November 26-30, 2012

## Program

Conference Program Conference Program (Japanese)

## Abstract

Abstract (1.5MB)
Monday, 26th November

13:30 -- 14:25 **Michiaki Inaba** (Kyoto)

Moduli of parabolic connections of spectral type

14:40 -- 15:35 **Frank Loray** (Rennes, France)

Lagrangian fibrations on moduli spaces of connections

15:50 -- 16:45 **Hiroshi Umemura** (Nagoya)

Can we quantify Galois theory?

Tuesday, 27th November

10:00 -- 10:55 **Takuro Mochizuki** (RIMS)

Harmonic bundle and Toda lattice with opposite sign [preprint]

11:05 -- 12:00 **Philip Boalch** (ENS, Paris, France)

Transformation groups for isomonodromy equations

13:20 -- 14:15 **Victor Novokshenov** (Ufa, Russia)

Truncated solutions of the Painleve equations and singularities
of the monodromy manifold

14:30 -- 15:25 **Andrei Kapaev** (SISSA, Trieste, Italy)

On an equation of isomonodromic deformations without the Painleve property
(Joint work with B.A. Dubrovin)

15:40 -- 16:35 **Nalini Joshi** (Sydney, Australia)

Geometry and asymptotics of the Painleve Equations

16:45 -- 17:15 **Kohei Iwaki** (RIMS)

Voros coefficients of Painleve equations and Parametric Stokes phenomena

Wednesday, 28th Nobember

9:40 -- 10:35 **Davide Guzzetti** (SISSA, Trieste, Italy)

A tabulation of the Painleve 6 transcendents

10:45 -- 11:35 **Irina Goryuchkina** (Keldysh Institute, Moscow, Russia)

Methods of plane power geometry and formal solutions to the sixth Painleve
equation>

11:45 -- 12:40 **Alexander Bruno** (Keldysh Institute, Moscow, Russia)

All elliptic expansions of solutions to the Painleve equations

13:30 -- Poster Session

18:30 -- Receptions

Thursday, 29th November

10:00 -- 10:55 **Hayato Chiba** (Kyushu)

Blow-up of vector fields and dynamical systems of compactified Painleve
equations

11:05 -- 12:00 **Peter A. Clarkson** (Kent, UK)

Discrete and continuous Painleve equations and semi-classical orthogonal
polynomials

13:20 -- 14:15 **Laura Desideri** (Lille, France)

Describing minimal surfaces using isomonodromic deformations equations

14:30 -- 15:25 **Tetsu Masuda** (Aoyama Gakuin)

An Explicit Formula for the Discrete Power Function Associated with Circle
Patterns of Schramm Type

15:40 -- 16:35 **Hajime Nagoya** (Kobe)

On the sixth quantum Painleve equation

16:45 -- 17:15 **Irfan Mahmood** (Angers, France)

Lax pair representation and Darboux transformation of noncommutative
Painleve's second equation

Friday, 30th November

10:00 -- 10:55 **Claude Mitschi** (Strasbourg, France)

The monodromy of parameterized linear differential systems

11:05 -- 12:00 **Reinhard Schäfke** (Strasbourg, France)

On Parameter asymptotics for the second Painleve equation

13:20 -- 14:15 **Toshio Oshima** (Tokyo)

Linear differential equations on the Riemann sphere

14:30 -- 15:25 **Hidetaka Sakai** (Tokyo)

Toward a classification of four-dimensional Painleve-type equations (Joint work
with H. Kawakami and A. Nakamura)

15:40 -- 16:10 **Takato Uehara** (Niigata)

The entropy values of automorphisms on rational surfaces

16:15 -- 16:45 **Masataka Kanki** (Tokyo)

Painleve equations modulo a prime number

## Abstract

**Frank Loray** (Rennes, France)

**Title: Lagrangian fibrations on moduli spaces of connections **

**Abstract**: We investigate the geometry of moduli spaces of rank 2 logarithmic
connections on curves. After recalling what is known on the Painlevé VI case
(genus 0 with 4 poles), we will describe some work in progress with Masa-Hiko Saito
on the Garnier case (genus 0 with n poles) and the relationship for n=6 with the
case of holomorphic connections on genus 2 curves (joint work with Viktoria Heu).

**Claude Mitschi** (Strasbourg, France)

**Title: The monodromy of parameterized linear differential systems**

**Abstract**: The talk is about joint work with Michael. F. Singer. I will describe various
properties of the monodromy for parameterized families of linear differential systems with
regular singularities only. These include an analogue of the weak Riemann-Hilbert problem
and of Tretkoff's density theorem, in the framework of Cassidy and Singer's parameterized
Picard-Vessiot theory. I will also show how ``projectively isomonodromic deformations"
may translate into a purely algebraic condition on the parameterized Picard-Vessiot group.
An example of such deformation was studied by S. Chakravarty, M. J. Ablowitz and Y. Ohyama who
showed that the Darboux-Halphen equation accounts for this type of deformation in the same
way as the Painlevé equations account for isomonodromic deformations.

**Davide Guzzetti** (SISSA, Italy)

**Title: A tabulation of the Painlevé 6 transcendents **

**Abstract**: In the last decades the Painlevé equations have emerged as
one of the central objects in pure mathematics and mathematical physics,
with applications in a variety of problems, such as number theory,
theory of analytic varieties (like Frobenius structures), random matrix
theory, orthogonal polynomials, non linear evolutionary PDEs,
combinatorial problems, etc.
The properties of the classical (linear) transcendental functions
have
been organised and tabulated in various classical handbooks. A
comparable organisation and tabulation of the properties of the
Painlevé functions is now needed.
I will show that today we can write an essentially complete table of
the critical behaviors and expansions of the Painlevé 6 functions, and
the corresponding connection formulae.
I will also describe the essential steps - based on the method of
monodromy preserving deformations - which have provided the table, and
comment on the properties of the tanscendents tabulated. The talk is
based on the table published in Tabulation of the Painlevé 6 transcendents, in Nonlinearity

**Peter Clarkson** (Kent, UK)

**Title: Discrete and continuous Painlevé equations and semi-classical orthogonal polynomials**

**Abstract**: In this talk I shall discuss the relationship between the Painlevé equations, discrete Painlevé equations and orthogonal polynomials with respect to semi-classical weights. It is well-known that orthogonal polynomials satisfy a three-term recurrence relation and for some semi-classical weights, these coefficients in the recurrence relation can be expressed in terms of solutions of a Painlevé equation or solutions of a discrete Painlevé equation. I shall illustrate how the properties of the orthogonal polynomials are related to properties of the continuous and discrete Painlevé equations.

**Andrei Kapaev** (St. Petersburg, Russia)

**Title: On an equation of isomonodromic deformations without the Painlevé property**

(Joint work with B.A. Dubrovin)

**Abstract**: We present a fourth order first degree nonlinear ODE which
governs isomonodromy deformations of a linear 2x2
matrix ODE with polynomial coefficients but does not possess
the Painlevé property. In fact, this nonlinear ODE is satisfied
by a 4-parameter Puiseux series in degrees of (t-a)^{1/3}.
We explain the natural origin of this equation, deduce its
Lax pair, describe some features of the set of the related
Riemann-Hilbert problems and discuss their solvability.

**Alexander Bruno** (Keldysh, Moscow, Russia)

**Title: All elliptic expansions of solutions to the Painlevé equations**

**Extended Abstract**

**Nalini Joshi** (Sydney, Australia)

**Title: Geometry and asymptotics of the Painlevé Equations**

**Abstract**:

**Victor Novokshenov** (Ufa, Russia)

**Title: Truncated solutions of the Painlevé equations and singularities of the monodromy manifold**

**Extended Abstract**

**Laura Desideri** (Lille, France)

**Title: Describing minimal surfaces using isomonodromic deformations equations**

**Abstract**: I will present a correspondence due to R. Garnier between minimal surfaces with a
polygonal boundary curve and a certain class of Fuchsian equations. In this
correspondence, the monodromy of an equation is prescribed by the edge directions of
the polygonal boundary curve of the associated minimal surface. We will see how
isomonodromic deformations can then provide us with an explicit description of
minimal disks, that can be used to solve the Plateau problem. We will then discuss
the possibility to extend this point of view to minimal annuli, whose associated
equations are defined on an elliptic curve.

**Reinhard Schäfke** (Strasbourg, France)

**Title: On Parameter asymptotics for the Second Painlevé Equation**

**Abstract**: We consider P II, i.e. d^{2}y/dx^{2}=2y^{3}+xy+a and study solutions
without poles in certain x-regions when the parameter a is large.
We prove Borel-summability of the formal solutions in the parameter
on regions bounded by Stokes curves.

The behavior of these Borel sums near the turning points can be studied
using composite asymptotic expansions and is connected to
the confluence of P II to P I.

The work is commen work with T. Aoki, T. Koike and Y. Takei.

**Philip Boalch** (Strasbourg, France)

**Title: Transformation groups for isomonodromy equations**

**Extended Abstract**

**Irina Goryuchkina** (Keldysh, Russia)

**Title: Methods of plane power geometry and formal
solutions to the sixth Painlevé equation**

**Abstract**:

**Irfan Mahmood** (Angers, France)

**Title: Lax pair representation and Darboux transformation of noncommut
ative Painlevé's second equation**

**Abstract**: Extension of the Painlevé equations to noncommutative spaces has been
extensively investigated in the theory of integrable systems. An interesting topic is
the exploration of some remarkable aspects of these equations, such as the Painlevé
property, the Lax representation and the Darboux transformation, and their connection
to well-known integrable equations. This paper addresses the Lax formulation, the Darboux
transformation and a quasideterminant solution of the noncommutative form of Painlevé's second equation introduced by Retakh and Rubtsov [V. Retakh, V. Rubtsov, Noncommutative Toda chain, Hankel quasideterminants and Painlevé II equation.

