The Wonderful Geometry of Matroids

I taught a course in the Fall of 2019 called The Wonderful Geometry of Matroids, with the goal of understanding algebraic invariants of matroids through connections with topology and algebraic geometry. More specifically, I discussed the characteristic polynomial and the Orlik-Solomon algebra in relation with the cohomology of the complement of a complex hyperplane arrangement, I outlined the proof of log concavity of the characteristic polynomial using the cohomology ring of the wonderful variety (or more generally the Chow ring of the matroid), and I used similar machinery to explain the proof that the correlation constant of a matroid is always less than 2. I also did a little bit of Kazhdan--Lusztig theory of matroids at the end.

Each 50 minute class period consisted of 25-30 minutes of lecture followed by an exercise session. Here are the exercises, along with brief summaries of each lecture.

At some point Lectures 12 and 13 got combined and I was too lazy to renumber all of the exercises, which is why there is no Lecture 13 below. A few of the lectures were a bit longer than the others, and therefore I did not write any exercises for those.

Lecture 1: Definition and basic examples
Lecture 2: Deletion, contraction, and dualization
Lecture 3: Direct sums and truncation
Lecture 4: Realizability
Lecture 5: The poset of flats
Lecture 6: The lattice of flats
Lecture 7: The Mobius function
Lecture 8: The characteristic polynomial
Lecture 9: Mobius inversion and the characteristic polynomial
Lecture 10: The Poincare polynomial
Lecture 11: The long exact sequence
Lecture 12: The Orlik-Solomon algebra
Lecture 14: Log concavity
Lecture 15: The wonderful compactification
Lecture 16: The wonderful compactification (continued)
Lecture 17: The Chow ring
Lecture 18: The top degree part of the Chow ring
Lecture 19: The Chow ring and the characteristic polynomial
Lecture 20: The Kahler package
Lecture 21: Hodge--Riemann implies log concavity
Lecture 22: Correlation bounds
Lecture 23: Rayleigh's theorem
Lecture 24: Correlation and the Chow ring
Lecture 25: Maximizing correlation
Lecture 26: Intersection cohomology
Lecture 27: The reciprocal plane
Lecture 28: Kazhdan--Lusztig polynomials of matroids
Lecture 29: Cycles and complete graphs