An undergraduate seminar during the fall of 2017.
The goal of this seminar is to understand the recent proof of Rota–Welsh Conjecture due to Adiprasito–Huh–Katz.
Briefly, the Rota–Welsh Conjecture is a combinatorial conjecture that posits that the coefficients of a certain polynomial associated with certain combinatorial objects, called matroids, form a log-concave sequence. Log-concavity of a sequence is a slight strengthening of unimodality of a sequence, i.e. if the sequence first increases then decreases. The basic case to keep in mind is that the coefficients of the chromatic polynomial of a graph are log-concave.
The recent work of Adiprasito–Huh–Katz establishes the Rota–Welsh Conjecture by developing a combinatorial Hodge theory for matroids in general. Their methods, though inspired by algebraic geometric considerations, are elementary in the sense that they do not need to directly invoke the algebro-geometric muse.
The primary reference for this seminar is the original paper
See also their Notices survey
Matt Baker wrote a blog article here and a survey
At some point, we will need to discuss matroids. Some references include:
J. G. Oxley, Matroid theory. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992.
E. Katz, Matroid theory for algebraic geometers. Simons Symposium Proceedings.
We meet every Tuesday in Mathematics Room 507 between 8:30PM and 10:30PM.