it is a theorem of de bruijn and erdos that $n$ points in the plane determine 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 of this theorem, which confirms a “top-heavy” conjecture of dowling and wilson in 1975. i will give a sketch of the key idea of the proof, which uses the hard lefschetz theorem and the decomposition theorem in algebraic geometry. i will also talk about a log-concave conjecture on the number of independent sets. this is joint work with june huh.