Tutte seminar - Dave Wagner

Proof of the monotone column permanent conjecture

Abstract:

Let A be a square matrix of real numbers which are weakly decreasing down each column, let J be the all-ones matrix of the same size, and let z be an indeterminate. In 1999, Jim Haglund, Ken Ono, and I conjectured that the permanent of the matrix zJ+A is a polynomial with only real roots. Last year, Petter Brändén, Jim, Mirkó Visontai and I proved a multivariate version of this. This is a relatively simple application (suitable as an introduction) of one of my favourite subjects -- the theory of multivariate stable polynomials.