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SUMMARY:Diagrams\, Nonabelian Hodge Spaces and Global Lie Theory
DTSTART;VALUE=DATE-TIME:20200302T130000Z
DTEND;VALUE=DATE-TIME:20200302T140000Z
DTSTAMP;VALUE=DATE-TIME:20221001T105800Z
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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