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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Reciprocity for Valuations of Theta Functions - Gr
egory Muller (University of Oklahoma)
DTSTART;TZID=Europe/London:20211112T134500
DTEND;TZID=Europe/London:20211112T141500
UID:TALK164815AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/164815
DESCRIPTION:The Gross-Siebert program associates a theta funct
ion 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 clus
ter varieties). Since mirror duality is a symmetri
c relation\, this provides two ways to associate a
n 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 t
he valuation n to the theta function associated to
m.Resolving a conjecture of Gross-Hacking-Keel-Ko
ntsevich\, we show that these two numbers are equa
l in a generality which covers all cluster algebra
s (specifically\, when the theta functions are giv
en by enumerating broken lines in a scattering dia
gram generated by finitely-many elementary incomin
g walls). Time permitting\, I will discuss applica
tions to tropicalizations of theta functions\, Don
aldson-Thomas transformations\, and localizations
of cluster algebras. This work is joint with Man-w
ai Cheung\, Tim Magee\, and Travis Mandel.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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