We prove the strongest version of Mason’s conjecture, that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials (CLC), whose bivariate restrictions have ultra log-concave coefficients.
I will talk about a number of polynomials which are CLC including homogenization of the generating polynomial of independent sets of a given matroid, or the multivariate Tuttle polynomial in certain regimes. If time permits I will talk about connections to High dimensional simplicial complexes.
Based on joint works with Nima Anari, Kuikui Liu and Cynthia Vinzant.
Note: Pre-seminar will start at 3:40pm. The main talk will start as usual at 4:10pm.