Reciprocity for Valuations of Theta Functions

Duration: 42 mins 16 secs

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Description:	Professor Gregory Muller (University of Oklahoma) 12th November 2021 \| 13?45 - 14:15


Created:	2021-11-15 15:47
Collection:	Cluster algebras and representation theory
Publisher:	Isaac Newton Institute for Mathematical Sciences
Copyright:	Professor Gregory Muller
Language:	eng (English)


Abstract:	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.

Abstract:

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.

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