Title: Ultrapotencias en espacios de Banach (minicurso)
By Abraham Rueda Zoca (UGR)
Abstract: El objetivo de este minicurso es presentar de forma relativamente elaborada y autocontenida la construcción del ultraproducto y la ultrapotencia de un espacio de Banach.Teniendo esto en cuenta y pensando en que este curso se dirige potencialmente a un estudiante medio de máster, con lo que el conocimiento previo que asumiremos es el contenido natural de un grado en matemáticas con un curso en Análisis Funcional, se introducirán rudimientos sobre ultrafiltros, así como la definición de límite a través de un ultrafiltro. Tras esto, se hará la construcción formal de ultraproducto y ultrapotencias en espacios de Banach, y se estudiará la relación entre la superreflexividad y la reflexividad de ultrapotencias así como el vínculo entre entre la convexidad uniforme y la convexidad estricta de la ultrapotencia
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Abstract: We discuss an eigenvalue estimate that holds on every embedded self-similar shrinker for mean curvature flow. This result is obtained via a Reilly-type formula and can be viewed as an analogue of the first eigenvalue estimate obtained by Choi and Wang for embedded minimal hypersurfaces in the round sphere. Our estimate generalizes earlier work of Ding and Xin on closed selfshrinkers by introducing and minimizing a new functional to treat the non-compact case. This is joint work with Simon Brendle.
Abstract: In my talk I will present an Amann-Zehnder type result for resonance systems of quasilinear elliptic equations with homogeneous Dirichlet boundary conditions, involving nonlinearities growing asymptotically (p, q)-linear at infinity. The proof relies on a cohomological linking in a product Banach space where the properties of cones of the sublevels are missing, differently from the single quasilinear equation. Critical groups of the energy functional have been computed at the origin, in spite of the lack of its C^2 regularity, to exclude that the found mini-max solution is trivial. Finally a local condition is given to guarantee that the found solution is not semi-trivial. This is a joint paper with S. Cingolani and G. Vannella.
By Rafael Chiclana (Michigan State University, USA)
Abstract: El lema de Johnson-Lindenstrauss es uno de los resultados más usados en Informática, ya que permite reducir la dimensión de un conjunto finito, preservando la distancia entre sus puntos. Recientemente, ha surgido una familia de funciones con propiedades más fuertes de preservación de distancias, conocidas como "terminal embeddings". En esta presentación, exploramos cómo construir estas aplicaciones y analizamos su regularidad.
Dates: November 25 (Monday), 11:30-12:30 and November 27 (Wednesday), 12:30-13:30.
Abstract:
The numerical analysis and computation of solutions of differential equations is always at the very centre of applied mathematics and approximation theory. Both boundary value problems (cf., e.g., B., Jódar and Rodríguez, 2019) and initial value problems need approximations by splines, polynomials, quasi-interpolants, radial basis functions etc. (see, e.g., B. and Jäger, 2022) of the local solutions for the algorithms to work efficiently.
A particularly fascinating special case of initial value problems is the situation of delay equation where the solution of the equations depends also on the past and not only on the current values of the solutions. The dynamics of these solutions are naturally much more difficult (non-linear) but there are interesting railway engineering applications which we shall discuss: namely, the current collecting systems of modern high-speed trains (AVE, Eurostar) from overhead wires supported in intervals by stiff springs can be reliably described by delay differential equations. We shall discuss the current situation of research and future challenges in different lectures.
Dates: November 25 (Monday), 11:30-12:30 and November 27 (Wednesday), 12:30-13:30.
Abstract:
The numerical analysis and computation of solutions of differential equations is always at the very centre of applied mathematics and approximation theory. Both boundary value problems (cf., e.g., B., Jódar and Rodríguez, 2019) and initial value problems need approximations by splines, polynomials, quasi-interpolants, radial basis functions etc. (see, e.g., B. and Jäger, 2022) of the local solutions for the algorithms to work efficiently.
A particularly fascinating special case of initial value problems is the situation of delay equation where the solution of the equations depends also on the past and not only on the current values of the solutions. The dynamics of these solutions are naturally much more difficult (non-linear) but there are interesting railway engineering applications which we shall discuss: namely, the current collecting systems of modern high-speed trains (AVE, Eurostar) from overhead wires supported in intervals by stiff springs can be reliably described by delay differential equations. We shall discuss the current situation of research and future challenges in different lectures.
Title: How minimal can be the linear structure of the set of norm attaining functionals of a Banach space?
