Videos and presentation materials from other INI events are also available.
Event  When  Speaker  Title  Presentation Material 

PEMW01 
11th September 2006 09:45 to 10:00 
Opening  
PEMW01 
11th September 2006 10:00 to 11:00 
Introduction to the Painlev\'e equations  
PEMW01 
11th September 2006 11:30 to 12:30 
Differential Galois theory and the Painlev\'e equations  
PEMW01 
11th September 2006 14:00 to 15:00 
Introduction to the Painlev\'e equations II  
PEMW01 
11th September 2006 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. 

PEMW01 
11th September 2006 16:30 to 17:30 
F Nijhoff  Integrable lattics and discrete Painlev\'e equations  
PEMW01 
12th September 2006 09:00 to 10:00 
Introduction to the Painlev\'e equations III  
PEMW01 
12th September 2006 10:00 to 11:00 
Differential Galois theory and the Painlev\'e equations II  
PEMW01 
12th September 2006 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. 

PEMW01 
12th September 2006 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. 

PEMW01 
12th September 2006 14:00 to 15:00 
Introduction to the Painlev\'e equations IV  
PEMW01 
12th September 2006 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. 

PEMW01 
12th September 2006 16:30 to 17:30 
F Nijhoff  Discrete Painlev\'e equations II  
PEMW01 
13th September 2006 09:00 to 10:00 
Random matrices and Painlev\'e equations  
PEMW01 
13th September 2006 10:00 to 11:00 
Differential Galois theory and the Painlev\'e equations III  
PEMW01 
13th September 2006 11:30 to 12:00 
T Tsuda 
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. 

PEMW01 
13th September 2006 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. 

PEMW01 
13th September 2006 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


PEMW01 
14th September 2006 09:00 to 10:00 
RiemannHilbert problems  
PEMW01 
14th September 2006 10:00 to 11:00 
Differential Galois theory and the Painlev\'e equations IV  
PEMW01 
14th September 2006 11:30 to 12:00 
G Casale 
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. 

PEMW01 
14th September 2006 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 

PEMW01 
14th September 2006 14:00 to 15:00 
Random matrices and Painlev\'e equations II  
PEMW01 
14th September 2006 15:30 to 16:30 
Hamiltonian structure of the Painlev\'e equations  
PEMW01 
14th September 2006 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 

PEMW01 
15th September 2006 09:00 to 10:00 
RiemannHilbert problems II  
PEMW01 
15th September 2006 10:00 to 11:00 
Classical solutions on the Painlev\'e equations: from PII to PV II  
PEMW01 
15th September 2006 11:30 to 12:00 
T Claeys 
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 

PEMW01 
15th September 2006 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. 

PEMW01 
15th September 2006 14:00 to 15:00 
Hamiltonian structure of the Painlev\'e equations II  
PEMW01 
15th September 2006 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 

PEMW02 
18th September 2006 10:00 to 11:00 
Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's function  
PEMW02 
18th September 2006 11:30 to 12:30 
Painleve systems arising from integrable hierarchies I will give an overview of a class of Lax representations for Painlev¥'e equations and their generalization in terms of Lie algebras. In that context discrete symmetries of Painlev¥'e systems are described by means of birational Weyl group actions. I will also discuss how they are related to integrable hierarchies associated with affine Lie algebras. 

PEMW02 
18th September 2006 14:00 to 15:00 
H Sakai 
Rational surfaces and discrete Painlev\'e equations This is an introductory talk on a connection between rational surfaces and discrete Painlev\'e equations, for nonexperts. Contents are as follows: 1.Introduction 2.Translation of discrete Painlev\'e equations into a language of Linear Algebra (Matrix) 3.Translation into a language of affine Weyl group 4.Classification 

PEMW02 
18th September 2006 15:30 to 16:30 
Diophantine integrability Discrete equations over the rational numbers (and more generally over number fields) will be considered. The height of a rational number a/b is max(a,b), where a and b are coprime. The height of the nth iterate of an equation appears to grow like a power of n for discrete equations broadly considered to be of Painlev\'e type, and exponentially for other equations. Methods for classifying equations according to this criterion will be described. Connections with other approaches, such as Nevanlinna theory, singularity confinement and algebraic entropy, will be discussed. 

