Integrable Systems Seminar

[Boris Dubrovin]

Dear All,

Next seminar will take place on

Tuesday, January 23 at 11-12 am, room 139 of SISSA Main Building

Prof. Vladimir Dragović (The University of Texas at Dallas)

Algebro-geometric approach to the Schlesinger systems: from Poncelet to
Painlevé  VI and beyond

Abstract: A new method of construction of algebro-geometric solutions of
rank two Schlesinger systems is presented. For an elliptic curve
represented as a ramified double covering of CP^1, a meromorphic
differential is constructed with the following property: the common
projection of its two zeros on the base of the covering, regarded as a
function of the only moving branch point of the covering, is a solution of
a Painlevé VI equation. This differential provides an invariant
formulation of a classical Okamoto transformation for the Painlevé VI
equations. A generalization of this differential to hyperelliptic curves
is also constructed. The corresponding solutions of the rank two
Schlesinger systems associated with elliptic and hyperelliptic curves are
constructed in terms of these differentials. The initial data for the
construction of the meromorphic differentials include a point in the
Jacobian of the curve, under the assumption that this point has
non-variable coordinates with respect to the lattice of the Jacobian while
the branch points vary.  This method is motivated by an observation of
Hitchin, who related the Poncelet polygons to algebraic solutions of a
Painlevé VI equation. The research has been partially supported by the NSF
grant 1444147. This presentation is based on a joint work with Vasilisa
Shramchenko:
Vladimir Dragović, Vasilisa Shramchenko, Algebro-Geometric Solutions of
the Schlesinger Systems and the Poncelet-Type Polygons in Higher
Dimensions, International Mathematics Research Notices, rnx015,
https://doi.org/10.1093/imrn/rnx015 <https://doi.org/10.1093/imrn/rnx015>, 2017

Everybody is welcome

Boris Dubrovin