Painleve' transcendents and two-dimensional topological field theory

Monodromy of certain Painleve' VI transcendents and reflection groups

We study the global analytic properties of the solutions of a particular family of Painleve' VI equations with the parameters $β=γ=0$, $δ={1\over2}$ and $α$ arbitrary. 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. This result is used to classify all the algebraic solutions of o…

On the algebraic independence of generic Painleve transcendents

We prove that if y" = f(y,y',t) is a generic Painleve equation from among the classes II to V then any collection of distinct solutions and their derivatives are algebraically independent over C(t). (Already proved by Nishioka for the single Painleve I equation). For generic Painleve VI we prove a slightly weaker statement.

泛型Painlevé超越的代数独立性：P_III和P_VI

Algebraic Independence of generic Painlevé Transcendents: P_III and P_VI

本文证明了泛型Painlevé方程III和VI的超越函数在C(t)上代数独立，并解决了Riccati方程解的代数独立性问题。

Poles Distribution of PVI Transcendents close to a Critical Point (summer 2011)

The distribution of the poles of branches of the Painleve' VI transcendents associated to semi-simple Frobenius manifolds is determined close to a critical point. It is shown that the poles accumulate at the critical point, asymptotically along two rays. The example of the Frobenius manifold given by the quantum cohomology of the two-dimensional complex projective space is also considered.

Towards a nonlinear Schwarz's list

This is basically the text of a survey talk (entitled 'Painleve, Klein and the icosahedron') given at Hitchin's 60th birthday conference. It discusses the search for and construction of algebraic solutions of the sixth Painleve differential equation, which may be viewed as a nonlinear analogue of the Gauss hypergeometric equation. Both algebraic and transcendental methods are used and the story involves affine Weyl groups, braid groups and cubic surfaces. Some emphasis is given to the interpretation of the sixth Painleve equation as the explicit form of the simplest nonabelian Gauss-Manin conn…

拓扑弦的复活结构与第一类 Painlevé 方程

Resurgent Structure of the Topological String and the First Painlevé Equation

本文给出拓扑弦配分函数上Stokes自同构的显式公式，揭示平坦坐标按Delabaere-Dillinger-Pham公式作为量子周期变换，并证明该公式既源于Painlevé I方程的非线性Stokes现象，也适用于任意Calabi-Yau三维流形。