## 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

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

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

Abstract:Random tilings are known to concentrate around limit shapes when the size of the domain grow large. The fluctuations around these shapes are believed to be described universal processes. An interesting conjecture is that the global fluctuations are described by the Gaussian Free Field. This conjecture is proved in certain integrable examples, but for many models it is still open. In this talk I will be discuss one approach to study such limit shapes and their fluctuation by means of orthogonal polynomials. This discussion will be centered around q-distributions for boxed plane partitions (or equivalently, q-distributions for lozenge tilings of a finite hexagon), where q-Hahn polynomials appear.

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

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

Abstract:The Painleve-IV equation has two families of rational solutions generated respectively by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann-Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painleve-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asymptotic arguments that suggest that in gen- eral solutions of Painleve-IV with large parameters behave either as an algebraic function or an elliptic function. Moreover, the results show that the elliptic approximation holds on the union of a curvilinear rectangle and, in the case of the generalized Okamoto rational solutions, four curvilinear triangles each of which shares an edge with the rectangle; the algebraic approximation is valid in the complementary unbounded domain. We compare the theoretical predictions for the locations of the poles and zeros with numerical plots of the actual poles and zeros obtained from the generating polynomials, and find excellent agreement.

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

Title and abstract: TBA

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

Title and abstract: TBA

**10 September: TBA<**

Title and abstract: TBA

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

Title and abstract: TBA

** Past seminars**

**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/INI Online Mathematical Sciences Seminars. **