The seminar will not be held between mid-July and mid-September. The next session will be in the third week of September. 

[04/07/2025]  Some Perspectives on the Invariant Subspace Problem, Francisco Javier González Doña, Universidad Carlos III de Madrid, Spain.

Abstract: Does every bounded linear operator on a Banach space have a nontrivial invariant subspace? Despite the simplicity of this question, the Invariant Subspace Problem remains one of the most important unsolved problems in Operator Theory.
In this talk, we will present classical results related to this problem, such as Lomonosov's Theorem for compact operators and the consequences of the Borel functional calculus for normal operators. Moreover, we will consider compact perturbations of normal operators on Hilbert spaces, a subclass where the problem remains open. We will also discuss recent advances in this direction and explore how these techniques relate to other aspects of Operator Theory, such as local spectral theory and unconventional functional calculi.

This is part of joint work with Eva A. Gallardo-Gutiérrez.

[20/06/2025] Asymptotics and zeros of Bergman polynomials for domains with reflection-invariant corners, Erwin Miña Díaz,  Mathematics Department at the University of Mississippi, MS, USA.

Abstract: I will present recent results on the strong asymptotic behavior and limiting zero distribution of polynomials $(p_n)_{n=0}^{\infty}$ that are orthogonal over a domain $D$ with piecewise analytic boundary. More specifically, $D$ is assumed to have the property that conformal maps $\varphi$ of $D$ onto the unit disk extend analytically across the boundary $L$ of $D$, and that $\varphi'$ has a finite number of zeros $z_1,\ldots,z_q$ on $L$. The boundary $L$ is then piecewise analytic with corners at the zeros of $\varphi'$. We prove that a Carleman-type strong asymptotic formula for $p_n$ holds outside a certain compact set $K$ that contains each corner of $L$ but otherwise sits entirely inside $D$.  Near each corner, $K$ consists of an analytic arc departing from the corner. As $n\to \infty$, the zeros of $p_n$ accumulate on $K$ and every boundary point of $K$ is a zero limit point.

This is joint work with Aron Wennman.

[06/06/2025]  Asymptotic analysis for a class of planar orthogonal polynomials on the unit disc, Leslie Diëgo Molag, Universidad Carlos III de Madrid, Spain.

Abstract:  We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure $d\mu(z) = (1-|z|^{2})^{\alpha-1}|z-x|^{\gamma}\mathbf{1}_{|z| < 1}d^{2}z,$ where $d^{2}z$ is the two dimensional area measure, $\alpha$ is a parameter that can grow with $n$, while $\gamma>-2$ and $x>0$ are fixed. This measure arises naturally in the study of characteristic polynomials of non-Hermitian random matrix ensembles and generalises the example of a Gaussian weight that was recently studied by several authors. We obtain asymptotics in all regions of the complex plane and via an appropriate differential identity, we obtain the asymptotic expansion of the partition function. The main approach is to convert the planar orthogonality to one defined on suitable contours in the complex plane. Then the asymptotic analysis is performed using the Deift-Zhou steepest descent method for the associated Riemann-Hilbert problem.

This is joint work with Alfredo Deaño, Ken McLaughlin and Nick Simm.

[23/05/2025] Eñe producto, producto de Hadamard, y fórmulas de monodromía, Ricardo Pérez-Marco, CNRS, IMJ-PRG, Université Patis Cité, Paris, Francia.

Resumen: Presentamos el eñe producto con sus propiedades algebraicas y analíticas, y su extensión a las funciones con singularidades aisladas. Ello conduce de forma natural a nuevas fórmulas integrales de monodromía para el eñe producto y para el producto de Hadamard.

[09/05/2025]  From classical sampling to average sampling in shift-invariant-like subspaces of Hilbert-Schmidt operators, Antonio García García, Universidad Carlos III de Madrid, Spain.

Abstract: The talk is divided into two parts. The first one consists of an introduction to classical sampling theory addressed to a non familiar audience in this topic. Furthermore, this contain will help to understand the second part of the talk.
Thus, in a second part we introduce  the average sampling in shift-invariant-like subspaces of Hilbert-Schmidt operators. Both a mathematical and a practical motivation for this sampling problem is given. We need to introduce some non usual concepts concerning this new setting. In particular, the Kohn-Nirenberg transform (or the Weyl transform) for  Hilbert-Schimdt operators on $L^2(\mathbb{R}^d)$. This is a unitary mapping between the space of square integrable functions in the phase space $\mathbb{R}^d\times \widehat{\mathbb{R}}^d$ and the Hilbert space $\mathcal{H}\mathcal{S}(\mathbb{R}^d)$ of Hilbert-Schmidt operators on $L^2(\mathbb{R}^d)$, which permits to take advantage of  some well established sampling results for shift-invariant subspaces in $L^2(\mathbb{R}^d)$ in order to prove similar sampling results in this new setting.

[25/04/2025] On semi-classical weight functions on the unit circle, Luana L. Silva Ribeiro, UNIFEI- Federal University of Itajubá, Brazil

Abstract: In this talk, we consider orthogonal polynomials on the unit circle associated with certain semi-classical weight functions. This means that the weight function satisfies the Pearson-type differential equation. We describe all semi-classical weight functions such that the Pearson equation involves two polynomials of degree at most 2. We also present structure relations for the orthogonal polynomials and non-linear difference equations for the Verblunsky coefficients. 
* Joint work with Cleonice F. Bracciali and Karina S. Rampazzi.

