MT 855: Graduate Combinatorics

Spring 2018

Time: T/Th 9:30-10:50

Location: Maloney 560

Professor: Josh Greene

Office: Maloney 527

Course description: This course will be a panorama of topics in combinatorics, some classical, some very recent, and all fascinating. It is targeted at advanced undergraduate students and graduate students, with a view towards Boston College graduate students writing theses in (algebraic) geometry, topology, and number theory. Very frequently, combinatorial problems surface in different areas, and my hope, in part, is to help you spot them lurking beneath the surface and to train your instincts for addressing them. Furthermore, I hope to emphasize connections between combinatorics and different areas of greater local interest.

I envision the following breakdown of the material:

* classical combinatorics (Ramsey theory, extremal graph theory)

* probabilistic methods (graphs of large girth and chromatic number)

* topological methods (Kneser's conjecture, curves on surfaces)

* algebraic / spectral methods (intersections of set systems, eigenvalues and expansion)

* algebraic geometry / polynomial methods (combinatorial Nullstellensatz, finite field Kakeya, bounds on capsets)

* Guth and Katz's recent work on the Erdos distinct distance problem, which beautifully synthesizes all of the above.

Lecture Notes

Lecture 1.