Rota's conjecture and positivity of algebraic cycles in permutohedral varieties

Abstract

Rota's log-concavity conjecture predicts that the coefficients of the characteristic polynomial of a matroid form a log-concave sequence. I will give a proof of the conjecture for representable matroids using intersection theory in toric varieties. The same approach to the conjecture in the general case (for possibly non-realizable matroids) leads to several intriguing questions on higher codimension algebraic cycles in the toric variety associated to the permutohedron.