• An error in A1 is now fixed, thanks to Nicole.
• By the polynomial Hopf algebra in Q3 of A2, I just mean the univariate polynomial Hopf algebra we discussed, a.k.a. the binomial Hopf algebra.
• Following the university directive, we will not meet March 17 and 19 and will resume online after that.
• Projects are due Tuesday April 14 at 1pm. I am happy to discuss the best way for you to submit your project.
• Class will be held in LEARN's "virtual classroom" (engine is Bongo). I will send out a link as I did for the test meeting.
• I will also send out a link for office hours on Tuesday. Also feel free to email with questions or to set up a video meeting.
• Bongo didn't work very well so now we are using Zoom. Links are going out by email.
• An email link will go out by email for presentations on Tuesday April 13.

Assignments

Assignments will be roughly biweekly and due on Thursdays.

1. Assignment 1 due Thursday February 13 in class. Solutions.
2. Assignment 2 due Thursday March 5 in class. Solutions.
3. Assignment 3 due Thursday April 2 at class time (you can email it to me). Solutions.

Class Summaries

These summaries are not meant to replace your own notes, but give an overview and useful references.

Part 0: Introduction

1. Lecture 1 summary.

Part 1: Hopf algebras in combinatorics

1. Lecture 2 summary.
2. Lecture 3 summary.
3. Lecture 4 summary.
4. Lecture 5 summary.
5. Lecture 6 summary.
6. Lecture 7 summary.
7. Lecture 8 summary.
8. Lecture 9 summary.
9. Lecture 10 summary.

Part 2: Renormalization by Hopf algebras

1. Lecture 11 summary.
2. Lecture 12 summary.
3. Lecture 13 summary.
4. Lecture 14 summary.
5. Lecture 15 summary.
6. Lecture 16 summary.
7. Lecture 17 summary and integration note.

Part 3: Sym and friends

1. Lecture 18 summary.
2. Lecture 19 summary.
3. Lecture 20: We filled in the following slides in today's virtual lecture NSym, duality of QSym and NSym, first commutative square, and NCSym.
4. Lecture 21: We filled in the following slides in today's virtual lecture second commutative square (corrected) and third commutative square. A reference on the third square is arXiv:0812.2419.
5. Lecture 22: We filled in the following slides in today's virtual lecture a commutative hexagon, third definition of combinatorial Hopf algebra, the chromatic symmetric function, and some links on categorification since we ran out of time to really cover categorification.