We will discuss a proof that the integral Hodge conjecture is false for a very general abelian variety of dimension ≥ 4. Associated to any regular matroid is a degeneration of principally polarized abelian varieties. We introduce a new combinatorial invariant of regular matroids, which obstructs the algebraicity of the minimal curve class, on the very general fiber of the associated degeneration. In concert with a result of Voisin, one deduces (via the intermediate Jacobian) the stable irrationality of a very general cubic threefold. This is joint work with Olivier de Gaay Fortman and Stefan Schreieder.

The SYZ conjecture provides a way to construct the mirror to a log Calabi-Yau manifold by counting specific rational curves inside the initial variety. In recent years, Keel-Yu and Gross-Siebert implemented the SYZ mirror construction using non-archimedean and logarithmic geometry respectively, and encoded the mirrors into a combinatorial object called a scattering diagram. A major open question is to compare these two approaches to the SYZ conjecture. In this talk, after reviewing these constructions and ideas, I will present a comparison result that expresses non-archimedean cylinder counts in terms of log Gromov-Witten invariants. In the surface case, I will explain how to deduce the exponential formula, a remarkable identity which relates the non-archimedean and logarithmic scattering diagrams, and implies the equivalence of the Keel-Yu and Gross-Siebert mirror constructions. The exponential formula is the first explicit formula relating non-archimedean Gromov-Witten invariants to punctured log Gromov-Witten invariants. If time permits, I will discuss a work in progress that extends the comparison to higher dimensions. Joint work with Tony Yue Yu, based on arXiv:2510.18319.