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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:20210412T031459Z
UID:indico-event-5713@indico.math.cnrs.fr
DESCRIPTION:Whereas the exponential map from a Lie algebra to a Lie group
can be viewed as the monodromy of a singular connection A dz/z on a di
sk\, the wild character varieties are the receptacles for the monodromy da
ta for arbitrary meromorphic connections on Riemann surfaces. This suggest
s one should think of the wild character varieties (or the full nonabelian
Hodge triple of spaces\, bringing in the meromorphic Higgs bundle moduli
spaces too) as global analogues of Lie groups\, and try to classify them.
As a step in this direction I'll explain some recent joint work with D. Ya
makawa that defines a diagram for any algebraic connection on a vector bun
dle on the affine line. This generalises the definition made by the spea
ker in the untwisted case in 2008 in arXiv:0806.1050 Apx. C\, related to
the « quiver modularity theorem »\, that a large class of Nakajima qu
iver varieties arise as moduli spaces of meromorphic connections on a triv
ial 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 th
e Okamoto symmetries of the Painlevé equations\, and recovers their speci
al solutions upon removing one node. The case of Painlevé 3 caused the mo
st difficulties.\n\nhttps://indico.math.cnrs.fr/event/5713/
LOCATION:IHES Centre de conférences Marilyn et James Simons
URL:https://indico.math.cnrs.fr/event/5713/
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