The Tropical Vertex

From CRCG-Wiki

One insight of mirror symmetry is the fact that the enumerative geometry of rational curves is related to the deformation theory of a "mirror variety". A pro-unipotent group of automorphisms of the algebraic 2-torus over rules the constructions of maximal degenerations (Kontsevich/Soibelman, Gross/S.). On the mirror side this group should have some enumerative geometry meaning. In the talk I will present joint work with M. Gross (UCSD) and R. Pandharipande (Princeton) showing that this group indeed cleverly organizes a certain type of Gromov-Witten invariants on all toric surfaces into an algebraic structure. This correspondence runs via tropical geometry.