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. INI 1 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. Related Links http://arxiv.org/abs/math-ph/0501074 - Related papers on the math arxivhttp://arxiv.org/abs/math-ph/0508062http://arxiv.org/abs/math.CA/0605201 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 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 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 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 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)
 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) Related Links http://arxiv.org/abs/math-ph/0603040 - Integrals of rational symmetric functions, two matrix models and biorthogonal polynomials (Harnad, Orlov)http://arxiv.org/abs/math-ph/0512056 - Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions (Harnad, Orlov)http://arxiv.org/abs/nlin.SI/0410043 - Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions (Bertola, Eynard, Harnad)http://arxiv.org/abs/nlin.SI/0204054 - Partition functions for Matrix Models and Isomonodromic Tau functions (Bertola, Eynard. Harnad) INI 1 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. 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 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 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 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