Math 262A (Spring 2022)
Summary
This is a graduate topics course in combinatorics. The course will focus on the polynomial method in extremal combinatorics.
To mangle a Feynman quote, you should always keep in the back of your mind your 12 favorite problems and your 12 favorite tricks for solving problems; eventually something will match up and everyone will call you a genius.
In extremal combinatorics, one of your 12 solution slots should be reserved for "the polynomial method". Time and again over the last couple of decades, short and elegant solutions of previously intractable problems have used this technique, but it is not an approach you might think of organically.
The goal of this course is to be able to attempt to find matches between the polynomial method and whatever your 12 favorite problems are. We will do this by covering a fairly standard list of proof highlights: the joints problem in incidence geometry, Dvir's proof of finite field Kakeya, the recent breakthroughs on the capset problem, etc..
The exact list of topics is subject to change / class interest / etc..
Contacts
The instructor is Freddie Manners (email fmanners); my office is AP&M 7343.
Class and office hours
Lectures are held on Mondays, Wednesdays and Fridays, 1100–1150. Lectures will be held in-person in AP&M 2402. They will be podcasted.
I will hold regular office hours as follows:—
| Mondays | Wednesdays |
|---|---|
| 10:00am – 11:00am | 12:30pm – 1:30pm |
A calendar is provided below for your convenience, and may be updated from time to time to reflect changes to this schedule.
Homework
I will post one or more problem sets on the course Canvas page. They will be significantly but not entirely optional. Any solutions can be graded via the course Gradescope page, including but not limited to those being graded for class credit.
Resources
In addition to this website, the course has a Canvas page and a Gradescope page. The sign-up code for Gradescope is listed on the Canvas home page.
There is no assigned course textbook. The union of the following sets of pre-existing notes will contain most of what is covered in the course and may be consulted for reference:—