Canonical coordinates for Calabi Yau manifolds I

In the two talks, aimed at a broad mathematical audience, I'll explain my conjecture, joint with Gross, Hacking, and Siebert, which says, informally, that if you live on a Calabi Yau, your world comes with an intrinsic global positioning system. More precisely: that the coordinate ring comes with a canonical vector space basis, such that the structure constants for the multiplication rule are given by counts of rational curves (on the mirror). Then I'll explain our partial results (joint with Kontsevich and Yu), and then spend most of the time discussing some of the myriad applications---to Teichmuller theory, birational geometry, moduli spaces, representation theory, cluster algebras, and mirror symmetry.