By Miguel Martín (UGR)
Abstract:
Ten years ago, the late Charles Read constructed a Banach space which contains no proximinal subspaces of finite codimension greater than one, solving a long standing open problem by Singer of the 1970's. Shortly after that, Martin Rmoutil showed that this space also satisfies that its set of norm attaining functionals contains no two dimensional subspace, solving thus a problem of Godefroy of 2000. There are extensions (and somehow simplifications) of Read's space by Kadets, Lopez, Martin, and Werner. None of these spaces is smooth, so their sets of norm attaining functionals contain nontrivial cones. Even though there are smooth a posteriori versions of Read spaces, their sets of norm attaining functionals also contain nontrivial cones.
The aim of this series of talks is to present a detailed account of the very recent construction of a Banach space whose set of norm attaining functionals contains no nontrivial cones (see http://arxiv.org/abs/2406.07273). Actually, the space satisfies that the intersection of any two dimensional subspace of its dual with the set of norm attaining functionals contains, at most, two straight lines (which is, of course, the minimal possibility). We will also show the relation between this example and the long standing open problem of whether finite-rank operators can be always approximated by norm attaining operators.
There is no needed background apart of basic knowledge of functional analysis and of the geometry of Banach spaces.
By Bojan Kuzma (University of Primorska, Eslovenia)
Abstract:
Unbounded, densely defined, closed operators are encountered in quantum physics when solving linear second-order partial differential equations with the method of variable separation (Sturm-Liouville problem) and as generators of strongly continuous one-parameter semigroups. Particularly important is their spectrum. We will present a recent result that classifies additive bijections preserving the spectrum of such operators and compare it to the classification of spectrum-preserving maps whose domain consists of all unbounded operators.
IMPORTANT: LECTURES RESCHEDULED TO MONDAYS AND WEDNESDAYS FROM NOVEMBER 18 TO DECEMBER 4 DUE TO CANCELLATION OF FIRST SESSION (EXCEPT FOR WEDNESDAY NOVEMBER 27, POSTPONED TO THURSDAY NOVEMBER 28).
Mini-course:
An accurate polynomial approximation on equispaced nodes via an interpolation-regression approach with applications.
Lecturer: Federico Nudo, Università della Calabria (Italy) & Université de Pau et des Pays de l'Adour (France).
ABSTRACT:
A very common problem in computational science is determining an approximation, within a fixed interval, of a function evaluated only at a finite set of points. A popular approach to solving this problem relies on polynomial interpolation, which involves determining a polynomial that corresponds to the function at the specified points. A case of particular practical interest is when these points are equispaced within the interval. Under these conditions, a known issue with polynomial interpolation is the Runge phenomenon, where interpolation errors increase near the ends of the interval. In 2009, J. Boyd and F. Xu demonstrated that the Runge phenomenon could be mitigated by interpolating the function over a specific subset of points close to the Chebyshev-Lobatto nodes, known as mock–Chebyshev nodes. However, this strategy entails discarding most of the available data. To enhance the accuracy of Boyd and Xu’s method while fully using all data points, S. De Marchi, F. Dell’Accio, and M. Mazza introduced a technique called the constrained mock-Chebyshev least squares approximation. In this method, the nodal polynomial plays a critical role in ensuring interpolation at mock-Chebyshev nodes. Extending this approach to the bivariate case, however, requires alternative techniques. A recent development by F. Dell’Accio, F. Di Tommaso, and F. Nudo uses the Lagrange multipliers method to define the constrained mock-Chebyshev least squares approximation on a uniform grid. This innovative technique, analytically equivalent to the earlier univariate method, also proves to be more accurate in numerical terms. This mini-course will explore the main properties of this approximation operator and its applications to numerical quadrature, differentiation problems, and the histopolation method.
Dates: every Monday and Wednesday from November 18 to December 4, 2024, at 12:30h, (except Wednesday 27th of November , rescheduled to Thursday 28th of November at 12:30h)
Place: Seminar 1 IMAG and Zoom virtual room no. 1 (Mondays), Seminar 2 IMAG and Zoom virtual room no. 2 (Wednesdays). On Thursday 28th of November, the session will take place in Seminar 1 IMAG and Zoom virtual room no.1.
Soluciones periódicas y subarmónicas para el modelo de la perla.
Por Alexander Gutiérrez (Universidad de Pereira, Colombia).
Abstract: Establecemos condiciones necesarias para la existencia de soluciones periódicas y subarmónicas de una ecuación diferencial que describe el movimiento de una perla sobre un aro circular que gira sujeto a una velocidad angular constante ω y un forzamiento periódico. Nuestro enfoque implica utilizando métodos de sub y super soluciones y algunas técnicas presentadas por Zanolin-Boscaggin y Ureña.
