Title: Enumeration of points, lines, planes, etc.
Speaker: Botong Wang (University of Wisconsin-Madison)
Abstract: It is a theorem of de Bruijn and Erdős that n points in the plane determines at least n lines, unless all the points lie on a line. This is one of the earliest results in enumerative combinatorial geometry. We will present a higher dimensional generalization to this theorem. Let $E$ be a generating subset of a $d$-dimensional vector space. Let $W_k$ be the number of $k$-dimensional subspaces that is generated by a subset of E. We show that $W_k\leq W_{d-k}$, when $k\leq d/2$. This confirms a "top-heavy" conjecture of Dowling and Wilson in 1974 for all matroids realizable over some field. The main ingredients of the proof are the hard Lefschetz theorem and the decomposition theorem. I will also talk about a proof of Welsh and Mason's log-concave conjecture on the number of k-element independent sets. These are joint works with June Huh.
Seminar URL: https://research.math.osu.edu/agseminar/