Math 261C (Spring 2024)
Summary
Math 261C is the third and final installment of the Probabilistic Combinatorics graduate sequence.
Since 261A/B already covered a lot of the core and fundamental material in probabilistic combinatorics, we will grant ourselves some license to choose some varied topics and areas of focus.
One such area I want to cover is entropy methods in combinatorics. Topics within this include:
- a recap on the fundamentals of entropy and information theory;
- the "entropy method" for obtaining combinatorial upper bounds (e.g., matchings and Minc's conjecture, hypergraph matchings, silly questions about chessboards);
- various applications of Shearer's lemma, and inequalities of Brascamp–Lieb type;
- entropy lower bounds and Sidorenko's conjecture;
- if I can find something sensible to say, meditations on the relationship to other techniques and the tensor power method;
- potentially some arithmetic applications.
Another area I plan to cover is the regularity lemma, its applications, and the correct way to think about it in terms of measure theory.
If these two topics do not cover the whole quarter, which seems very possible, I will choose some further ones as we go.
There is no required textbook for the course. Probabilistic combinatorics by Alon and Spencer remains the bible in this area, but it remains to be seen how much it will be relevant to the topics above.
Contacts
The instructor is Freddie Manners (email fmanners; office AP&M 7343). There is no TA for this class.
Class and office hours
Lectures are held on Tuesdays and Thursdays, 09:30am–10:50am. They will be held in-person in APM 2402.
There will be no class in Week 7 (Tuesday May 14 and Thursday May 16).
I will hold office hours twice per week (except week 7); the usual times are Tuesdays 1–2 PM and Thursdays 3–4 PM. Changes may occur from time to time; check the course calendar below for authoritative times.
Grading
There are two compenents to the grading of this class.
First, I will set two homework assignments, one due at the end of week 5 (Friday May 3) and the other at the end of week 10 (Friday June 7). These deadlines will be adhered to. You are not necessarily expected to complete all of the problems; students should contact the instructor if they wish to clarify the expectations around homework grading.
Second, all students will complete a brief, 20-30 minute presentation during class on a topic they find interesting. This need not be a fully-fledged lecture: a good presentation would focus on a particular paper or result, relate the background to the problem and explain why it is interesting, and describe some of the key ideas or tools in the paper, likely without giving full technical details. I intend this task to be approachable and not a significant drain on your time. The topic should be somehow relevant to Probabilistic Combinatorics, but potentially very broadly construed. Talk to the instructor for more precise suggestions or feedback if you wish. Arrangements for booking slots during class time will be made during the quarter; there will be a lot of flexibility offered, and certainly you may choose a slot towards the end of the quarter after qualfying exams have finished.
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.