The goal of this lecture series is to give an introduction to the program for understanding mirror symmetry introduced by myself and Siebert. In particular, we have recently given general constructions for mirrors of log Calabi-Yau manifolds with maximally degenerate boundary and Calabi-Yau manifolds equipped with a maximally unipotent degeneration. The key ingredient of these constructions is logarithmic Gromov-Witten theory, constructed by myself and Siebert as well as Abramovich and Chen.
The lectures will begin with a general introduction to logarithmic algebraic geometry and log and punctured Gromov-Witten theory. I will then explain the mirror constructions in terms of canonical wall structures as well as the more direct construction of "Intrinsic Mirror Symmetry".
Finally, I will discuss ongoing work with Keel, Hacking and Siebert, applying these constructions to understand compactifications of the moduli space of K3 surfaces .