Abstract : Recently, problems arising in statistical and probabilistic models with an underlying integrable
structure have been found to possess deep links to continuous and discrete Painlevé equations.
The theory of Painlevé equations has therefore come to play an increasingly important role in the
study of such problems. The way in which Painlevé equations appear, and the types of equations
that appear in these problems pose deep questions on the side of the theory of Painlevé equations.
This mini-symposium aims to bring together experts both in Painlevé equations and the broad
range of problems in which they appear, and illustrate this interplay.
Organizer(s) : Anton Dzhamay, Alexander Stokes, Tomoyuki Takenawa, Ralph Willox
[04296] The Identification Problem for Discrete Painlevé Equations
Format : Talk at Waseda University
Author(s) :
Anton Dzhamay (University of Northern Colorado and BIMSA)
Abstract : We describe a refined version of the discrete Painlevé equations identification problem. We emphasize that, in addition to determining the surface type of the equation, it is important to determine the actual translation element, up to conjugation, and to keep in mind possible special point configurations that can affect the symmetry group of the equation. We illustrate this by a variety of examples that appear in applications, especially in the theory of orthogonal polynomials.
[04597] Orthogonal polynomials and discrete Painlevé equations on the $D_5^{(1)}$ Sakai surface
Format : Talk at Waseda University
Author(s) :
Alexander Stokes (The University of Tokyo)
Anton Dzhamay (University of Northern Colorado)
Galina Filipuk (University of Warsaw)
Abstract : We show that two recurrences coming from the theory of orthogonal polynomials are transformable to discrete Painlevé equations, which share the same surface type $D_5^{(1)}$ in the Sakai classification scheme but are non-equivalent.
The surfaces associated with these recurrences do not have the full parameter freedom for their type, and we find the symmetry groups of these examples as subgroups of the extended affine Weyl group of type $A_3^{(1)}$ from the generic case.
[03019] Orthogonal polynomials, Schur flow and Painlevé equations
Format : Talk at Waseda University
Author(s) :
Walter Van Assche (KU Leuven)
Abstract : We give a brief introduction to orthogonal polynomials on the unit circle and show how an exponential modification of the weight function leads to the Ablowitz-Ladik lattice equations for the recurrence coefficients (Verblunsky coefficients) of these polynomials. As shown by Periwal and Shevitz (1990) these orthogonal polynomials appear in unitary matrix models. The Verblunsky coefficients satisfy the discrete Painlevé II equation and the ratio of these coefficients satisfy Painlevé III. The Lax pair can be written in terms of the CMV matrix, which is a pentadiagonal infinite matrix similar to the Jacobi matrix for orthogonal polynomials on the real line.
[04527] Symmetries of discrete Nahm systems and Normalizers in Coxeter groups
Format : Online Talk on Zoom
Author(s) :
yang shi (Flinders university)
Giorgio Gubbiotti (Universita degli Studi di Milano)
Abstract : It is known that discrete Nahm systems arise as autonomous versions of
Sakai's classification of discrete Painlev\e equations. Here we study the groups of symmetries of these systems using the theory of normalizers of Coxeter groups developed by Brink and Howlett (Invent. Math, 1999).
[04366] On the bilinear equations of the Painlev\'e transcendents
Format : Talk at Waseda University
Author(s) :
Hidetaka Sakai (University of Tokyo)
Tatsuya Hosoi (University of Tokyo)
Abstract : The sixth Painlev\'e equation is a basic equation with three fixed singular points, corresponding to Gauss's hypergeometric differential equation among linear equations. Similar to hypergeometric equation, for nonlinear equations, we would like to determine the equation from the local behavior around the three singularities. In this talk, the sixth Painlev\'e equation is derived by imposing the condition that it is of type (H) at each three singular points for quadratic 4th-order differential equation.
[04932] Large-degree asymptotics of Generalized Hastings-McLeod functions
Format : Talk at Waseda University
Author(s) :
Robert Buckingham (University of Cincinnati)
Abstract : The Generalized Hastings-McLeod functions form an infinite sequence of solutions to the inhomogeneous Painleve-II equation. The functions have recently arisen in a variety of random matrix and interacting particle system problems. Using Riemann-Hilbert analysis and the nonlinear steepest-descent method, we establish the leading-order asymptotic behavior inside and outside the pole region as the inhomogeneous term tends to infinity. This is joint work with Kurt Schmidt.
[04515] Spaces of initial values for equations with the quasi-Painleve property
Format : Online Talk on Zoom
Author(s) :
Thomas Kecker (University of Portsmouth)
Abstract : Considering differential equations and Hamiltonian systems with the property that all movable singularities of all their solutions in the complex plane are algebraic poles (quasi-Painlevé property), we generalise the concept of the Okamoto's space of initial values for these types of equations. Starting from a general equation with analytic coefficient functions, the construction of this space yields certain differential conditions on these functions that are equivalent to the resonances found e.g. by the (quasi-)Painlevé test.
