SEMINARIO DE ECUACIONES DIFERENCIALES
Conferenciante: Alexis Molino (Universidad de Almería)
Abstract: Abstract_Alexis_Molino
SEMINARIO GOYA
PRIMERA CHARLA
Conferenciante: Salameh Sedaghat (Imam Khomeini International University, Iran)
Título: Analytical properties and applications of Eta functions
Abstract: This talk is dedicated to introducing the properties and application of Eta functions. We derive the properties of the Eta function, such as the generating function, integral representation, and the Laplace transform. Also, some properties of the Eta-based functions are introduced. Next, we prove that these functions are the analytical eigensolutions of regular fractional Sturm-Liouville problems (RFSLPs) with the fractional differential operators both of Riemann-Liouville and Caputo type. These eigenfunctions are orthogonal with respect to the weight function associated with RFSLPs and form a set of non-polynomial bases.
SEGUNDA CHARLA
Conferenciante: Marlon J. Recarte (Universidad Nacional Autónoma de Honduras)
Título: Bernstein-type operators on the unit disk
Abstract: We construct and study sequences of linear operators of Bernstein-type acting on bivariate functions defined on the unit disk. To this end, we study Bernstein-type operators under a domain transformation, we analyse the bivariate Bernstein-Stancu operators, and we introduce Bernstein-type operators on disk quadrants by means of continuously differentiable transformations of the function. We state convergence results for continuous functions and we estimate the rate of convergence. Finally some interesting numerical examples are given, comparing approximations using the shifted Bernstein-Stancu and the Bernstein-type operator on disk quadrants.
SEMINARIO DE GEOMETRÍA
Conferenciante: Philipp Käse (Technische Universität Darmstadt)
Abstract: In 1841 Delaunay characterized surfaces of constant mean curvature \(H=1\) in Euclidean 3-space invariant under rotation. The result was generalized by several authors to screw-motion invariant CMC surfaces in \(\mathbb{E}(\kappa,\tau)\). In this more general setting CMC tubes can arise in addition to the Delaunay surfaces. In this talk I want to present existence conditions and talk about further properties of these tubes such as embeddedness and foliation.
SEMINARIO DE ÁLGEBRA
Conferenciante: Carlos Jesús Moreno Ávila (Universitat Jaume I)
Abstract: Let \(C\) be a curve on \(\mathbb{P}^2\) with only one place at infinity at a point \(p\in\mathbb{P}^2\), and let \(S_{C,\infty}\) be its semigroup at infinity, i.e. the additive submonoid of \((\mathbb{N},+)\) consisting of the orders ---with negative sign--- of the poles of the regular functions around (but not in) \(p\). After a theorem by Abhyankar and Moh, we can associate to \(C\) a so-called \(\delta\)-sequence in \(\mathbb{N}_{>0}\) which is a system of generators of \(S_{C,\infty}\), by no means unique.
Curves with only one place at infinity are relevant, for instance they play an important role in the study of the Jacobian conjecture. However, not so much is known about \(S_{C,\infty}\) from a combinatorial point of view. In this talk we review the previous concepts and results and we see some properties of the \(\delta\)-sequences; in particular, we introduce the notion of minimal \(\delta\)-sequence as that with least length among all the \(\delta\)-sequences generating the same semigroup at infinity. In addition, we show how we can compute the remaining \(\delta\)-sequences and prove that a minimal \(\delta\)-sequence encodes a family of infinitely many rational surfaces which are Mori dream surfaces. Finally, we see algorithm procedures to compute minimal \(\delta\)-sequences and then families of Mori dream surfaces.
This talk is based on a joint work with C. Galindo, F. Monserrat and J.-J. Moyano-Fernández.
SEMINARIO DE ECUACIONES DIFERENCIALES
Conferenciante: Ángel Arroyo (Universidad Complutense de Madrid)
Abstract: abstract angel arroyo
YOUNG RESEARCHERS MATH SEMINAR
Conferenciante: Gerardo García Moreno (Universidad de Granada)
Abstract: General Relativity is the most succesful theory we have for describing gravitational phenomena. Its range of applicability is vast: from solar system scales to cosmological scales. It is well-known that GR allows the existence of time machines: devices that are able to generate Closed-Timelike-Curves (CTCs). Although these kind of objects are allowed in GR, most researchers believe that they cannot occur on macroscopic scales. However, it is widely believed that a full quantum theory of gravity would be required to fully understand the mechanisms beneath this "Chronology Protection Conjecture". In this talk, I will describe the status of this conjecture within the context of analogue gravity. Analogue gravity is a research programme which investigates analogues of general relativistic gravitational fields within other physical systems, typically condensed matter systems, with the aim of gaining new insights into their corresponding problems. There exists a huge catalogue of analogue models: from classical and quantum fluids to optic systems.
