1. A Linear Algebra setting for the study of monomiality and operator methods
   
   Luis Verde-Star, department of Mathematics, Universidad Autónoma Metropolitana (México)
   
   Date: Tuesday, September 30, 2025. From 16:00 to 17:00, room 2.2.D08.
   
   Abstract: In this talk we will discuss some properties of an algebra ${\mathcal L}$ of infinite generalized Hessenberg matrices that provides a convenient setting for the study of the monomiality concept and the operator methods that are often used to study sequences of polynomials.
   A polynomial sequence $\{p_n(t)\}_{n\ge 0}$ has the monomiality property if there exist linear operators $P$ and $M$ that satisfy
   
   P p_n(t) = n\, p_{n-1}(t), \quad M p_n(t)=p_{n+1}(t), \qquad n\ge 0.
   
   The elements of the algebra ${\mathcal L}$ can be considered as sequences of polynomials, sequences of formal Laurent sequences, or linear operators on the vector space of polynomials and the vector space of formal Laurent sequences.
   We will present some examples of the monomiality operators $P$ and $M$ for Appell and Sheffer sequences and also for some orthogonal polynomial sequences, such as the Hermite, Laguerre, Charlier, and Meixner polynomials.
   
   

Reference:
L. Verde-Star, A linear algebra approach to monomiality and operational methods, Linear Algebra Appl. 726 (2025), 1-31.

1. Characterizations of Sheffer- Dunkl polynomials
   
   Judit Mínguez Ceniceros, department of Mathematics and Computation, Universidad de la Rioja
   
   Date: Tuesday, June 10, 2025. From 16:00 to 17:00, room 2.2.D08.
   
   Abstract: In this talk we define Sheffer-Dunkl sequences of polynomials using umbral calculus and provide some properties and examples as Bernoulli-Dunkl or Euler-Dunkl polynomials. In this way some classical operators as the derivative operator or the difference operator are replaced as analogous operators in the Dunkl universe.  Moreover, Sheffer-Dunkl polynomials can be characterized using different Stieltjes integrals extending Thorne Theorem and Sheffer Theorem to the Dunkl context.
   
   

2. Reproducing Kernel Banach Spaces and Abstract Neural Networks
   
   Raúl Felipe Sosa, Department of Mathematics, Universidad Carlos III de Madrid.

Date: Tuesday, May 27, 2025. From 16:00 to 17:00, room 2.2.D08.

Abstract:  We can say that we are living in the era of artificial intelligence, which has emerged as a new paradigm impacting all aspects of life. Following this fundamental trend, mathematicians have sought theoretical frameworks to understand, from a mathematical perspective, the underlying nature of deep learning models. One such framework is that of Reproducing Kernel Banach Spaces (RKBS), which generalize the well-known concept of Reproducing Kernel Hilbert Spaces (RKHS), originally developed in the context of Hilbert spaces. The connection between RKBS and fully connected neural networks with a single hidden layer (MLP with one hidden layer) is far from trivial. It is related to an existence result of sparse solutions for the variational problem of minimizing an abstract loss functional defined on the corresponding RKBS. In the first part of the talk, we will briefly review the results that lead to this fascinating relationship.On the other hand, since RKBS can only be naturally associated with single-hidden-layer neural networks, we introduce a generalization of the RKBS concept, now related to deep neural networks (with many layers), which also allows, in a certain sense, the composition of generalized RKBS to form what we call abstract neural networks. In the second part of the talk, we will present the results we have obtained in this direction.

3. Group equivariant non-expansive operator with respect to the Hilbert metric
   
   Raúl Felipe Parada, Centro de Investigación en Matemáticas, A.C., Guanajuato. México

Date: Tuesday, May 13, 2025. From 16:00 to 17:00, room 2.2.D08.

Abstract: The notion of GENEO (group equivariant non-expansive operator) was introduced by P. Frosini and G. Jablonski in 2016. Since, it has quickly found applications, for instance, in Topological Data Analysis (TDA), Machine Learning, and Artificial Intelligence. Formally, GENEOs are maps between perception pairs. A perception pair $(B(X), G)$ consists of a Banach space $B(X)$ of real-valued functions defined on a set $X$, and a subgroup $G$ of the group of all bijections on $X$. Usually, $B(X)$ is formed by bounded functions, and the norm used is the supremum norm. Moreover, it is required that $B(X)$ is closed under composition with elements of $G$, and that the norm on $B(X)$ is $G$-invariant. Given two perception pairs $(B(X), G)$ and $(B(Y), H)$, and a group homomorphism $T: G \longrightarrow H$, we say that a map $F: B(X) \longrightarrow B(Y)$ is a GENEO, if $F$ is non-expansive with respect to the Hilbert metric and $T$-equivariant. The latter means that $F(f \circ g)=F(f) \circ T(g)$. In this work, we aim to study GENEOs between spaces of real-valued functions equipped with the Hilbert metric (more precisely, a pseudo-metric) defined via the cone of non-negative functions. This framework allows us to construct GENEOs using the class of homogeneous and order-preserving maps.

