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SUMMARY:Diagrams\, Nonabelian Hodge Spaces and Global Lie Theory
DTSTART:20200302T130000Z
DTEND:20200302T140000Z
DTSTAMP:20260721T071300Z
UID:indico-event-5713@indico.math.cnrs.fr
CONTACT:cecile@ihes.fr
DESCRIPTION:Speakers: Philip Boalch (Paris Diderot)\n\nWhereas the exponen
 tial map from a Lie algebra to a Lie group can be viewed as the monodromy
   of a singular connection A dz/z on a disk\, the wild character variet
 ies are the receptacles for the monodromy data for arbitrary meromorphic c
 onnections on Riemann surfaces. This suggests one should think of the wild
  character varieties (or the full nonabelian Hodge triple of spaces\, brin
 ging in the meromorphic Higgs bundle moduli spaces too) as global analogue
 s of Lie groups\, and try to classify them. As a step in this direction I'
 ll explain some recent joint work with D. Yamakawa that defines a diagram 
 for any algebraic connection on a vector bundle on the affine line.  This
  generalises the definition made by the speaker in the untwisted case in 2
 008 in arXiv:0806.1050 Apx. C\, related to the « quiver modularity the
 orem »\, that a large class of Nakajima quiver varieties arise as moduli
  spaces of meromorphic connections on a trivial vector bundle the Riemann 
 sphere\, proved in the simply-laced case and conjectured in general in op.
 cit. (published in Pub. Math. IHES 2012)\, and proved in general by Hiroe
 -Yamakawa (Adv. Math. 2014).  In particular this construction of diagrams
  yields all the affine Dynkin diagrams of the Okamoto symmetries of the Pa
 inlevé equations\, and recovers their special solutions upon removing one
  node. The case of Painlevé 3 caused the most difficulties.\n\nhttps://in
 dico.math.cnrs.fr/event/5713/
LOCATION:Centre de conférences Marilyn et James Simons (IHES)
URL:https://indico.math.cnrs.fr/event/5713/
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