Philip Engel, from the University of Illinois, presents a proof that the integral Hodge conjecture fails for a very general abelian variety of dimension four or higher, delivered as part of a workshop on Hodge theory and o-minimality at the Institute for Advanced Study. Engel builds the argument from regular matroids, showing how each one gives rise to a degeneration of principally polarized abelian varieties. He introduces a new combinatorial invariant of regular matroids that blocks the minimal curve class on the generic fiber of this degeneration from being algebraic. Combined with a theorem of Claire Voisin, applied through the intermediate Jacobian, the result also yields the stable irrationality of a very general cubic threefold. The work is joint with Olivier de Gaay Fortman and Stefan Schreieder. This is a specialist research talk aimed at algebraic geometers already fluent in Hodge theory, matroid combinatorics, and moduli of abelian varieties, tracing a chain of reasoning from discrete combinatorial structures to a deep classical question about algebraic cycles.