PEMW02 
18th September 2006 16:30 to 17:30 
W Van Assche 
Discrete Painleve equations for recurrence coefficients of orthogonal polynomials The recurrence coefficients of certain semiclassical orthogonal polynomials satisfy discrete Painlevé equations. The Freud equation for the recurrence coefficients of the orthogonal polynomials for the weight exp(x^4+ t x^2) is in fact a special case of discrete Painlevé I, the Verblunsky coefficients of orthogonal polynomials on the unit circle with weight exp(K cos t) satisfy discrete Painlevé II, the recurrence coefficients of generalized Charlier polynomials can be written in terms of a solution of discrete Painlevé II, and a qdeformation of the Freud polynomials on the exponential lattice has recurrence coefficients that satisfy a qdiscrete Painlevé I equation. Unfortunately, these nonlinear recurrence relations are not suited for computing the recurrence coefficients starting from two initial conditions, since minor deviations from the correct initial values quickly leads to major deviations from the correct value. For the Freud equations for the weight exp(x^4) Lew and Quarles showed that there is a unique solution of the discrete Painlevé I equation which starts at 0 and remains positive for all n. This positive solution is in fact a fixed point in a metric space of sequences, and it can be found by successive iterations of a contractive mapping. This procedure give a numerically stable way to compute the recurrence coefficients. We will show that a similar result is also true for the discrete Painlevé II equation and for the qdiscrete Painlevé I equation. In both cases the fixed point solution is precisely the solution that gives the recurrence coefficients of the corresponding orthogonal polynomals. Related Links


PEMW02 
19th September 2006 09:00 to 10:00 
Critical infinitedimensional diffusions and nonlinear equations for their transition probabilities Introducing the time in random matrix ensembles, Dyson has shown that its spectrum evolves according to nonintersecting Brownian motions held together by a drift term. For large size random matrices, the universal edge, gap and bulk scalings applied to such diffusions lead to the Airy, Pearcey and Sine processes. The transition probabilities for these infinitedimensional random processes are governed by nonlinear equations, which I plan on describing. 

PEMW02 
19th September 2006 10:00 to 11:00 
Universality of Painleve functions in random matrix models Several types of critical phenomena take place in the unitary random matrix ensembles (1/Z_n) e^{n Tr V(M)} dM defined on nbyn Hermitian matrices M in the limit as n tends to infinity. The first type of critical behavior is associated with the vanishing of the equilibrium measure in an interior point of the spectrum, while the second type is associated with the higher order vanishing at an endpoint. The two types are associated with special solutions of the Painlev\'e II and Painlev\'e I equation, respectively. The quartic potential is the simplest case where this behavior occurs and serves as a model for the universal appearance of Painlev\'e functions in random matrix models. Related Links


PEMW02 
19th September 2006 11:30 to 12:00 
C Klein 
Dissipationless shocks and Painleve equations The Cauchy problem for dissipationless equations as the Korteweg de Vries (KdV) equation with small dispersion of order $\epsilon^2$, $\epsilon\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations of wavelength of order $\epsilon$. Near the gradient catastrophe of the dispersionless equation ($\epsilon=0$), a multiscales expansion gives an asymptotic solution in terms of a fourth order generalization of Painlev\'e I. At the leading edge of the oscillatory zone, a corresponding multiscales expansion yields an asymptotic description of the oscillations where the envelope is given by a solution to the Painlev\'e II equation. We study the applicability of these approximations for several PDEs and random matrix models numerically. 

PEMW02 
19th September 2006 12:00 to 12:30 
Finite order meromorphic solutions and the discrete Painleve equations Let w(z) be an admissible finiteorder meromorphic solution of the secondorder difference equation w(z+1)+w(z1)=R(z,w(z)) where R(z,w(z)) is rational in w(z) with coefficients that are meromorphic in z. Then either w(z) satisfies a difference linear or Riccati equation or else the above equation can be transformed to one of a list of canonical difference equations. This list consists of all known difference Painlevé equations of the form w(z+1)+w(z1)=R(z,w(z)), together with their autonomous versions. This suggests that the existence of finiteorder meromorphic solutions is a good detector of integrable difference equations. 

PEMW02 
19th September 2006 14:00 to 15:00 
Value distribution of solutions of Painleve differential equations We present a survey of the value distribution on nonrational solutions of Painlevé differential equations in terms of the Nevanlinna theory. We consider their growth and, in particular, their deficiencies and ramification indices with respect both to constant and to small moving targets. An important role here is played by the fact that the second main theorem of the Nevanlinna theory for Painlevé solutions reduces to an asymptotic equality. In addition, we make some remarks concerning the value distribution of solutions of higher order Painlevé equations. List of open problems will be presented as well. 

