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.