Algebraic Combinatorics Seminar - David Wagner

Title: Proof of the monotone column permanent conjecture

Abstract: 

In 1993, Jim Haglund conjectured the following.  If  A  is a  square matrix of real numbers which are weakly decreasing down each column, and J is the all-ones matrix of the same size, then the permanent of the matrix xJ+A is a polynomial with only real roots.  Jim, Ken Ono and I proved this for 0-1 matrices in 1995.  Using the multivariate generalization of polynomials with only real roots developed by Borcea and Branden, in 2010 Branden, Haglund, Visontai and I proved a multivariate generalization of the whole thing.  I will present the main ideas of the proof.