Abstract:
We announce some results which might bring a new insight into the classification of algebraic solutions to the sixth Painleve equation. The main results consist of the rationality of parameters, trigonometric Diophantine conditions, and what the author calls the Tetrahedral Theorem regarding the absence of algebraic solutions in certain situations. The method is based on fruitful interactions between the moduli theoretical formulation of Painleve VI and dynamics on character varieties via the Riemann-Hilbert correspondence.

Abstract:
Various Schlesinger transformations can be combined with a direct pull-back of a hypergeometric 2x2 system to obtain $RS^2_4$-pullback transformations to isomonodromic 2x2 Fuchsian systems with 4 singularities. The corresponding Painleve VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. This paper demonstrates a direct computation of Schlesinger transformations acting on several apparent singular points, and presents an algebraic procedure (via syzygies) of computing algebraic Painleve VI solutions without deriving full RS-pullback transformations.

Abstract:
In this paper, we classify all values of the parameters $\alpha$, $\beta$, $\gamma$ and $\delta$ of the Painlev\'e VI equation such that there are rational solutions. We give a formula for them up to the birational canonical transformations and the symmetries of the Painlev\'e VI equation.

Abstract:
Algebraic solutions of the sixth Painleve equation can be computed using pullback transformations of hypergeometric equations with respect to specially ramified rational coverings. In particular, as was noticed by the second author and Doran, some algebraic solutions can be constructed from a rational covering alone, without computation of the pullbacked Fuchsian equation. But the same covering can be used to pullback different hypergeometric equations, resulting in different algebraic Painleve VI solutions. This paper presents computations of explicit RS-pullback transformations, and derivation of algebraic Painleve VI solutions from them.

Abstract:
The solutions of the (nonlinear) Painleve VI differential equation having icosahedral linear monodromy group will be classified up to equivalence under Okamoto's affine F4 Weyl group action and many properties of the solutions will be given. There are 52 classes, the first ten of which correspond directly to the ten icosahedral entries on Schwarz's list of algebraic solutions of the hypergeometric equation. The next nine solutions are simple deformations of known PVI solutions (and have less than five branches) and five of the larger solutions are already known, due to work of Dubrovin and Mazzocco and Kitaev. Of the remaining 28 solutions we will find 20 explicitly using (the author's correction of) Jimbo's asymptotic formula. Amongst those constructed there is one solution that is 'generic' in that its parameters lie on none of the affine F4 hyperplanes, one that is equivalent to the Dubrovin--Mazzocco elliptic solution and three elliptic solutions that are related to the Valentiner three-dimensional complex reflection group, the largest having 24 branches.

Abstract:
In this letter we establish a connection of Picard-type elliptic solutions of Painleve VI equation with the special solutions of the non-stationary Lame equation. The latter appeared in the study of the ground state properties of Baxter's solvable eight-vertex lattice model at a particular point, $\eta=\pi/3$, of the disordered regime.

Abstract:
A starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2x2 isomonodromic Fuchsian systems associated to the Painleve VI equation. Up to birational automorphisms of the monodromy manifold, we find three transformations. Two of them are identified as the action of known quadratic or quartic transformations of the Painleve VI equation. The third transformation of the monodromy manifold gives a new transformation of degree 3 of Picard's solutions of Painleve VI.

Abstract:
Every finite branch solutions to the sixth Painleve equation around a fixed singular point is an algebraic branch solution. In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. The proof of this result relies on algebraic geometry of Painleve VI, Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces, resolutions of Kleinian singularities, and power geometry of algebraic differential equations. In the course of the proof we are also able to classify all finite branch solutions up to Backlund transformations.

Abstract:
We present an explicit method to perform similarity reduction of a Riemann-Hilbert factorization problem for a homogeneous GL (N, C) loop group and use our results to find solutions to the Painleve VI equation for N=3. The tau function of the reduced hierarchy is shown to satisfy the sigma-form of the Painleve VI equation. A class of tau functions of the reduced integrable hierarchy is constructed by means of a Grassmannian formulation. These solutions provide rational solutions of the Painleve VI equation.