## 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 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

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

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

Abstract: The Gaussian ensembles, originally introduced by Wigner may be generalized to an n-point ensemble called the beta-Hermite ensemble. As with the original ensembles we are interested in studying the local behavior of the eigenvalues. At the edges of the ensemble the rescaled eigenvalues converge to the Airy_beta process which for general beta is characterized as the eigenvalues of a certain random differential operator called the stochastic Airy operator (SAO). In this talk I will give a short introduction to the Stochastic Airy Operator and the proof of convergence of the eigenvalues, before introducing another interesting eigenvalue process. This process can be characterized as a limit of eigenvalues of minors of the tridiagonal matrix model associated to the beta-Hermite ensemble as well as the process formed by the eigenvalues of the SAO under a restriction of the domain. This is joint work with Angelica Gonzalez.

**Thereafter talks will be held monthly on following dates:**

**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 *

Joint work with A.I. Aptekarev and W. Van Assche

Abstract: Hermite–Padé approximants of type II are vectors of rational functions with common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector is given by a pair of Cauchy transforms of smooth complex measures supported on the real line. The convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals (Angelesco systems) or two intervals that coincide under the condition that the ratio of the measures is a restriction of the Cauchy transform of a third measure (Nikishin systems). In this work we consider the case where the supports form two overlapping intervals (in a symmetric way) and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap. We derive Szegő-type formulae for the asymptotics of the approximants, identify the convergence and divergence domains (the divergence domains appear for Angelesco systems but are not present for Nikishin systems), and show the presence of overinterpolation (a feature peculiar for Nikishin systems but not for Angelesco systems).

**26 November**:title and abstract: TBA

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

** Past seminars**

**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. **