[02754] On the (quasi-)Painleve equations
Format : Online Talk on Zoom
Author(s) :
Galina Filipuk (University of Warsaw)
Abstract : Painleve equations are second order nonlinear differential equations solutions of which have no movable critical points. They appear in many applications. For solutions of quasi-Painleve equations algebraic singularities are allowed. The so-called geometric approach may help in many cases to understand the nature of singularities. In this talk I shall present some recent results on the geometric approach for the Painleve and quasi-Painleve equations. This is a joint work with A. Stokes.
[03574] A dynamical systems approach to map enumeration
Format : Online Talk on Zoom
Author(s) :
Joceline Lega (University of Arizona)
Abstract : Freud orbits are trajectories of discrete Painlevé equations that describe the evolution of recurrence coefficients for orthogonal polynomials. We will introduce a nonlinear transformation, which converts a Freud orbit into a solution that converges towards a fixed point along a center manifold. Subsequent analysis will lead to an asymptotic expansion whose terms are generating functions for map enumeration. Examples of map counts will be provided. This is joint work with Nick Ercolani and Brandon Tippings.
[03010] An affine Weyl group action on the basic hypergeometric series
Format : Talk at Waseda University
Author(s) :
Takao Suzuki (Kindai University)
Abstract : Recently, we formulated a higher order generalization of the $q$-difference Painlevé equations called the $q$-Garnier system in a framework of an extended affine Weyl group of type $A^{(1)}_{2n+1}\times A^{(1)}_1\times A^{(1)}_1$.
On the other hand, the $q$-Garnier system admits a particular solution in terms of the basic hypergeometric series ${}_{n+1}\phi_n$.
In this talk, we investigate an action of the extended affine Weyl group on ${}_{n+1}\phi_n$.
[03311] On the growth properties of some families of birational maps
Format : Online Talk on Zoom
Author(s) :
Giorgio Gubbiotti (Universita degli Studi di Milano)
Michele Graffeo (Politecnico di Milano)
Abstract : We characterise the growth and integrability properties of a family of elements in the Cremona group of a complex projective space in dimension three using techniques from algebraic geometry. This collection consists of maps obtained by composing the standard Cremona transformation $\mathrm{c}_3\in\mathrm{Bir}(\mathbb{P}^3)$ with projectivities permuting the fixed points of $\mathrm{c}_3$ and the points over which $\mathrm{c}_3$ performs a divisorial contraction. Time permitting, we discuss the possible extensions of this construction to higher dimensions.
[05042] Laguerre (q-Laguerre) Weight Recurrence and Geometric Theory of Painlevé equations
Format : Online Talk on Zoom
Author(s) :
Jie Hu (Jinzhong University（晋中学院）)
Abstract : The Sakai geometric theory of Painlevé equations is useful for identify difference or differential equations as corresponding Painlevé equations. In this talk we consider two examples of recurrence relations appearing in study of recurrence coefficients of semi-classical orthogonal polynomials, namely, polynomials with Laguerre weight and with the deformed q-Laguerre weight. Using the geometry, we give explicit change of variables matching these equations with some standard discrete Painlevé equations.
[04054] Folding transformations for q-Painleve equations
Format : Online Talk on Zoom
Author(s) :
Mikhail Bershtein (Kavli IPMU, Landau Institute and Skoltech)
Abstract : Folding transformation of the Painleve equations is an algebraic (of degree greater than 1) transformation between solutions of different equations. In 2005 Tsuda, Okamoto and Sakai classified folding transformations of differential Painleve equations. These transformations are in correspondence with automorphisms of affine Dynkin diagrams. We give a complete classification of folding transformations of the q-difference Painleve equations, these transformations are in correspondence with certain subdiagrams of the affine Dynkin diagrams (possibly with automorphism). The method is based on Sakai's approach to Painleve equations through rational surfaces.
Based on joint work with A. Shchechkin [arXiv:2110.15320]
[04739] Laguerre (q-Laguerre) Weight Recurrence and Geometric Theory of Painlevé equations
Format : Online Talk on Zoom
Author(s) :
Jie HU (Jinzhong University)
Abstract : Sakai's geometric theory of Painlevé equations is used to identify difference or differential equations with corresponding Painlevé equations. In this talk we will consider two classes of examples of recurrence coefficients of semi-classical orthogonal polynomials: Laguerre weight and the deformed $q$-Laguerre weight. We will gives the identification procedure on how to deduce related solutions of canonical discrete Painlevé equations from coefficients. Meanwhile, we also give the explicit birational function of variables achieving that reduction.
