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SUMMARY:Towards Batyrev duality for cluster varieties - Tim Magee\, Birmin
 gham
DTSTART:20190528T150000Z
DTEND:20190528T160000Z
UID:TALK122602@talks.cam.ac.uk
CONTACT:Mark Gross
DESCRIPTION:In 1993 Batyrev gave a now-classic method for constructing (co
 njecturally) mirror families of Calabi-Yau varieties. In his picture\, the
  CY varieties\nare hypersurfaces of toric varieties\, and mirror duality s
 tems from a much\nsimpler duality of the ambient toric varieties. Recently
  cluster varieties\nhave emerged as ``the new toric varieties''. While the
 y are non-toric\, many\ntoric constructions generalize cleanly to the clus
 ter world\; and like toric\nvarieties\, cluster varieties are tractable al
 gebro-geometric objects that\nform a furtive testing ground for new ideas.
  Tropical versions of the\ncombinatorial gadgets used in the Batyrev const
 ruction (and a\ngeneralization by Batyrev-Borisov) appear naturally in the
  cluster world.\nThis led my collaborators (Bossinger\, Cheung\, Frías-Me
 dina\, and Nájera\nChávez) and I to ask:\nCan the Batyrev(-Borisov) cons
 truction of mirror families of CY\nsubvarieties of toric varieties be gene
 ralized to give a construction of\nmirror families of CY subvarieties of c
 luster varieties?\nThis is part of a long-term project branching out in se
 veral directions.\nIn an ongoing work\, M.-W. Cheung and I pursue this que
 stion in the simplest\nsetting available-- so-called finite-type cluster v
 arieties without frozen\ndirections. I'll briefly review the Batyrev const
 ruction and cluster\nvarieties\, then discuss what we have proved so far a
 nd how we intend to proceed.
LOCATION:CMS MR13
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