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Seminars (PEM)

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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 second-order non-linear 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 higher-order 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 non-terminating ${}_{10}\phi_9$-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
Yang-Baxter maps and integrable difference equations

The connection between integrable partial difference equations and Yang-Baxter maps is explained and some integrable partial difference equations on a 2-dim grid as coupled Yang-Baxter 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 second-order non-linear 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 higher-order 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 blown-up from (P^1)^N, we construct a tropical (or subtraction-free birational) representation of Weyl groups as a group of pseudo isomorphisms of the varieties. Furthermore, we introduce a geometric framework of tau-functions 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) q-difference 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 quad-graphs to discrete Painlev\'e equations

We investigate the Lie point and generalized symmetries of certain integrable equations on quad-graphs. 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 so-called Riccati-type 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), Ablowits-Segal'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.

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PEMW01 14th September 2006
09:00 to 10:00
Riemann-Hilbert 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 $dh_i = \sum h_j \omega_i^j$ ; $\omega_i^j$ 1-forms 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 $dh_i = \sum h_j \omega_i^j \mod H$; $\omega_i^j$ 1-forms with coefficients in $L_i$} ] \item[--] extension by a first integral of a codimension one foliation,

\hspace*{1cm}[ {\sl there is an integrable 1-form $\omega$ 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 Riemann-Hilbert problem associated with the polynomials. Moreover, we extend the analysis to non-symmetric 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.

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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: Riemann-Hilbert 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.

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PEMW01 15th September 2006
09:00 to 10:00
Riemann-Hilbert 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 pole-free 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 Riemann-Hilbert problem through the approach of a vanishing lemma. In addition, by applying the Deift/Zhou steepest-descent method to this Riemann-Hilbert 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.

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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: Riemann-Hilbert 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 non-experts. 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 semi-classical 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 q-deformation of the Freud polynomials on the exponential lattice has recurrence coefficients that satisfy a q-discrete Painlevé I equation. Unfortunately, these non-linear 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 q-discrete Painlevé I equation. In both cases the fixed point solution is precisely the solution that gives the recurrence coefficients of the corresponding orthogonal polynomals.

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PEMW02 19th September 2006
09:00 to 10:00
Critical infinite-dimensional diffusions and non-linear equations for their transition probabilities

Introducing the time in random matrix ensembles, Dyson has shown that its spectrum evolves according to non-intersecting 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 infinite-dimensional random processes are governed by non-linear 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 n-by-n 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.

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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 wave-length of order $\epsilon$. Near the gradient catastrophe of the dispersionless equation ($\epsilon=0$), a multi-scales 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 multi-scales 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 finite-order meromorphic solution of the second-order difference equation w(z+1)+w(z-1)=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(z-1)=R(z,w(z)), together with their autonomous versions. This suggests that the existence of finite-order 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 non-rational 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 non-elementary 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 2-dimensional integrable quantum field theory

The Painlev\'e equations are related to two-point correlation functions of certain "interacting" spinless scaling fields in free fermionic models of 2-dimensional quantum field theory (QFT). This relation leads to non-trivial predictions for the solutions to some of the connection problems associated to Painlev\'e equations. Indeed, short-distance and large-distance 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 tau-functions 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 tau-function 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 Anti-self-dual Yang-Mills equation to Painleve equations in Mason-Woodhouse'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 (Christoffel-Darboux 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 two-matrix 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)

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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(PII--PV). The Painlev\'e equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past twenty-five 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 Yablonskii-Vorob'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 Riemann-Hilbert approach to the asymptotics of isomonodromic problems

In the Joint work with M. Bertola (math-ph/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 semi-classical orthogonal polynomials via the `Riemann-Hilbert method'. In this talk I will explain what these admissible Boutroux curves are and how they are related to the `Riemann-Hilbert 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 self-dual Yang-Mills 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
Non-linear 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 three-component KP hierarchy.

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PEMW02 22nd September 2006
12:00 to 12:30
Riemann-Hilbert problems associated with Hurwitz Frobenious manifolds
PEMW02 22nd September 2006
14:00 to 15:00
Two constructions for the isomonodromic tau-functions
PEMW02 22nd September 2006
15:30 to 16:30
Monodromy-free Schrodinger equations and Painlev\'e transcendents

A Schroedinger operator with meromorphic potential is called monodromy-free 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 so-called "finite-gap" operators, but in general the description of all monodromy-free operators is open even in the class of rational potentials, although in some special cases the answer is known (Duistermaat-Grunbaum, Gesztesy-Weikard, Oblomkov).

In the talk I will describe a class of Schroedinger operators with trivial monodromy, constructed in terms of the Painleve-IV 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
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons