Various Aspects on the Painlevé Equations

November 26-30, 2012

RIMS, Kyoto University, Japan

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 25 no.12 (2012) 3235-.

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. d2y/dx2=2y3+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.