An old theorem, due to Mikhalkin, says that the number of
rational plane curves of degree d through 3d-1 points is equal to a
count of tropical curves (combinatorial objects which are more amenable
to computations). There are two natural directions for generalising this
result: extending to higher genus curves and allowing for more general
conditions than passing through points. I'll discuss a generalisation
which does both, which on the tropical side relates to the refined
invariants of Blechman and Shustin. At the end I will mention some
recent work connecting this story to mirror symmetry for log Calabi-Yau
surfaces. This is joint work with Patrick Kennedy-Hunt and Ajith
Urundolil Kumaran.
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