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Reciprocity for Valuations of Theta Functions

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CARW03 - Interdisciplinary applications of cluster algebras

The Gross-Siebert program associates a theta function on X to each boundary valuation on Y, where X and Y are a pair of mirror dual affine log Calabi-Yau varieties with maximal boundary (such as cluster varieties). Since mirror duality is a symmetric relation, this provides two ways to associate an integer to a pair m and n of boundary valuations on X and Y (respectively).1) Apply the valuation m to the theta function associated to n.2) Apply the valuation n to the theta function associated to m.Resolving a conjecture of Gross-Hacking-Keel-Kontsevich, we show that these two numbers are equal in a generality which covers all cluster algebras (specifically, when the theta functions are given by enumerating broken lines in a scattering diagram generated by finitely-many elementary incoming walls). Time permitting, I will discuss applications to tropicalizations of theta functions, Donaldson-Thomas transformations, and localizations of cluster algebras. This work is joint with Man-wai Cheung, Tim Magee, and Travis Mandel.

This talk is part of the Isaac Newton Institute Seminar Series series.

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