Orthogonal Polynomials, Special Functions, Operator Theory and Applications

Orthogonal Polynomials, Special Functions, Operator Theory and Applications (OPSFOTA) virtual seminar series

Organisers: Ana F. Loureiro (University of Kent), Thomas Bothner (King's College London), Adri Olde Daalhuis (University of Edinburgh), Walter Van Assche (KU Leuven, Belgium), Jani Virtanen (University of Reading)

Thursdays 15.30 -16.30 BST beginning Thursday 18 June.

Each week we will present a seminar on the topics of orthogonal polynomials and special functions, operator theory and their remarkable applications to other fields of mathematics including analytic number theory, random matrix theory, numerical and computational methods and mathematical physics. The focus of discussion will be on the fruitful interactions between numerical analysis, approximation theory, operator theory, spectral methods, combinatorics and the modern theory of orthogonal polynomials and special functions.

The series is organised by the OPSFOTA research network group funded by LMS. Due to the funding limitations it was restricted to researchers that are based in geographically close institutions: in the UK, University of Kent, University of Reading, King’s College London, Imperial College, University of Sussex, University College London and elsewhere in Europe KU Leuven (Belgium), Radboud Universiteit (the Netherlands) and Université Lille (France). The current situation has given the us the opportunity to open the seminar series to anyone who would like to participate.

To register to attend the seminar please complete this form. Registration will close at 12.00 on the day of the seminar. The meetings will be held using Zoom and joining details will be sent after 12.00 on the day of the seminar..

Seminars and Speakers

9 July: Fredrik Johansson: (INRIA Bordeaux, France) Special functions in Fungrim

Abstract: I will present Fungrim (http://fungrim.org), a project to build a database of formulas and theorems for special functions. A central goal of Fungrim is to make the content mathematically rigorous and easily computer-readable; this means, among other things, using a symbolic representation with well-defined semantics and providing explicit conditions on the domain of variables. As an extension of the effort to define the semantics and test the entries in Fungrim for correctness, I am presently developing a new library for exact real and complex arithmetic (http://fredrikj.net/calcium/) which will be discussed briefly.

16 July: Sergei Denisov : (U. Wisconsin, USA) Continuity of weighted operators, Muckenhoupt weights, and Steklov problem for orthogonal polynomials.

Abstract: We consider weighted operators acting on L^p and show that they depend continuously on the weight in Muckenhoupt space A_p, which slightly improves the recent work of Pattakos and Volberg. Then, we apply this result to estimate L^p_w norm of polynomials orthogonal on the unit circle when the weight w belongs to A_2 and p>2. We obtain the asymptotics of the polynomial entropy as well. This talk is based on the joint work with M. Alexis (UW-Madison) and A. Aptekarev (Keldysh Institute, Moscow).

23 July: TBC: (TBC) TBC

Abstract: TBC

30 July: Manuela Girotti : (U. Montreal, Canada) Fredholm Determinant Solutions of the Painlevé II Hierarchy and Gap Probabilities of Determinantal Point Processes.

Abstract: We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr (2018). In addition, we obtain asymptotics at infinity for the Painlevé transcendents and large gap asymptotics for the corresponding point processes. This is a joint work with Mattia Cafasso (Univ. Angers) and Tom Claeys (UC Louvain).

Past seminars

2 July: Emma Bailey (University of Bristol, UK) Moments of moments

A recording of this talk can be found here

25 June: Grzegorz Swiderski (KU Leuven, Belgium) About essential spectra of unbounded Jacobi matrices

This talk was not recorded

18 June: Håkan Hedenmalm (KTH Stockholm, Sweden) Gaussian analytic functions and operator symbols of Dirichlet type

A recording of this talk can be found here

This seminar series is supported as part of the ICMS/INI Online Mathematical Sciences Seminars.