Painleve Equations and Monodromy Problems: An Introduction
Monday 11th September 2006 to Friday 15th September 2006
08:30 to 09:45  Registration  
09:45 to 10:00  Opening  INI 1  
10:00 to 11:00  Introduction to the Painlev\'e equations  INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30  Differential Galois theory and the Painlev\'e equations  INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00  Introduction to the Painlev\'e equations II  INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
Asymptotics of Painlev\'e equations The classical secondorder nonlinear Painleve equations appear in many scientific applications as mathematical models. Their roles as models demand asymptotic information about their highly transcendental solutions. We will introduce methods to (i) find asymptotic behaviours of the solutions in various limits; (ii) show how to extend such techniques to gain global information; (iii) find error estimates; (iv) extend these methods to higherorder Painleve equations; and (v) if time permits, discuss asymptotics of discrete Painleve equations. 
INI 1  
16:30 to 17:30 
F Nijhoff ([Leeds]) Integrable lattics and discrete Painlev\'e equations 
INI 1  
17:30 to 18:15  Wine and Beer Reception and Poster Session  
18:30 to 19:30  Dinner at Selwyn College (Residents only) 
09:00 to 10:00  Introduction to the Painlev\'e equations III  INI 1  
10:00 to 11:00  Differential Galois theory and the Painlev\'e equations II  INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:00 
Symmetry groups underlying Bailey's transfomations for $^{10}\phi^9$ series The concept of symmetry groups associated with two term transformations for basic hypergeometric series is well known, and most of them have been studied and identified (J. Math. Phys. 1999:6692+ and references therein). One two term identity for which the invariance group, to our knowledge, was not written down explicitly is Bailey's four term transformation for nonterminating ${}_{10}\phi_9$series considered as a two term transformation between a linear combination of such series which we call $\Phi$. It is shown that the invariance group of this transformation is the Weyl group of type $E_6$. We demonstrate that the group associated with a three term transformation between $\Phi$series, each admitting Bailey's two term transformation, is the Weyl group of type $E_7$. We do this by giving a description of the root system of type $E_7$ that allows to find a transformation between equivalent three term identities in an easy way. We also show how one can find a prototype of each of the five essentially different three term identities between $\Phi$series. 
INI 1  
12:00 to 12:30 
YangBaxter maps and integrable difference equations The connection between integrable partial difference equations and YangBaxter maps is explained and some integrable partial difference equations on a 2dim grid as coupled YangBaxter maps are presented. Also higher dimensional analogues of these coupled maps are discussed. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00  Introduction to the Painlev\'e equations IV  INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
Asymptotics of Painlev\'e equations II The classical secondorder nonlinear Painleve equations appear in many scientific applications as mathematical models. Their roles as models demand asymptotic information about their highly transcendental solutions. We will introduce methods to (i) find asymptotic behaviours of the solutions in various limits; (ii) show how to extend such techniques to gain global information; (iii) find error estimates; (iv) extend these methods to higherorder Painleve equations; and (v) if time permits, discuss asymptotics of discrete Painleve equations. 
INI 1  
16:30 to 17:30 
F Nijhoff ([Leeds]) Discrete Painlev\'e equations II 
INI 1  
18:30 to 19:30  Dinner at Selwyn College (Residents only) 
09:00 to 10:00  Random matrices and Painlev\'e equations  INI 1  
10:00 to 11:00  Differential Galois theory and the Painlev\'e equations III  INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:00 
T Tsuda ([Kobe]) Tropical representation of Weyl groups associated with certain rational varieties Starting from certain rational varieties blownup from (P^1)^N, we construct a tropical (or subtractionfree birational) representation of Weyl groups as a group of pseudo isomorphisms of the varieties. Furthermore, we introduce a geometric framework of taufunctions as defining functions of exceptional divisors on the varieties. In the case where the corresponding root system is of affine type, our construction yields (higher order) qdifference Painleve equations. This is a joint work with Tomoyuki Takenawa. 
