Course 1: “Asymptotic methods for Orthogonal Polynomials and more general Padé approximants”.
Lecturer: Marco Bertola (Concordia University, Montreal, Canada).
Course overview: The goal of the course is to familiarize the students with the description of the asymptotic of orthogonal polynomials that employs the so-called “Riemann Hilbert Method”. The method was introduced in the late nineties and saw the main application in the first proof of “universality” results in Random Matrix Theory. In the literature, the specific technique is variably referred to as “nonlinear steepest descent method” or “Deift-Zhou method”.
A particular appeal of the method is that it applies to a wider range of problems which include asymptotic studies for nonlinear integrable wave equations like the Korteweg-deVries equation or the nonlinear Schrödinger equation, used in oceanography and nonlinear optics, respectively.
The goal of the course is, however, focused on the case of orthogonal polynomials, or rather denominators of Padé approximations (namely the so-called “non-hermitean” orthogonality). Of special interest are the cases of “semiclassical” orthogonal polynomials introduced by Shohat, Maroni, Marcellán, Rocha, and their asymptotics. The method combines input from potential theory as well as analysis, and is able to provide “strong-asymptotic results” for orthogonal polynomials, in principle allowing for a complete asymptotic expansion for large degree valid point-wise in the plane.
Time permitting, we will touch upon applications of the method beyond orthogonal polynomials and to the newly introduced notion of Padé approximations on higher genus Riemann surfaces.
A rough breakdown of the course is as follows:
- Padé approximations and Orthogonal Polynomials;
- Generalities about Riemann–Hilbert problems (RHPs); formulation of Padé approximation problems in terms of RHPs.
- Large degree asymptotic analysis:
- Elements of potential theory and the “g-function” mechanism;
- Reduction of the RHP to a “small–norm” problem;
- Construction of “parametrices”; approximate solutions;
- Some aspects of geometry of Riemann surfaces needed for the construction of parametrices;
- Padé approximants on Riemann surfaces (time permitting)
- Other applications of the method and interesting problems (time permitting)
The method is seeing continued developments and new applications to this day, with an ever growing literature; the first instances where the method was developed are:
– P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Bull. Amer. Math. Soc. (N.S.), 26(1):119–123, 1992.
– P. Deift, “Orthogonal Polynomials and Random Matrices: a Riemann–Hilbert approach”; Courant Institute Lectures, (’98);
– P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math., 52(11):1335–1425, 1999.
Course 2: “Connections between Orthogonal Polynomials, Sparse Interpolation, Exponential Analysis, Padé Approximation and Gaussian Integration”.
Lecturer: Annie Cuyt (Universiteit Antwerpen, Belgium).
Course overview: This course explores the deep connections between various mathematical tools and techniques: orthogonal polynomials, Padé approximation, Gaussian quadrature, exponential analysis, and sparse interpolation. Despite appearing in different contexts, all of these methods are linked through a common structure—Hankel matrices.
The course explains how orthogonal polynomials can be formally defined using a linear functional, and how these polynomials naturally relate to Padé approximants of a power series with the linear functional’s scalar values as coefficients. When, in addition, these values are moment integrals with respect to a weight function, the Padé approximants can be interpreted as Gaussian quadrature rules, where the nodes and weights have a direct connection to the polynomials orthogonal with respect to the given linear functional.
Further, the problem of reconstructing the Gaussian nodes and weights from the given moments, is shown to be equivalent to exponential analysis or the Prony problem—a classic inverse problem. In the context of computer algebra, this is referred to as sparse interpolation. The discussion also highlights how the nodes can be obtained from the generalized eigenvalues of a Hankel structured matrix pencil, and how the weights are computed by solving a linear system with a Vandermonde structure.
Finally, it is noted that these relationships and methods can be extended to the multivariate setting, provided the appropriate generalizations are applied.
Course 3: “From Jacobi polynomials to random tilings”.
Lecturer: Arno Kuijlaars (Katholieke Universiteit Leuven, Belgium).
In the first part of the course, we discuss the asymptotic behavior of Jacobi polynomials with varying parameters. This is used as a toy model to illustrate the powerful Riemann-Hilbert analysis The second part of the course will cover a recent application of Jacobi polynomials with varying non-standard parameters to random tiling problems.
Provisional outline:
1. Jacobi polynomials with classical parameters
2. Riemann-Hilbert problem
3. Jacobi polynomials with non-standard parameters
4. Lozenge tilings of hexagon
5. Eynard-Mehta formula
6. Asymptotics of Jacobi polynomials.
Course 4: “Multivariate orthogonal polynomials and applications”.
Lecturer: Lidia Fernández (Universidad de Granada, Spain).
Course overview: Multivariate orthogonal polynomials are not just a simple generalization of the polynomials in a single variable, but they are actually very complex mathematical objects with singular properties. The first part of the course will cover the general properties of orthogonal polynomials in several variables including, for example, recurrence relations, Jacobi matrices or Christoffel-Darboux formulae. Some interesting examples will be analyzed, pointing out the different bases and their particular properties. The final part of the course will focus the applications of these polynomials, such as the relation with optics and aberrations.
Course 5: “Diagrammatic and harmonic analysis methods for
orthogonal polynomials”.
Lecturer: Luis Velázquez (Universidad de Zaragoza, Spain).
Course overview: Orthogonal polynomials have links with the study of random walks -more generally, Markov chains- which have been traditionally used to get probabilistic information about such random systems via standard methods from the theory of orthogonal polynomials. These lectures will show some payoffs of this connection which take the form of diagrammatic techniques in orthogonal polynomial theory. Among other things, this diagrammatic approach sheds light on another classical connection which links orthogonal polynomials and harmonic analysis, providing new results in both areas. The course will show some of these results, as well as some open problems posed by this new look at orthogonal polynomial theory.