PEMW02 
19th September 2006 15:30 to 16:30 
A family solutions of a degenerate Garnier system near a singularity The two dimensional Garnier system is obtained from isomonodromic deformation of a Fuchsian differential equation with two deformation parameters. Applying successive limiting procedure to it, H. Kimura computed a degeneration scheme consisting of degenerate Garnier systems written in the Hamiltonian form. Among them, we consider a degenerate Garnier system (G) which is a two variable version of the first Painleve equation. We present a three parameter family of asymptotic solutions of (G) near a singular locus. 

PEMW02 
19th September 2006 16:30 to 17:30 
The sixth Painleve equation: a chaotic dynamical system We show that the Poincare return map of the sixth Painleve equation is chaotic along almost every loop, called a nonelementary loop, in the domain of definition. For each such map we construct a natural invariant Borel probability measure and establish some dynamical properties of it such as positivity of the entropy, ergodicity, hyperbolicity, and so on. We also give an algorithm to calculate the entropy in terms of a reduced word of the loop. This is a joint work with my research student Takato Uehara. 

PEMW02 
20th September 2006 09:00 to 10:00 
Solving Painleve connection problems using 2dimensional integrable quantum field theory The Painlev\'e equations are related to twopoint correlation functions of certain "interacting" spinless scaling fields in free fermionic models of 2dimensional quantum field theory (QFT). This relation leads to nontrivial predictions for the solutions to some of the connection problems associated to Painlev\'e equations. Indeed, shortdistance and largedistance expansions can be obtained in QFT from conformalperturbation theory and form factors, respectively. These expansions areunambiguous once the normalisations of the fields have been fixed, and fully calculable. In turn, they give expansions, including the normalisation, for Painlev\'e transcendents near some critical points, as well as the relative normalisation of the associated taufunctions near these critical points. As an example, I will explain how this works in the Dirac theory on the Poincar\'e disk, giving in particular predictions concerning connection problems in certain degenerate cases of Painlev\'e VI that are excluded from the general formula of M. Jimbo of 1982. 

PEMW02 
20th September 2006 10:00 to 11:00 
Bergman taufunction and determinants of Laplacians in flat conical metrics over Riemann surfaces  
PEMW02 
20th September 2006 11:30 to 12:00 
Y Murata 
On matrix Painlev\'e equations Reconstructing the reduction process of Antiselfdual YangMills equation to Painleve equations in MasonWoodhouse's work, we can obtain matrix type ordinary differential equations MPS (Matrix Painleve Systems). MPS are characterized by Young diagrams of weight 4 and constant matrix P, and are classified into 15 types. 15 MPS are transformed into Painleve systems and other degenerated equations. This correspondence explains various degeneration phenomena of Painleve equations. Furthermore, MPS include linear 2 systems which are equivalent to hypergeometric or confluent hypergeometric equations. This part is a joint work with N.M.J.Woodhouse. 

PEMW02 
21st September 2006 09:00 to 10:00 
Matrix integrals as isomonodromic tau functions It is well known that the tau functions associated to special solutions of the Painleve equations may be expressed as matrix integrals (e.g. gap probabilities for sine kernel, airy kernel or Bessel kernel determinantal ensembles). The partition functions for many types of matrix models are also known to be isomondromic tau functions, as are various types of correlation functions. More generally, for a wide variety of generalized orthogonal polynomial (ChristoffelDarboux kernel) ensembles, with orthogonality support taken on quite general curve segments in the complex plane, the matrix integrals representing partition functions, gap probabilities and expectation values of spectral invariant functions can all be interpreted on the same footing, and shown to be isomonodromic tau functions. This result also extends to twomatrix integrals, which are associated with the isomonodromic systems corresponding to sequences of biorthogonal polynomials. (This talk is based on joint work wih: Marco Bertola, Bertrand Eynard and Alexander Orlov) Related Links


PEMW02 
21st September 2006 10:00 to 11:00 
Rational solutions and associated special polynomials associated for the Painlev\'e equations In this talk I shall discuss rational solutions and associated polynomials for the second, third, fourth and fifth Painlev\'e equations(PIIPV). The Painlev\'e equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past twentyfive years, which have arisen in a variety of physical applications and may be thought of as nonlinear special functions.Rational solutions of the Painlev\'e equations are expressible as the logarithmic derivative of special polynomials. For PII these special polynomials are known as the {\it YablonskiiVorob'ev polynomials\/}. The locations of the roots of these polynomials is shown to have a highly regular triangular structure in the complex plane. The analogous special polynomials for PIII, PIV and PV are derived and I shall show that the roots of these special polynomials also have a highly regular structure. 

