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Timetable (PEMW02)

Painleve Equations and Monodromy Problems: Recent Developments

Monday 18th September 2006 to Friday 22nd September 2006

Monday 18th September 2006
08:30 to 09:45 Registration
09:45 to 10:00 Opening INI 1
10:00 to 11:00 A Bobenko ([Berlin])
Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's function
INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 M Noumi ([Kobe])
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.

INI 1
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 15:00 H Sakai ([Tokyo])
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

INI 1
15:00 to 15:30 Tea
15:30 to 16:30 R Halburd ([Loughborough])
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.

INI 1
16:30 to 17:30 W Van Assche ([Leuven])
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.

Related Links

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)
Tuesday 19th September 2006
09:00 to 10:00 P van Moerbeke ([Louvain])
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.

INI 1
10:00 to 11:00 A Kuijlaars ([Leuven])
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.

Related Links

INI 1
11:00 to 11:30 Coffee
11:30 to 12:00 C Klein ([Leipzig])
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.

INI 1
12:00 to 12:30 R Korhonen ([Joensuu])
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.

INI 1
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 15:00 I Laine ([Joensuu])
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.

INI 1
15:00 to 15:30 Tea
15:30 to 16:30 S Shimomura ([Keio])
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.

INI 1
16:30 to 17:30 K Iwasaki ([Kyushu])
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.

INI 1
18:30 to 19:30 Dinner at Selwyn College (Residents only)
Wednesday 20th September 2006
09:00 to 10:00 B Doyon ([Oxford])
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.

INI 1
10:00 to 11:00 D Korotkin ([Concordia])
Bergman tau-function and determinants of Laplacians in flat conical metrics over Riemann surfaces
INI 1
11:00 to 11:30 Coffee
11:30 to 12:00 Y Murata ([Nagasaki])
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.

INI 1
12:00 to 12:30 S Barannikov ([ENS Paris])
TBA
INI 1
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 15:00 Informal discussions 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)
Thursday 21st September 2006
09:00 to 10:00 J Harnad ([Montreal])
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)

Related Links

INI 1
10:00 to 11:00 P Clarkson ([Kent])
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.

INI 1
11:00 to 11:30 Coffee
11:30 to 12:00 M Feigin ([Glasgow])
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.

INI 1
12:00 to 12:30 H Kimura ([Kumamoto])
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.

INI 1
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 14:30 MY Mo ([Montreal])
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'.

INI 1
14:30 to 15:00 EYM Chiang ([Hong Kong])
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.

INI 1
15:00 to 15:30 Tea
15:30 to 16:30 M Mazzocco ([Manchester])
The Hamiltonian structure of the second Painlev\'e hierarchy
INI 1
16:30 to 17:30 N Woodhouse ([Oxford])
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.

INI 1
19:30 to 18:00 Conference Dinner in the Old Library at Emmanuel College
Friday 22nd September 2006
09:00 to 10:00 B Dubrovin ([SISSA-ISAS])
On oscillatory behaviour in PDEs and certain Painleve'-type transcendents
INI 1
10:00 to 11:00 B Malgrange ([Institut Fourier])
Non-linear differential Galois theory
INI 1
11:00 to 11:30 Coffee
11:30 to 12:00 S Kakei ([Tokyo])
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.

Related Links

INI 1
12:00 to 12:30 V Shramchenko ([Oxford])
Riemann-Hilbert problems associated with Hurwitz Frobenious manifolds
INI 1
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 15:00 I Strachan ([Glasgow])
Two constructions for the isomonodromic tau-functions
INI 1
15:00 to 15:30 Tea
15:30 to 16:30 A Veselov ([Loughborough])
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.

INI 1
16:30 to 16:45 Closing INI 1
18:30 to 19:30 Dinner at Selwyn College (Residents only)
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons