# Theta functions for affine log CY varieties

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Sean Keel , University of Texas at Austin
Fine Hall 322

Gross, Hacking, Siebert and I conjecture that the vector space of regular functions on an affine log CY (with maximal boundary) comes with a canonical basis, generalizing the monomial basis on a torus, in which the structure constants for the multiplication rule are given by counts of rational curves on the mirror. Instances are a basis of the Cox ring of a Fano canonically determined by a single choice of anti-canonical divisor, one example of which gives a canonical basis for every irreducible representation of a semi-simple group (without doing any representation theory!). I'll explain the conjecture, these applications, and then some of the ideas in my recent construction, joint with Tony Yu, of the algebra in dimension two using some simple ideas from Berkovich analytic geometry.