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75
ON THE REDUCTIONS AND CLASSICAL SOLUTIONS OF THE SCHLESINGER EQUATIONS.
, 2006
"... To the memory of our friend Andrei Bolibruch Abstract. The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n+1 poles. They can be considered as a family of commuting timedependent Hamiltonian systems on the direct product of n copies of m × ..."
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To the memory of our friend Andrei Bolibruch Abstract. The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n+1 poles. They can be considered as a family of commuting timedependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we address the problem of reduction of particular solutions of “more complicated” Schlesinger equations S (n,m) to “simpler ” S (n ′,m ′ ) having n ′ < n or m ′ < m. Contents
Finite orbits of the braid group action on sets of reflections, www.arxiv.org mathph/0409026
, 2004
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From Klein to Painlevé via Fourier, Laplace and Jimbo
, 2004
"... We will describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation, generalising that of Dubrovin–Mazzocco. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL2(C) out of triples of generators of ..."
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We will describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation, generalising that of Dubrovin–Mazzocco. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL2(C) out of triples of generators of threedimensional complex reflection groups. (This involves the Fourier–Laplace transform for certain irregular connections.) Then we adapt a result of Jimbo to produce the Painlevé VI solutions. (In particular this solves a Riemann–Hilbert problem explicitly.) Each step will be illustrated using the complex reflection group associated to Klein’s simple group of order 168. This leads to a new algebraic solution with seven branches. We will also prove that, unlike the algebraic solutions of Dubrovin–Mazzocco and Hitchin, this solution is not equivalent to any solution coming from a finite subgroup of SL2(C). The results of this paper also yield a simple proof of a recent theorem of Inaba– Iwasaki–Saito on the action of Okamoto’s affine D4 symmetry group as well as the correct connection formulae for generic Painlevé VI equations. Klein’s quartic curve
Stokes Matrix for the Quantum Cohomology of Cubic Surfaces
"... We prove the conjectural relation between the Stokes matrix for the quantum cohomology of X and an exceptional collection generating D b coh(X) when X is a smooth cubic surface. The proof is based on a toric degeneration of a cubic surface, the Givental’s mirror theorem for toric manifolds, and the ..."
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We prove the conjectural relation between the Stokes matrix for the quantum cohomology of X and an exceptional collection generating D b coh(X) when X is a smooth cubic surface. The proof is based on a toric degeneration of a cubic surface, the Givental’s mirror theorem for toric manifolds, and the PicardLefschetz theory. 1
Painlevé equations and complex reflections
 PROCEEDINGS OF THE CONFERENCE IN HONOR OF F. PHAM
, 2002
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New modular spaces of pointed curves and pencils of flat connections
 Michigan Math. J
"... Abstract. It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad (H∗(M0,n+1)) of the moduli spaces of n–pointed stable curves of genus zero. In this paper we establish a similar relationship between the pencils of formal flat co ..."
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Abstract. It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad (H∗(M0,n+1)) of the moduli spaces of n–pointed stable curves of genus zero. In this paper we establish a similar relationship between the pencils of formal flat connections (or solutions to the Commutativity Equations) and homology of a new series Ln of pointed stable curves of genus zero. Whereas M0,n+1 parametrizes trees of P 1 ’s with pairwise distinct nonsingular marked points, Ln parametrizes strings of P 1 ’s stabilized by marked points of two types. The union of all Ln’s forms a semigroup rather than operad, and the role of operadic algebras is taken over by the representations of the appropriately twisted homology algebra of this union. 0. Introduction and plan of the paper One of the remarkable basic results in the theory of the Associativity Equations (or Frobenius manifolds) is the fact that their formal solutions are the same as cyclic algebras over the homology operad (H∗(M0,n+1)) of the moduli spaces of n– pointed stable curves of genus zero. This connection was discovered by physicists,
The geometry of dual isomonodromic deformations.
, 2003
"... The JMMS equations are studied using the geometry of the spectral curve of a pair of dual systems. It is shown that the equations can be represented as timeindependent Hamiltonian flows on a Jacobian bundle. 1 ..."
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The JMMS equations are studied using the geometry of the spectral curve of a pair of dual systems. It is shown that the equations can be represented as timeindependent Hamiltonian flows on a Jacobian bundle. 1
RiemannHilbert problem associated to Frobenius manifold structures on Hurwitz spaces: irregular singularity
 Duke Math. J
, 2008
"... Abstract. Solutions to the RiemannHilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferen ..."
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Abstract. Solutions to the RiemannHilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferentials defined on the underlying Riemann surface. The relationship between different classes of Frobenius manifolds structures on Hurwitz spaces (real doubles, deformations) is described at the level of the corresponding RiemannHilbert problems. 1
Purely nonlocal Hamiltonian formalism for systems of hydrodynamic type
, 812
"... We study purely nonlocal Hamiltonian structures for systems of hydrodynamic type. In the case of a semiHamiltonian system, we show that such structures are related to quadratic expansions of the diagonal metrics naturally associated with the system. ..."
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We study purely nonlocal Hamiltonian structures for systems of hydrodynamic type. In the case of a semiHamiltonian system, we show that such structures are related to quadratic expansions of the diagonal metrics naturally associated with the system.