## Orthogonal Polynomials, Special Functions, Operator Theory and Applications

**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 GMT 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

**Talks will now be held monthly on following dates:**

**26 November**:** Benjamin Eichinger**, Johannes Kepler University, Linz, Institute of Analysis (Austria)

**Title:** Stahl-Totik regularity for orthogonal rational functions with real poles and GMP matrices

**Abstract:**There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a real, compactly supported measure and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for infinity. We extend aspects of this theory in the setting of rational functions with poles on the extended real line, obtaining a formulation which allows multiple poles and proving an invariance with respect to Möbius transformations. As for Jacobi matrices, we obtain a characterization of Stahl-Totik regularity of a GMP matrix in terms of its matrix elements; this yields an alternative proof of a theorem about a Cesàro-Nevai property of regular Jacobi matrices on finite gap sets. If time permits, we will discuss a related Chebyshev extremal problem and asymptotics of the extremal rational function.

This talk is based on a joint work with M. Lukić and G. Young.

**10 or 17 December**: title and abstract:TBA

** Past seminars**

**29 October: Maxim Yattselev (Indiana University-Purdue University Indianapolis, USA)**

Title: *On Hermite–Padé approximants for a pair of Cauchy transforms with overlapping symmetric supports *

A recording of this talk is available here.

**1 October** Diane Holcomb (KTH, Sweden)

Title: *The stochastic Airy operator and an interesting eigenvalue process.*

**17 September: Margit Rösler (Paderborn University, Germany)**

Title: *Intertwining operators between Dunkl operators of type Bn*

Lecture slides can be found here

**03 September: Giorgio Mantica (Università Degli Studi Dell’insubria, Italy)**

*Computing the isospectral torus of finite sets of intervals via mappings of polynomial*

A recording of this talk can be found here.

**27 August : W. Riley Casper (California State University, Fullerton, USA)**

*Fourier Algebras for Orthogonal Matrix Polynomials*

A recording of this talk can be found here.

**20 August: Part I** **Robert Buckingham (The University of Cincinnati, USA,) and Part II Peter Miller (U. Michigan, USA)**

*Large-degree asymptotics of rational Painleve-IV solutions by the isomonodromy method.*

Part I - Robert Buckingham: recording here

Part II - Peter Miller: recording here

**13 August : Maurice Duits (KTH, Sweden) **

*Limit shape fluctuations for q-distributions for boxed plane partitions via q-Hahn polynomials*

A recording of this talk can be found here

**06 August:** **Robert Milson : (Dalhousie University, Canada)**

*The adelic grassmannian and exceptional Hermite polynomials.*

A recording of this talk can be found here

** 30 July: Manuela Girotti : (U. Montreal, Canada) **

*Fredholm Determinant Solutions of the Painlevé II Hierarchy and Gap Probabilities of Determinantal Point Processes.*

A recording of this talk can be found here

** 23 July: Roozbeh Gharakhloo: (Colorado State, US)**

*A Riemann-Hilbert approach to asymptotic analysis of Toeplitz+Hankel determinants*

A recording of this talk can be found here

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

A recording of this talk can be found here

**9 July: Fredrik Johansson: (INRIA Bordeaux, France) **

*Special functions in Fungrim*

A recording of this talk can be found here

**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 Online Mathematical Sciences Seminars. **