In my two talks I'll explain the general features of my conjecture, joint with Gross, Hacking and Siebert, that the algebra of regular functions on an affine log CY (with maximal boundary), comes with a natural vector space basis, generalizing the characters of an algebraic torus, for which the structure constants for the multiplication rule are positive integers, counting holomorphic discs (on the mirror).

In particular, I'll explain how using these "theta functions" one can generalize the basic constructions of toric geometry. E.g. a single choice of anti-canonical normal crossing divisor on a Fano Y (conjecturally) canonically determines for each line bundle L on Y a basis of sections parameterized by the integer points of a "polytope" (more precisely, a piecewise integer affine  manifold with boundary).  The talks will be at a very general level -- in particular I won't assume any familiarity with cluster varieties or mirror symmetry