Math 264B (Winter 2021)
Summary
This is a graduate course in combinatorics. The focus is on Additive Combinatorics or more generally Approximate Algebraic Structure.
Roughly speaking, we are in the business of taking an algebraic object (e.g., a group), and replacing its axioms with versions that hold "most of the time" or "some of the time". For example, what happens to a group if we only insist (ab)c = a(bc) for 99% of triples (a,b,c)? Or 1% of triples?
We will focus fairly heavily on the classical questions of additive combinatorics (sumsets, additive energy etc.) as this is where the richest theory exists. Topics in this area include:
- doubling constants and approximate groups;
- Ruzsa's triangle inequality and Plunnecke's inequalities;
- the Balog–Szemeredi–Gowers theorem;
- Freiman homomorphisms;
- Freiman's theorem;
- Gowers norms and the inverse theorem;
- Roth's theorem and Szemeredi's theorem;
- approximate polynomials;
- no-three-in-a-line configurations;
- ...
Contacts
The instructor is Freddie Manners (email fmanners; my office is AP&M 7343, but don't go there).
Class and office hours
Lectures are held on Mondays, Wednesdays and Fridays, 0900–0950. Lectures will be held via Zoom. See Canvas for the Zoom links.
I will hold regular office hours as follows:—
| Mondays | Wednesdays |
| 10:00 AM – 11:00 AM | 1:30 PM – 2:30 PM |
A calendar is provided below for your convenience, and may be updated from time to time to reflect changes to this schedule.
Please note that lectures may be recorded in Zoom and reposted to the class.
Virtual attendance in class is mandatory and carries credit; see below. However, exceptions can be made if your timezone or circumstances make this logistically difficult. In that case, you can arrange to attend the lectures "doubly virtually" at another time. You must contact the lecturer to arrange this option, or discuss other logistical difficulties.
Grading
The grading for the class will be based on attendance (5% of grade; credit is capped at 80% attendance) and homework assignments (95% of grade).
Homework assigments will be conducted asynchronously: they may be submitted at any time until the last day of instruction, March 12. However, suggested unofficial due dates are given below.
Note that homework is held to high standards of academic integrity. Discussing homework with your peers is acceptable, but your solutions must be written by you alone, in your own words. Soliciting help with homework on the internet is a serious AI violation.
Undergraduates taking the course are strongly encouraged to discuss grading expectations with the instructor at the start of the quarter, and to attend office hours as often as possible.
While homework can — technically — be submitted at any time, the following table provides expectations about when homework will be released, and ideally returned.
| No | Release Date | Return Date (?) |
| 1 | Monday January 11 | Around January 25 |
| 2 | Monday January 25 | Around February 8 |
| 3 | Monday February 8 | Around February 22 |
| 4 | Monday February 22 | March 12 (last day of class) |