INI 1  
12:00 to 12:30 
Symmetries and group invariant reductions of integrable equations on quadgraphs to discrete Painlev\'e equations We investigate the Lie point and generalized symmetries of certain integrable equations on quadgraphs. After introducing symmetry group techniques, we give a number of illustrative examples of discrete Painlev\'e equations arising as group invariant solutions of the relevant integrable lattice equations. The associated isomonodromic deformation problems are constructed through the symmetry reduction as well. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Special solutions of the Painleve equations: P1 to P5 The Painleve functions are transcendental in general, but the Painleve equations have special solutions which are not transcendental. In this talk (2 hours), we give a definition of `special' solutions of the Painleve equations. In Umemura's sense, special solutions are divided into two classes. One is algebraic solutions and the other is socalled Riccatitype solutions. In order to give a precise definition of special solutions, we review Okamoto's initial value spaces. And the Backlund transformation groups play an important role to classify complete list of special solutions. In the first lecture we show complete list of special solutions for P1 to P5. In the second lecture, we study a representation of special solutions. Moreover we may research other class of special solutions beyond Umemura's classical solutions, such as Boutroux's solutions (P1), AblowitsSegal's solutions (P2), symmetric solutions (P1,2,4) and Picard's solution (P6). This lecture is a bridge between Umemura's lecture and Boalch's lecture. Algebraic solutions of P6 are shown in Boalch's talk. Related Links

INI 1  
15:00 to 15:30  Tea  
15:30 to 17:30  Informal discussions  INI 1  
18:30 to 19:30  Dinner at Selwyn College (Residents only) 
09:00 to 10:00  RiemannHilbert problems  INI 1  
10:00 to 11:00  Differential Galois theory and the Painlev\'e equations IV  INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:00 
G Casale ([Tokyo]) Local irreducibility of the first Painlev\'e equation Let $K \supset \mathbb{C}(x)$ be an ordinary differential extension of fields. An order two differential equation $$y'' = E(x,y,y') \in \mathbb{C}(x,y,y')$$ is said to be reducible over $K$ if there exist two independent first integrals of the vectors field $ \frac{\partial}{\partial x} + y' \frac{\partial}{\partial y} + E(x,y,y') \frac{\partial}{\partial y'}$ in a differential extension $L$ of the partial differential field $K(y,y')$ (with derivations $\frac{\partial}{\partial x}$, $\frac{\partial}{\partial y}$, $\frac{\partial}{\partial y'}$) such that $K(y,y')=L_0 \subset L_1 \subset \ldots \subset L_p = L$ with intermediate extensions $L_i \subset L_{i+1}$ \begin{itemize} \item[] algebraic extension, \item[] strongly normal extension, \hspace*{1cm}[ {\sl typical s.n.\,extensions are $L_{i+1} = L_i(h_1,\ldots,h_q)$ with $dh_i = \sum h_j \omega_i^j$ ; $\omega_i^j$ 1forms with coefficients in $L_i$} ] \item[] codimension one strongly normal extension, \hspace*{1cm}[ {\sl there is $H \in K_i$ such that $K_{i+1} = K_i(< h_1,\ldots,h_q>)$ with $dh_i = \sum h_j \omega_i^j \mod H$; $\omega_i^j$ 1forms with coefficients in $L_i$} ] \item[] extension by a first integral of a codimension one foliation, \hspace*{1cm}[ {\sl there is an integrable 1form $\omega$ with coefficients in $L_i$ and $K_{i+1} = K_i (< h >)$ with $dh \wedge \omega = 0$} ] \end{itemize} where $(<\ >)$ stands for `differential fields generated by'. An equation is said locally irreducible if it is irreducible over any field $K$.\\ In this talk, local irreducibility of the first Painlev\'e equation, $ y'' = 6y^2+x $, is investigated. The main tool used is the Galois groupoid of $P_1$ over $K$ defined by B. Malgrange. On gets a characterization of reducible equation : the transversal differential dimension of the Galois groupoid of a reducible equation is one. From the computation of the Galois groupoid of $P_1$, we prove that the transversal differential dimension of the Galois groupoid of $P_1$ is two. This computation involves \'E. Cartan classification on structural equation of pseudogroups on $\mathbb{C}^2$ and special weights on dependent variables following H. Umemura. 
