This is a first course in matroid theory, with an emphasis on studying certain algebraic invariants of matroids that arise from topology and algebraic geometry. We will begin with cryptomorphic definitions, standard constructions, and connections to graph theory and linear algebra. We will then shift our focus to the Tutte polynomial, the Chow ring, and other invariants which have been at the center of recent developments. As our primary goal, we will work to understand the 2015 proof of the long-standing Rota-Heron-Welsh conjecture by Adiprasito, Huh, and Katz. Possible further topics include Mnëv universality, algebraic matroids, tropical geometry, matroid polytopes, and Kazhdan-Lusztig polynomials.
Each class period will consist of a 25-30 minute lecture followed by an exercise session. You are expected to attend and participate in the exercise sessions, to think about most of the exercises, and to submit written solutions to some of the exercises.
Lecture notes and exercises are posted here.Solutions to starred exercises will be graded on the following scale:
Written solutions to starred exercises may be submitted at any time but no later than April 30. Exercises will be graded and returned to you within two weeks.
Course grades will be assigned on the following scale:
Matroid Theory, 2nd ed., by James Oxley
Matroids: A Geometric Introduction by Gary Gordon and Jennifer McNulty
Lectures on matroids and oriented matroids by Victor Reiner
The geometry of matroids by Federico Ardila (Notices Amer. Math. Soc. 65 (2018), no. 8, 902–908)
Hodge theory in combinatorics by Matthew Baker (Bull. Amer. Math. Soc. 55 (2018), no. 1, 57-80)