Title: Proof of the monotone column permanent conjecture
|University of Waterloo
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.