INI 1  
12:00 to 12:30 
Painlev\'e I asymptotics for orthogonal polynomials with respect to a varying quartic weight We study polynomials that are orthogonal with respect to a varying quartic weight $\exp(N(x^2/2+tx^4/4))$ for $t < 0$, where the orthogonality takes place on certain contours in the complex plane. Inspired by developments in 2D quantum gravity, Fokas, Its, and Kitaev, showed that there exists a critical value for $t$ around which the asymptotics of the recurrence coefficients are described in terms of exactly specified solutions of the Painlev´e I equation. In this paper, we present an alternative and more direct proof of this result by means of the Deift/Zhou steepest descent analysis of the RiemannHilbert problem associated with the polynomials. Moreover, we extend the analysis to nonsymmetric combinations of contours. Special features in the steepest descent analysis are a modified equilibrium problem and the use of $\Psi$functions for the Painlev´e I equation in the construction of the local parametrix. Related Links 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00  Random matrices and Painlev\'e equations II  INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30  Hamiltonian structure of the Painlev\'e equations  INI 1  
16:30 to 17:30 
Algebraic solutions of the Painlev\'e equations I will survey what is known about the algebraic solutions of the Painleve VI equation. This gives an opportunity to see 'in action' lots of the Painleve/isomonodromy technology: RiemannHilbert correspondence, nonlinear monodromy actions, affine Weyl group symmetries, quadratic/Landen/folding transformations and precise asymptotic formulae. Moreover I will try to emphasise some of the mysteries and open problems of the subject. Related Links 
INI 1  
19:30 to 18:00  Conference Dinner in the Hall at Magdalene College 
09:00 to 10:00  RiemannHilbert problems II  INI 1  
10:00 to 11:00  Classical solutions on the Painlev\'e equations: from PII to PV II  INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:00 
T Claeys ([Leuven]) A real polefree solution of the 4th order analogue of the Painlev\'e I equation and critical edge points in random matrix ensembles (joint work with M. Vanlessen) We consider the following fourth order analogue of the Painlev\'e I equation, \[ x=Ty\left(\frac{1}{6}y^3+\frac{1}{24}(y_x^2+2yy_{xx}) +\frac{1}{240}y_{xxxx}\right). \] We give an overview of how to prove the existence of a real solution $y$ with no poles on the real line, which was conjectured by Dubrovin. We obtain our result by proving the solvability of an associated RiemannHilbert problem through the approach of a vanishing lemma. In addition, by applying the Deift/Zhou steepestdescent method to this RiemannHilbert problem, we obtain the asymptotics for $y(x,T)$ as $x\to\pm\infty$. Furthermore, we explain how functions associated with the $P_I^2$ equation appear in a double scaling limit near singular edge points in random matrix models. Related Links 
INI 1  
12:00 to 12:30 
On the middle convolution for Fuschian systems In the talk I shall explain the algorithm of Dettweiler and Reiter who generalized the Katz middle convolution functor. Middle convolution is an operation for Fuchsian systems of differential equations which preserves rigidity (and, hence, the number of accessory parameters) but changes the rank and monodromy group. In the simplest case of the sixth Painleve equation which describes monodromy preserving deformations of the rank 2 Fuchsian system with four singularities on the projective line the algorithm is applied to derive the Okamoto birational transformation. Next I shall discuss the invariance of deformation equation under middle convolution, which is a join work with Yoshishige Haraoka. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00  Hamiltonian structure of the Painlev\'e equations II  INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
Algebraic solutions of the Painlev\'e equations II I will survey what is known about the algebraic solutions of the Painleve VI equation. This gives an opportunity to see 'in action' lots of the Painleve/isomonodromy technology: RiemannHilbert correspondence, nonlinear monodromy actions, affine Weyl group symmetries, quadratic/Landen/folding transformations and precise asymptotic formulae. Moreover I will try to emphasise some of the mysteries and open problems of the subject. Related Links 
INI 1  
16:30 to 16:45  Closing  INI 1  
18:30 to 19:30  Dinner at Selwyn College (Residents only) 
13:00 to 17:00  Informal discussion (D Monaco) 