Abstract:The Hodge-Riemann relations (HRR) for graded Artinian Gorenstein (AG) algebras are an algebraic analogue of a certain property of the cohomology ring of a smooth complex projective algebraic variety which strengthen the strong Lefschetz property (SLP). In terms of Macaulay duality, the HRR are signature conditions on the higher Hessian matrices, whereas the SLP is a collection of non-degeneracy conditions. Recently P. Brändén and J. Huh introduced a class of real homogeneous polynomials called strictly Lorentzian polynomials, which turn out to characterize (in some sense) the Macaulay dual generators of AG algebras satisfying HRR in degree i = 1. In codimension two, strictly Lorentzian polynomials are defined by certain log concavity conditions on their coefficients. In this talk, I will focus on the codimension two case and discuss some new results, including a notion of higher log concavity and a new class of higher strictly Lorentzian polynomials which characterize the Macaulay dual generators of AG algebras satisfying HRR in degree i > 1. This is joint work with P. Macias-Marques, A. Seceleanu, and J. Watanabe.
Chris McDaniel (Endicott College) Higher Lorentzian Polynomials, Higher Hessians, and the Hodge-Riemann relations for codimension two graded Artinian Gorenstein algebras
Lake Hall 509/511