4. Reconstruction of Functions through an enrichment of the Crouzeix–Raviart Finite Element
   
   Federico Nudo, University of Padova
   
   Date: Friday, April 4, 2025. From 16:00 to 17:00, room 2.2.D08.
   
   Abstract: The reconstruction of functions is a fundamental task in various applications, ranging from computer graphics to remote sensing. In this work, we introduce a new family of general weighted quadratic polynomial-enriched finite elements based on the Crouzeix–Raviart finite element, aiming to develop an approximation operator that improves reconstruction accuracy. This approach employs additional weighted degrees of freedom to enhance both the flexibility and accuracy of the standard Crouzeix–Raviart finite element. Numerical experiments demonstrate that the proposed enriched elements achieve significantly higher accuracy compared to the traditional Crouzeix–Raviart finite element.
   
   

5. Special polynomials of mixed type: monomiality principle, degeneracy, and computational features.
   
   Yamilet Quintana-Mato, Department of Mathematics, Universidad Carlos III de Madrid.

Date: Friday, March 7, 2025. From 16:00 to 17:00, room 2.2.D08.

Abstract: The objective of this talk is to present an overview of the research activity I intend to develop in the coming years within the Department of Mathematics at Universidad Carlos III de Madrid (UC3M). This research work, planned for the short and medium term, falls within the fields of approximation theory and special functions, with an emphasis on studying the algebraic and analytical properties of various classes of special functions. In particular, the focus will be on special polynomials defined through generating functions and series expansions, such as the so-called special polynomials of mixed type, for which concepts like degeneracy and the monomiality principle have been explored, among others.  Part of this talk is based on the research project I submitted as part of my application for the position of Associate Professor in the area of Applied Mathematics at UC3M.

6. Markov and Stieltjes type-theorems for Sobolev orthogonality.
   
   Hector Pijeira-Cabrera, Departamento de Matemáticas, Universidad Carlos III de Madrid.
   
   Date: Friday, February 21, 2025. From 16:00 to 17:00, room 2.2.D08.
   
   Abstract: Introduction to the theory of orthogonal polynomials in close relation to rational approximation. Extensions of the Markov and Stieltjes theorems. Markov's theorem for Sobolev-type orthogonality. Progress in the search for a Stieljes-type theorem for the Sobolev case.
   
   

7. Transformaciones fuertes de Darboux y el álgebra $\mathcal{D}(W)$.
   
   Ignacio Bono, Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina.
   
   Date: Wednesday, Friday September 11 from 16:00 to 17:00, room 2.2.D08.
   
   Abstract: Dado un peso matricial $W$, podemos asociarle un producto interno, una secuencia única de polinomios ortogonales mónicos $P_n$, y el álgebra $\mathcal{D}(W)$ de todos los operadores diferenciales $D$ que tienen como funciones propias la secuencia $P_n$, es decir, $P_{n}(x) \cdot D = \Lambda_{n}(D) P_{n}(x)$. En esta charla, discutiremos el Problema de Bochner de Matriz, que tiene como objetivo clasificar todas las $N \times N$ matrices de peso $W$ cuya secuencia asociada de polinomios matriciales ortogonales sean funciones propias de un operador diferencial matricial de segundo orden. También introduciremos el concepto de una transformación Darboux fuerte entre dos matrices de peso y exploraremos cómo se relacionan sus álgebras asociadas.
   
   

8. Sampling in shift-invariant-like subspaces of Hilbert-Schmidt operators: 
   Why and how to do it?
   
   Antonio G. Garcia, Universidad Carlos III de Madrid.
   
   Date: Thursday September 5 at 16:00, classroom 2.2.D08.  

Abstract: The aim in this talk is to introduce the average sampling in shift-invariant-like subspaces of Hilbert-Schmidt operators. A practical motivation for this sampling problem is given by the channel estimation problem in wireless communications. The concept of translation of an operator allows to consider the analogue of shift-invariant subspaces in the class of Hilbert-Schmidt operators. Thus, we extend the concept of average sampling to this new setting, and we obtain the analogous sampling formulas. The key point here is the use of the Kohn-Nirenberg transform (or the Weyl transform), 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 of Hilbert-Schmidt operators on $L^2\left(\mathbb{R}^d\right)$, which permits to take advantage of some well established sampling results. The talk will start with a brief on sampling theory addressed to a non familiar audience in this topic.