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SUMMARY:The Mano Decompositions and the Space of Monodromy Data of the $q$
-Painlevé V I Equation
DTSTART;VALUE=DATE-TIME:20190613T144500Z
DTEND;VALUE=DATE-TIME:20190613T154500Z
DTSTAMP;VALUE=DATE-TIME:20200716T041331Z
UID:indico-contribution-3770@indico.math.cnrs.fr
DESCRIPTION:Speakers: Jean-Pierre Ramis (Université Paul Sabatier)\nThe t
alk is based upon a joint work with Y. OHYAMA and J. SAULOY. Classically t
he space of Monodromy data (or character variety) of PV I (the sixth Painl
evé diﬀerential equation) is the space of linear representations of the
fundamental group of a 4-punctured sphere up to equivalence of representa
tions. If one ﬁxes the local representation data it “is” a cubic sur
face. We will describe a $q$-analog: the space of $q$-Monodromy data of th
e $q$-Painlevé V I equation. For the $q$-analogs of the Painlevé equatio
ns (which are non-linear $q$-diﬀerence equations)\, according to H. SAKA
I work\, “everything” is well known on the “left side” of the ($q$
-analog of the) Riemann-Hilbert map (the varieties of “initial condition
s”)\, but the “right side” (the $q$-analogs of the spaces of Monodro
my data or character varieties) remained quite mysterious. \nWe will prese
nt a complete description of the space of Monodromy data of $q$−PV I (so
me local data being ﬁxed). It is a “modiﬁcation” of an elliptic su
rface and we will explicit some “natural” parametrizations. This surfa
ce is analytically\, but not algebraically isomorphic to the Sakai surface
of ”initial conditions”. Our description uses a new tool\, the Mano d
ecompositions\, which are a $q$-analog of the classical pants decompositio
ns of surfaces. We conjecture that our constructions can be extended to th
e others $q$-Painlevé equations. This involves $q$-Stokes phenomena.\n\nh
ttps://indico.math.cnrs.fr/event/4521/contributions/3770/
LOCATION:Le Bois-Marie Centre de Conférences Marilyn et James Simons
URL:https://indico.math.cnrs.fr/event/4521/contributions/3770/
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