Curve Counting Correspondences: Bridging Tropical and Algebraic Geometry

Curve counting has a long history in mathematics. Modern approaches have led to powerful enumerative frameworks such as Gromov-Witten and Pandharipande-Thomas invariants. In parallel, tropical geometry has provided new techniques on curve counting problems.

In 2003, Mikhalkin proved that the number of rational algebraic curves on a toric surface passing through ddd points in general position coincides with its tropical analogue. A higher dimensional correspondence was established by Nishinou-Siebert in 2004, and a higher genus generalisation was proved by Bousseau in 2017. More recently, new methods have been developed to study such curve counting correspondences for non-toric surfaces, while their extension to higher dimensions remains largely open.

In this talk, we will focus on the correspondence between algebraic and tropical curve counting for rational curves in the projective plane, as well as on the relation between tropical curve counting and the enumeration of Newton subdivisions.