Marta Mazzocco

Monodromy of certain Painlevé–VI transcendents and reflection groups

Abstract.We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β=γ=0, δ=1/2 and 2α=(2μ-1)2 with arbitrary μ, 2μ≠∈ℤ. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space.

Picard and Chazy solutions to the Painleve\\\' VI equation

Abstract. We study the solutions of a particular family of Painlevé VI equations with parameters

Rational solutions of the Painlevé VI equation

\beta=\gamma=0, \delta=\frac{1}{2}

The geometry of the classical solutions of the Garnier systems

and

The Hamiltonian structure of the second Painlevé hierarchy

2\alpha=(2\mu-1)^2

Existence and uniqueness of tri-tronquée solutions of the second Painlevé hierarchy

, for

A remark on the Hankel determinant formula for solutions of the Toda equation

2\mu\in{\mathbb Z}

Non-symmetric basic hypergeometric polynomials and representation theory for confluent Cherednik algebras

. We show that in the case of half-integer

Cubic and Quartic Transformations of the Sixth Painlevé Equation in Terms of Riemann–Hilbert Correspondence

\mu

Dualities in the q-Askey Scheme and Degenerate DAHA: Dualities in the q-Askey Scheme and Degenerate DAHA

, all solutions can be written in terms of known functions and they are of two types: a two-parameter family of solutions found by Picard and a new one-parameter family of classical solutions which we call Chazy solutions. We give explicit formulae for them and completely determine their asymptotic behaviour near the singular points