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Title:ÌýProof of the monotone column permanent conjecture

Abstract:Ìý

In 1993, Jim Haglund conjectured the following.Ìý IfÌý AÌý is aÌý 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. ÌýJim, Ken Ono and I proved this for 0-1 matrices in 1995.Ìý 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.Ìý I will present the main ideas of the proof.Ìý