I will describe, in language aimed at a second-year graduate student in mathematics, my conjecture, joint with Gross and Hacking, that affine Calabi-Yau manifolds (with maximal boundary) have "theta functions" — a canonical vector space basis for the ring of regular functions. Informally: If you live on a CY manifold, there are intrinsic quantities in your world whose values determine your position. This has loads of applications, as many of the fundamental objects in representation theory, combinatorics, geometry, topology, and string theory are CY. I will explain many corollaries of our partial results — the proof in dimension two, and, with Kontsevich, for cluster varieties of all dimensions. These corollaries include:

(1) A construction of a canonical basis for every irreducible representation of a semi-simple lie group, which we speculate is the canonical basis of Lusztig.

(2) A canoncal basis of functions on the character variety of a punctured Riemann surface.

(3) A construction of the Knutson-Tau "hives": lattice polytopes whose number of integer points are the Littleood-Richardson coefficients counting multiplies of irreducible factors in tensor products of irreducible representations as in (1).

(4) The classical theta functions for polarized Elliptic curves.

(5) Proof of the (corrected) Fock-Goncharov dual basis conjecture for cluster varieties.

All by a unified and conceptually simple construction that applies to any CY manifold, and in particular which has nothing a priori to do with representation theory, or Riemann surfaces, elliptic curves, or cluster varieties, and which (conjecturally) gives a simple synthetic construction of the mirror to an affine CY (with maximal boundary).