I'll explain my recent construction, with Hacking
and Gross, of the mirror family to a log Calabi-Yau
surface, and some of its many applications both to
symplectic topology, and classical
algebraic geometry. For example as a corollary
of the construction we discover: The complement of a plane
cubic has "theta-functions" -- a canonical basis of polynomial
functions, analogous to (but considerably rich than) classical
theta functions for Abelian varieties, together with a rule for multiplying them determined by counts of plane rational curves
meeting the cubic at a single point.
We also conjecture that this coordinate ring is the symplectic homology
ring of this (open) CY surface, so in particular the symplectic homology
has the same theta function structure.