Activities

• IMAG Functional Analysis Seminar
• Title: Dynamical properties of weighted shifts on sequence spaces
• By Tomáš Raunig (Charles University, Czech Republic)
• Abstract: In this talk, we will explore dynamical properties of weighted shifts on sequence spaces, a central class of operators in linear dynamics. After a brief overview of linear dynamics on Fréchet spaces, we will recall the notions of generalized hyperbolicity, (periodic) shadowing, and (uniform) topological expansivity. We then discuss how these concepts manifest in the setting of weighted shifts, highlighting an example of a generalized hyperbolic shift without the shadowing property, a characterization of uniform topological expansivity under natural assumptions, and a weight-based criterion for the periodic shadowing property. These results provide answers to three recent open questions in the field.

Seminario de Geometría
Conferenciantes: Francisco Torralbo y José María Espinar
Título: Usos de ChatGPT en el aula. Dos casos concretos

• IMAG Functional Analysis Seminar
• Title: Tingley's Problem(s): Positive answers and insightful results
• By Pedro Saavedra Ortiz (UGR)
• Abstract: Banach spaces play a key role in mathematics due to their structural richness and wide applications. Classical results link isometries of normed spaces, unit balls, and spheres to linear isometries, with Tingley's Problem (1987) focusing on whether sphere isometries extend isometrically and R-linearly - a question still open in dimensions ≥ 3. A variant of this problem studies spheres of positive elements, reducing even further the available information about the considered spaces. In this talk, we will show the interplay between Algebra, Analysis and Geometry that is suggested by Tingley's Problem, and we will explore some of the tools utilised to provide a positive answer in certain spaces (such as C(K) with K compact or JBW*-algebras).

• IMAG Functional Analysis Seminar
• Title: The Daugavet property is equivalent to the polynomial Daugavet property
• By Sheldon Dantas (UGR)
• Abstract: In this talk we will show that the Daugavet property implies the polynomial Daugavet property. The central argument relies on a geometric characterization due to Shvidkoy: we will prove that this characterization, originally formulated in terms of the weak topology, remains valid when replacing it with the weak polynomial topology. Finally, we will discuss other results obtained with close related techniques. This talk is based on a recent joint work with Miguel Martín and Yoël Perreau (University of Tartu, Estonia).

Author : Dong-Hwi Seo, University of Granada

Title : Free boundary minimal surfaces in a ball and eigenvalue problems

Where : Seminario 1, IMAG

When : September 19th, 12h - 13h

Abstract : Eigenvalue problems have played a pivotal role in the study of minimal surface theory. In this talk, I will provide an overview, starting with basic concepts and progressing to recent development. Specifically, I will explain free boundary minimal surfaces in the unit ball and their connection to eigenvalue problems. Key conjectures in this area will be highlighted, along with a discussion of my recent contribution.

Author : David Kalaj, University of Montenegro

Title : The Schwarz lemma for holomorphic and minimal disks at the boundary.

Where : Seminario 1 (IMAG)

When : 12h - 14h

Abstract : We first prove a Boundary Schwarz lemma for holomorphic disks on the unit ball in ${\mathbb{C}}^n$. Further by using a Schwarz lemma for minimal conformal disks of Forstnerič and Kalaj, we prove the boundary Schwarz lemma for such minimal disks.

Conferenciante : Antonio Hidalgo (Max Planck Institute for Mathematics in the Sciences)

Título : Pathological solutions of 2D Navier-Stokes equations with gradients in Hardy spaces.

Localización : Aula A24, Facultad de Ciencias

Resumen : In this talk, we will review the main results and limitations in the use of convex integration for the unforced Euler and Navier-Stokes equations in 2D. We will then present a new sharp result that establishes a threshold between rigidity and flexibility in the 2D Navier-Stokes equations. This is a joint work with Jan Burczak.

FIRST TALK: 11:30h

Author : Rabia Aktas-Karaman, Ankara University, Turkey

Title : Fourier transforms of orthogonal polynomials on the cone and on the paraboloid

Abstract : The purpose of this paper is to obtain Fourier transforms of multivariate orthogonal polynomials on the cone such as Laguerre polynomials on the cone and Jacobi polynomials on the cone and to define two new families of multivariate orthogonal functions by using Parseval's identity. Also, the obtained results are expressed in terms of the continuous Hahn polynomials. Moreover, we present similar results for the orthogonal polynomials on the paraboloid.

References:
1. Koelink, H.T. On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc., 124(3), 887-898, 1996.
2. Aktas Karaman, R., Area, I. Fourier transforms of orthogonal polynomials on the cone, Numerical Algorithms, https://doi.org/10.1007/s11075-025-02112-x , 2025.
3. Güldogan Lekesiz, E., Aktas, R., Area, I. Fourier transform of the orthogonal polynomials on the unit ball and continuous Hahn polynomials, Axioms, 11(10), 558, 2022.
4. Cetin, H.O., Aktas Karaman, R. Fourier transforms of orthogonal structures on the paraboloid, submitted for publication

SECOND TALK:

Author : Miguel Piñar, Universidad de Granada

Title : A fully diagonalized spectral method on the unit ball

Abstract : Our main objective in this work is to demonstrate how orthogonal Sobolev polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary Schr\"odinger equation on the unit ball can be studied from a variational perspective. In this variational formulation, a Sobolev inner product naturally arises. As test functions, we consider the linear space of polynomials satisfying the boundary conditions on the sphere, and a basis of mutually orthogonal polynomials with respect to the Sobolev inner product is provided. The basis of the proposed method is provided in terms of spherical harmonics and univariate Sobolev orthogonal polynomials. The connection formula between these orthogonal Sobolev polynomials and classical orthogonal polynomials on the ball is established. Consequently, the Sobolev Fourier coefficients of a function satisfying the boundary value problem are recursively derived. Finally, numerical experiments were presented.

Location: Seminario 2, IMAG.
When: Thursday, September 11th 2025

• IMAG Functional Analysis Seminar
• Title: El problema de Mackey-Gleason-Bunce-Wright: Morfismos de estructuras unitarias
• By Gerardo Martín Escolano (UGR)
• Abstract: En esta charla hablaremos sobre la reciente resolución del problema de Mackey-Gleason-Bunce-Wright y su aplicación en el estudio los llamados "piecewise Jordan homomorphisms" sobre el conjunto de unitarios de una JBW$^*$-álgebra.