Title: Matroids and the integral Hodge conjecture

Abstract: Associated to any regular matroid of rank g on k elements, one can associate a multivariable semistable degeneration of principally polarized abelian g-folds over a k-dimensional base. I will discuss joint work with de Gaay Fortman and Schreieder, proving that a combinatorial invariant of the matroid obstructs the algebraicity of the minimal curve class, on the very general fiber of the associated degeneration. Corollaries include the failure of the integral Hodge conjecture for abelian varieties of dimension ≥ 4 and the stable irrationality of very general cubic threefolds.