PEMW02 
21st September 2006 11:30 to 12:00 
M Feigin 
Degenerate Gaussian Unitary ensembles and Painlev\'e IV We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painleve IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable. This is a joint work with Y.Chen. 

PEMW02 
21st September 2006 12:00 to 12:30 
Degeneration and symmetry of the Schlesinger system from the point of view of Twistor theory Using the idea of Mason and Woodhouse, I will describe the isomonodromic deformation of the linear systems of differential equations on P^1. The deformation equations are defined on the Grassmannian manifold Gr(2,N). Using the method of constructing the confluence process for the general hypergeometric system on the Grassmaniann, we will describe the degeneration (confluence) of the isomonodromic deformation of the above system in a explicit way. Some symmetric property will be also discussed for the degenerated Schlesinger system. 

PEMW02 
21st September 2006 14:00 to 14:30 
MY Mo 
The RiemannHilbert approach to the asymptotics of isomonodromic problems In the Joint work with M. Bertola (mathph/0605043), we have studied properties of a special type of algebraic curves, which we called admissible Boutrox curves. We have shown that these curves can be used to compute the asymptotics of semiclassical orthogonal polynomials via the `RiemannHilbert method'. In this talk I will explain what these admissible Boutroux curves are and how they are related to the `RiemannHilbert method'. 

PEMW02 
21st September 2006 14:30 to 15:00 
Estimates of difference operators in the complex plane We present new results that relates the growth of logarithmic derivatives and difference quotients of meromorphic functions in the complex plane. 

PEMW02 
21st September 2006 15:30 to 16:30 
The Hamiltonian structure of the second Painlev\'e hierarchy  
PEMW02 
21st September 2006 16:30 to 17:30 
N Woodhouse 
Twistors and monodromy It has been known for some time that the six Painlevé equations are reductions of the selfdual YangMills equations under the action of various subgroups of the conformal group. The twistor theory of this result is reviewed, and also its application to the construction of classical solutions and special geometries. Two generalizations are described, which are related by an extended form of Harnad's duality. One gives a twistor description of the solution of the general isomonodromy problem with any number of irregular singularities; the second corresponds to a problem with two singularities, a regular one at the origin and an irregular one at infinity. The two are related by a simple operation on the corresponding bundle over twistor space. 

PEMW02 
22nd September 2006 09:00 to 10:00 
On oscillatory behaviour in PDEs and certain Painleve'type transcendents  
PEMW02 
22nd September 2006 10:00 to 11:00 
Nonlinear differential Galois theory  
PEMW02 
22nd September 2006 11:30 to 12:00 
S Kakei 
From the KP hierarchy to the Painlev\'e equations There are many examples of similarity reductions that connect soliton equations to the Painleve equations. In this talk, we will reformulate known examples from the viewpoint of the KP hierarchy and give a unified framework. As a result, the generic Painleve VI is obtained as a similarity reduction of the threecomponent KP hierarchy. Related Links


PEMW02 
22nd September 2006 12:00 to 12:30 
RiemannHilbert problems associated with Hurwitz Frobenious manifolds  
PEMW02 
22nd September 2006 14:00 to 15:00 
Two constructions for the isomonodromic taufunctions  
PEMW02 
22nd September 2006 15:30 to 16:30 
Monodromyfree Schrodinger equations and Painlev\'e transcendents A Schroedinger operator with meromorphic potential is called monodromyfree if all solutions of the corresponding Schroedinger equation are meromorphic for all values of energy (so the corresponding monodromy in the complex plane is trivial). A nice class of examples is given by the socalled "finitegap" operators, but in general the description of all monodromyfree operators is open even in the class of rational potentials, although in some special cases the answer is known (DuistermaatGrunbaum, GesztesyWeikard, Oblomkov). In the talk I will describe a class of Schroedinger operators with trivial monodromy, constructed in terms of the PainleveIV transcendents and their higher analogues determined by the periodic dressing chains. We will discuss also a new interpretation and a fundamental role of the Stieltjes relations in this problem. 

PEM 
25th September 2006 16:00 to 17:00 
G Casale  The Galois groupoid of the Picard PVI solutions  
PEM 
27th September 2006 11:30 to 12:30 
T Tsuda  Universal characters and an extension of the KP hierarchy  
PEM 
27th September 2006 16:00 to 17:00 
Klein's problem and the Painleve equations 