Counting real and complex curves in toric varieties - a tropical view
Tropical geometry provides a new view on counting real and complex curves and led to progress on Hilbert's 16th problem about the topology of real plane curves. Another important application is to mirror symmetry where tropical curves and disks provide a more tractable version of their complex analogues that help understand how such structures relate both sides of mirror symmetry (Gross-Siebert program). In a joint work with Travis Mandel, I use logarithmic Gromov--Witten theory to prove that tropical curve counts in toric varieties match up with log Gromov--Witten invariants under a non-superabundance assumption. This generalizes prior works of Mikhalkin and Siebert--Nishinou in particular by allowing for psi-class condition (tangency conditions). One of the most fascinating features of tropical curves is that they seem to know about the real and complex count of curves simultaneously by means of q-deformed Gromov--Witten invariants. I will present new results in this direction.