Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Prerequisites
No specific classes are required, but the course presupposes mathematical maturity at the level of a first-year math graduate student.
Course Description
This course examines classical and modern developments in graph theory and additive combinatorics, with a focus on topics and themes that connect the two subjects. The course also introduces students to current research topics and open problems.
A foundational result in additive combinatorics is Roth’s theorem, which says that every subset of {1, 2, …, n} without a 3-term arithmetic progression contains o(N) elements. You will see a couple of different proofs of Roth’s theorem: (1) a graph theoretic approach and (2) Roth’s original Fourier analytic approach. A central idea in both approaches is the dichotomy of structure versus pseudorandomness, and it is one of the key themes of the course.
Topics
- Forbidding subgraphs
- Szemerédi’s regularity lemma
- Pseudorandom graphs
- Graph limits
- Roth’s theorem
- Structure of set addition
- The sum-product problem
Grading
The final grade will be determined by the minimum of the student’s performance in the two categories:
- Problem sets: 6 problem sets
- Writing assignments: (1) course notes and (2) Wikipedia contributions
There will be no exams. For borderline grades, participation may play a factor in determining the final grade. In addition, there will be a list of open problems for which any significant progress/resolution may, at the discretion of the instructor, result in a grading bonus, overriding the above grading criteria.