MATH 261C
Instructor: Jacques Verstraete
General Information
Welcome to Math 261C. You will find course materials, announcements, and any lecture notes on this site. There will be two homework assignments. The homework assignment details are given below, and homework questions are contained at the end of each chapter of the notes. You will be required to submit written solutions to selected problems, on which your grade for the course is based - see below. It is your responsibility to be aware of the parameters of academic integrity and student conduct. Any suspected cases are reported to the integrity office.
All lectures will be delivered remotely using zoom. The lectures will be recorded and posted to canvas (there are currently some issues with this). Lectures here follow more or less what was written by the instructor during zoom sessions. Course notes are below.
Syllabus and Notes
We will be covering a selection of research topics in extremal and probabilistic combinatorics and combinatorial number theory.
Notes are available here (includes exercises, updated 6/5)
Some of the following topics will be covered:
- Equidistribution theory and rational approximation
- Invariant linear equations
- Roth`s Theorem and Sarközy`s Theorem
- Hardy-Littlewood circle method, Waring`s problem
- Geometry of sumsets, Plünnecke-Ruzsa Inequality, Freiman`s Theorem
- Sum-product theorems, restricted sumsets
- Quadratic sieve, Product representations
If time permits:
- Pseudorandom graphs and hypergraphs, regularity lemmas
- Counting independent sets, Container method and applications
- Ordered graphs and hypergraphs, Füredi-Hajnal, Marcus-Tardos
- Rödl nibble, differential equations method, existence of designs, applications
Books
Material not found in books will be covered, however, some helpful books include:
- Lovász, L. Large networks and graph limits.
- Nathanson, M. Additive number theory. Inverse problems / geometry of sumsets.
- Vaughan, R. C. The Hardy-Littlewood method.
- Tao, T; Vu, V. Additive combinatorics.
Homework and Office Hours
An office hour will be held at noon after class on Wednesdays via zoom. Note that office hours are not recorded.
For homework assignments, each student will complete three questions, one from each of Chapters 0, 1 and 2 of the notes. The homework should be prepared in TeX or LaTeX and compiled and submitted as a pdf file. Each student should by Monday April 20th send me the three questions they plan to submit.
The homework will be due for electronic hand in on Monday April 27th. In the online notes, the problems in blue are the ones that you should select questions from, one from each of Chapter 0, 1 and 2. If you feel strongly you would like to try a different question, please let me know.
Your grade for the course is based on the submitted homework.