[11/04/2025]  Hypergeometric polynomials via free probability, Andrei Martínez-Finkelshtein, Bayor University (USA) and University of Almería (Spain).

Abstract: Mathematics is a highly interconnected field, and ideas that were initially developed in one context can sometimes find unexpectedly fruitful applications in seemingly unrelated domains. A striking example of this is the recent application of tools from free probability theory to the study of the zeros of polynomials.
One such concept is the finite free convolution of polynomials, which was introduced relatively recently. This concept becomes particularly appealing when applied to hypergeometric polynomials. Remarkably, these polynomials can be represented as a finite free convolution of more elementary building blocks. This representation, combined with the preservation of real zeros and interlacing properties through free convolutions, provides an effective tool for analyzing when all roots of a particular hypergeometric polynomial are real and when they exhibit monotonicity with respect to parameters. Consequently, this approach offers a fresh perspective on the zero properties of hypergeometric polynomials.
Furthermore, this representation remains valid even in the asymptotic regime, allowing us to express the limit zero distribution of generalized hypergeometric polynomials as a free convolution of more "elementary" measures. This convolution can be expressed analytically by combining some integral transforms of these measures, and it turns out that in the case of hypergeometric polynomials, some of these transforms take a particularly simple form.
These results are demonstrated through applications to some families of multiple (or Hermite-Pad\'e) orthogonal polynomials that can be expressed in terms of generalized hypergeometric functions.
This is a joint work with R. Morales (Baylor University) and Daniel Perales (Texas A&M University).

[28/03/2025]  Generalizing Classical Polynomials via Moment Derivates.Victor Soto Larrosa, Universidad de Alcalá, España.

Abstract: This talk aims to generalize the second-order differential equations that generate classical polynomial families such as Jacobi, Laguerre, Hermite, Bessel, and Romanovski by employing the moment derivative. The objective is to construct generalized polynomial solutions that satisfy these moment differential equations, extending the classical framework to include broader functional equations, such as fractional differential equations and q-difference equations. This approach not only recovers the classical polynomials as special cases but also reveals a natural confluence between the generalized and classical systems under certain specifications of the sequence of moments. Numerical results will be presented to illustrate this convergence, highlighting the versatility and potential applications of the moment derivative approach.

[14/03/2025]  HIGHER-ORDER DIFFERENTIAL OPERATORS HAVING BIVARIATE ORTHOGONAL POLYNOMIALS AS EIGENFUNCTIONS, Misael Marriaga, Universidad Rey Juan Carlos, Spain.

Abstract: We introduce a systematic method for constructing higher-order partial differential equations for which bivariate orthogonal polynomials are eigenfunctions. Using the frame- work of moment functionals, the approach is independent of the orthogonality domain’s geometry, enabling broad applicability across different polynomial families. Applications to classical weight functions on the unit disk and triangle modified by measures defined on lower-dimensional manifolds are presented.

[28/02/2025] On numerically generating Sobolev orthogonal polynomials, Niel Van Buggenhout, Departamento de Matemáticas, Universidad  Carlos III de Madrid, Spain.

Abstract: Sobolev orthogonal polynomials  satisfy a long recurrence relation that can be represented by a Hessenberg matrix. The problem of generating a finite sequence of Sobolev orthogonal polynomials can be reformulated as a matrix problem, that is, a Hessenberg inverse eigenvalue problem, where the Hessenberg matrix of recurrences is generated from certain known spectral information. Via the connection to Krylov subspaces we show that the required spectral information is the Jordan matrix containing the eigenvalues of the Hessenberg matrix and the normalized first entries of its eigenvectors. Using a suitable quadrature rule the Sobolev inner product is discretized and the resulting quadrature nodes form the Jordan matrix and associated quadrature weights are the first entries of the eigenvectors. We propose two new numerical procedures to compute Sobolev orthonormal polynomials based on solving the equivalent Hessenberg inverse eigenvalue problem.

[14/02/2025]  Polinomios ortogonales de Sobolev para resolver la ecuación de Schrödinger estacionaria, Teresa E. Pérez-Fernández, Instituto de Matemáticas IMAG  y Departamento de Matemática Aplicada, Universidad de Granada, Spain.

Abstract:  En la formulación variacional del problema de valores en la frontera del oscilador armónico, aparecen de forma natural productos escalares de Sobolev. En esta charla estudiamos las sucesiones de polinomios ortogonales de Sobolev asociados a este producto escalar y sus representaciones en términos de los polinomios de Gegenbauer, y mostramos un algoritmo para generar estos polinomios de forma recursiva. También obtenemos la asintótica relativa exterior entre estos polinomios de Sobolev y los polinomios clásicos de Legendre. Además analizamos la funciones test del problema en términos de familias de polinomios de Sobolev, y daremos la solución del problema en términos de sumas de Fourier-Sobolev. Finalmente, mostraremos un ejemplo numérico de método espectral definido. 
Este es un trabajo conjunto con L. Fernández, F. Marcellán y M. A. Piñar.

• L. Fernández, F. Marcellán, T. E. Párez, M. A. Piñar, Sobolev orthogonal polynomials and spectral methods in boundary value problems. Appl. Numer. Math. 200 (2024), 254-272.

• L. Fernández, F. Marcellán, T. E. Párez, M. A. Piñar, Sobolev orthogonal polynomials for solving the Schrödinger equation with potentials, In: Castillo, K., Duráan, A.J. (eds) Orthogonal Polynomials and Special Functions. Coimbra Mathematical Texts 3 . Springer, Cham. 2024.

• Talk slides.