After briefly reviewing the role of time machines and the Chronology Protection Conjecture in General Relativity and introducing the analogue gravity program, I will jump to discuss the content of our last paper: arXiv:2201.11072 [gr qc]. We examine carefully whether it is possible to simulate these kind of spacetimes with CTCs within analogue gravity. Although I will focus the discussion on a concrete analogue system, sound waves propagating on top of a Bose Einstein condensate, I will explain the extension of our results to other kind of analogues.
SEMINARIO DE GEOMETRÍA
Conferenciante: José M. Manzano
Abstract: In this talk, we will consider an arbitrary orientable Riemannian surface \(M\) and an open relatively compact domain \(\Omega\subset M\) with piecewise regular boundary. Given a Killing submersion \(\pi:\mathbb{E}\to M\), we will discuss some properties of the divergence lines spanned by a sequence of minimal graphs over \(\Omega\), as well as how they produce certain laminations in \(\pi^{-1}(\Omega)\) whose leaves are vertical surfaces (after considering a subsequence). We will apply these results to give a general solution to the Jenkins-Serrin problem over \(\Omega\) under natural necessary assumptions. This is a joint work with Andrea del Prete and Barbara Nelli.
SEMINARIO DE ÁLGEBRA
Conferenciante: Patricio Almirón (IMAG-Universidad de Granada)
Resumen: El estudio local de singularidades de curvas planas aisladas es un vasto tema de investigación que ha sido desarrollado durante siglos. En el estudio local de una singularidad C de una curva plana aislada, un objeto algebraico importante es el conjunto de valores S de multiplicidades de intersección de C con otras curvas. Este conjunto de valores tiene una estructura natural de semigrupo, que en el caso de los irreducibles es un semigrupo numérico (un semigrupo sobre los números naturales con complemento finito). Por otro lado, se puede asociar un enlace algebraico a la singularidad que permite clasificar sus propiedades topológicas. El leitmotiv de la charla será explicar cómo se pueden utilizar las propiedades combinatorias del semigrupo de valores para entender la topología local del enlace asociado y viceversa.
DIFFERENTIAL EQUATIONS SEMINAR
Speaker: David Gómez Castro (Universidad Complutense de Madrid)
Abstract: The theory of viscosity solutions was developed to deal with Hamilton-Jacobi type problems in the late 80s and 90s by Crandall, Lions, and others. The surname "viscosity" comes from their construction through the vanishing viscosity method. In many scenearios have well-posedness and select the physical solution in many settings. Furthermore, their stability properties makes them suitable to study different approximations: finite differences and asymptotic limits as $t \to \infty$. The aim of this talk is to introduce this notion of solution and show its usefulness to study the time limit of Aggregation-Diffusion equations. The talk presents joint work with: J.A. Carrillo, A. Fernández-Jiménez, and J.L. Vázquez
DIFFERENTIAL EQUATIONS SEMINAR
Conferenciante: José Antonio Carrillo (Mathematical Institute, University of Oxford)
Abstract: We will discuss several recent results for aggregation-diffusion equations related to partial concentration of the density of particles. Nonlinear diffusions with homogeneous kernels will be reviewed quickly in the case of degenerate diffusions to have a full picture of the problem. Most of the talk will be devoted to discuss the less explored case of fast diffusion with homogeneous kernels with positive powers. We will first concentrate in the case of stationary solutions by looking at minimisers of the associated free energy showing that the minimiser must consist of a regular smooth solution with singularity at the origin plus possibly a partial concentration of the mass at the origin. We will give necessary conditions for this partial mass concentration to and not to happen. We will then look at the related evolution problem and show that for a given confinement potential this concentration happens in infinite time under certain conditions. We will briefly discuss the latest developments when we introduce the aggregation term. This talk is based on a series of works in collaboration with M. Delgadino, J. Dolbeault, A. Fernández, R. Frank, D. Gómez-Castro, F. Hoffmann, M. Lewin, and J. L. Vázquez.