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SUMMARY:Reciprocity for Valuations of Theta Functions - Gregory Muller (Un
 iversity of Oklahoma)
DTSTART:20211112T134500Z
DTEND:20211112T141500Z
UID:TALK164815@talks.cam.ac.uk
DESCRIPTION: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 aff
 ine log Calabi-Yau varieties with maximal boundary (such as cluster variet
 ies). Since mirror duality is a symmetric relation\, this provides two way
 s to associate an integer to a pair m and n of boundary valuations on X an
 d Y (respectively).1) Apply the valuation m to the theta function associat
 ed to n.2) Apply the valuation n to the theta function associated to m.Res
 olving a conjecture of Gross-Hacking-Keel-Kontsevich\, we show that these 
 two numbers are equal in a generality which covers all cluster algebras (s
 pecifically\, when the theta functions are given by enumerating broken lin
 es 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.
LOCATION:Seminar Room 1\, Newton Institute
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