**Tetsu Masuda** (Aoyama, Japan)

**Title: An Explicit Formula for the Discrete Power Function Associated with Circle Patterns of Schramm Type **

**Abstract**: We present an explicit formula for the discrete power function introduced by Bobenko, which is expressed in terms of the hypergeometric τ functions for the sixth Painlevé equation.
The original definition of the discrete power function imposes strict conditions on the domain and the value of the exponent.
However, we show that one can extend the value of the exponent to arbitrary complex numbers except even integers and the domain to a discrete analogue of
the Riemann surface.
Moreover, we show that the discrete power function is an immersion when the real part of the exponent is equal to one.

**Michiaki Inaba** (Kyoto, Japan)

**Title: Moduli of parabolic connections of spectral type**

**Abstract**: We will give a moduli space of stable regular singular parabolic connections
of a given spectral type on smooth projective curves.
This moduli space has a relative symplectic structure.
Moreover, we will prove the geometric Painlevé property of the isomonodromic
deformation defined on this moduli space.

**Takuro Mochizuki** (RIMS, Japan)

**Title: Harmonic bundle and Toda lattice with opposite sign**

**Abstract**: Ceccotti and Vafa observed the existence and uniqueness of
global solutions of *tt*^{*}-equations for some significant
models in physics, which they call ''magical solutions''.
Their method was the reduction to Painleve III.
Recently, Guest and Lin intensively studied its generalization,
and established it in the case of two known functions.
In this talk, we shall explain a generalization of
the results of Guest-Lin from the viewpoint of
Kobayashi-Hitchin correspondence for harmonic bundles.
We shall also discuss some related issues
such as the existence of integral structure.

**Hidetaka Sakai** (Tokyo, Japan)

**Title: Toward a classification of four-dimensional Painlevé-type equations**

(Joint work with H. Kawakami and A. Nakamura)

**Abstract**:

**Hayato Chiba** (Kyushu, Japan)

**Title: Blow-up of vector fields and dynamical systems of compactified Painlevé equations**

**Abstract**:

**Hajime Nagoya** (Kobe, Japan)

**Title: On the sixth quantum Painlevé equation**

**Abstract**:

**Toshio Oshima** (Tokyo, Japan)

**Title: Linear differential equations on the Riemann sphere**

**Abstract**:

**Hiroshi Umemura** (Nagoya, Japan)

**Title: Can we quantify Galois theory?**

**Abstract**: Today after a long pursuit, which goes back to Galois, we can fairly well understand
Galois theory of differential equations, the works of Drach and Vessiot. Of course there remains yet a task of founding the entire theory on a firm basis using a lucid general language.

We know the hypergeometric function as well as its *q*-analogue. The Galois group of the
hypergeometric function is a linear algebraic group *G*. When we pass from the classical hypergeometric function to the *q*-analogue, the Galois group of the
*q*-hypergeometric function is also a linear algebraic group and we do not encounter a quantification of the algebraic group *G*.

During the description of general Galois theory, we discovered a new framework that
would allow us to quantify differential Galois theory. We show how it works by examples.

**Masataka Kanki** (Tokyo, Japan)

**Title: Discrete Painlevé equations modulo a prime number**

**Abstract**: We introduce some of the recent results on the discrete integrable
equations over finite fields, which are developed jointly with J. Mada,
K. M. Tamizhmani and T. Tokihiro. Dynamical systems over non-Archimedean
fields are of great interest in the theory of arithmetic dynamics.
Integrable dynamical systems, in particular, the discrete Painlevé
equations, are shown to have a property that is similar to the good
reduction modulo a prime number from a field of p-adic numbers. We
observe that this property - which we call an 'almost good reduction' -
is an integrability detector, and an arithmetic analogue of the
singularity confinement test. Applications of our method to other
integrable equations are also discussed.

**Kohei Iwaki** (RIMS, Japan)

**Title: Voros coefficients of Painlevé equations and Parametric Stokes phenomena**

**Abstract**: In this talk we analyze the Painlevé equations with a large parameter
from the view point of the exact WKB analysis. We consider a connection
problem concerned with "parametric Stokes phenomena", which is a kind of
Stokes phenomena occurring when the parameters (which is not the
independent variable or the large parameter) vary. In order to describe
the discontinuous change of the asymptotic behavior of solutions
explicitly, we introduce the notion of "Voros coefficients" for Painlevé
equations. We derive some explicit connection formulas throgh the analysis
of the Voros coefficients.

**Takato Uehara** (Niigata, Japan)

**Title: The entropy values of automorphisms on rational surfaces**

**Abstract**: This talk is concerned with automorphisms on compact complex surfaces with
positive entropy. According to a result of S. Cantat, a surface admitting
an automorphism with positive entropy must be either a K3 surface, an
Enriques surface, a complex torus or a rational surface. Among these
surfaces, rather few examples for rational surfaces were known. In order to
describe many examples, I determine the set of entropy values of rational
surface automorphisms in terms of Weyl groups.