Natural operations on holomorphic forms

Demostramos que las únicas operaciones diferenciales naturales entre las formas holomórficas en una variedad compleja son las que se obtienen usando combinaciones lineales, el producto exterior y el diferencial exterior. Para llevar a cabo esta tarea, primero desarrollamos los fundamentos de la teoría de los haces holomórficos naturales sobre una variedad fija, haciendo explícita su estructura galoisiana al probar una equivalencia categórica a la Galois.We prove that the only natural differential operations between holomorphic forms on a complex manifold are those obtained using linear combinations, the exterior product and the exterior differential. In order to accomplish this task we first develop the basics of the theory of natural holomorphic bundles over a fixed manifold, making explicit its Galoisian structure by proving a categorical equivalence à la Galois.• Ministerio de Economía y Competitividad. Proyecto ICMAT Severo Ochoa SEV-2015-0554, para Alberto Navarro Garmendia • Ministerio de Economía y Competitividad. Proyecto MTM2013-42135-P (I+D+i), para Alberto Navarro Garmendia • Ministerio de Economía y Competitividad. Proyecto ANR-12-BS01-0002 (I+D+i), para Alberto Navarro Garmendia • Junta de Extremadura y Fondos FEDER. Ayuda, para para José Navarro Garmendia • Ministerio de Economía y Competitividad. Proyecto MTM2013-42135-P (I+D+i), para Tomás Carlos Tejero PrietopeerReviewe

Equidistribution of orbits of isometries on compact Riemannian manifolds

Sea S un semigrupo finito generado de isometrías de una compacta variedad de Riemann. Presentamos un argumento simple para demostrar que, si existe un punto x cuya órbita Sx es densa, entonces la órbita de cualquier punto está equidistribuida.Let S be a finitely generated semigroup of isometries of a compact Riemannian manifold. We present a simple argument to prove that, if there exists a point x whose orbit Sx is dense, then the orbit of any point is equidistributed.• Junta de Extremadura y Fondo Europeo de Desarrollo Regional. Ayuda (GR18001 and IB18087

Divergence-free tensors associated to a metric

En esta tesis se estudian los invariantes locales que se pueden asociar a una métrica pseudo-Riemanniana. En particular, se trata de describir los espacios de invariantes tensoriales con divergencia nula. A continuación, se utiliza esta teoría para probar algunos teoremas sobre la fundamentación de las ecuaciones de campo de la Relatividad General y el Electromagnetismo, que son los resultados centrales de la memoria. DESARROLLO TEÓRICO: El documento comienza con dos capítulos preliminares, en los que se analiza la definición de construcción local asociada a las métricas. La exposición presentada difiere de la que se encuentra en la literatura, de modo que se incluyen demostraciones completas de los resultados principales. El tercer capítulo contiene una re-elaboración de los fundamentos de la teoría de tensores naturales asociados a las métricas. En particular, se prueba una fórmula original para el cálculo de tensores homogéneos asociados a una métrica y un campo auxiliar. Dicha fórmula, inspirada en un resultado de Stredder, se utiliza sistemáticamente en el resto de capítulos de la tesis. El cuarto y quinto capítulos contienen los resultados fundamentales de la tesis. Se trata de sendas caracterizaciones del tensor de Einstein y del tensor de energía del campo magnético, respectivamente. La principal novedad de dichos resultados es la utilización de un argumento de análisis dimensional para caracterizar dichos tensores. Cabe destacar también que, en el quinto capítulo, se da una interpretación del tensor de energía de una (p+2)-forma como el tensor de energía del campo electromagnético en una teoría de p-branas cargadas, interpretación que es original. En el último capítulo se estudian, de modo sistemático, los tensores con divergencia nula construidos a partir de una métrica, usando segundas derivadas. El resultado más importante que se consigue es una generalización de un célebre teorema debido a Lovelock. CONCLUSIÓN: La principal conclusión, a la vista de los resultados de los capítulos cuatro y cinco, es que las ecuaciones de campo de la relatividad y el electromagnetismo son, en cierto sentido, "las únicas posibles". En concreto, dichas ecuaciones se caracterizan por ser las únicas que poseen ciertas leyes de conservación (divergencia nula) y cuya dependencia de las unidades de escala es la que observamos en la gravitación newtoniana.In this thesis, we study local invariants associated to a pseudo-Riemannian metric; in particular, we aim to describe the spaces of invariant tensors that are divergence-free. Then, we use this theory to prove some results concerning the foundations of the field equations of General Relativity and the Theory of Electromagnetism, that are the most significant theorems of the thesis

On invariant operations of Fedosov structures

In this paper we study invariant local operations that can performed on a Fedosov manifold, with a particular emphasis on tensor-valued operations (also known as natural tensors). Our main result describes the spaces of homogeneous natural tensors as certain finite dimensional linear representations of the symplectic group

Reflexive functors of modules in commutative algebra

Funtores reflexivos de módulos aparecen de forma natural en Geometría algebraica, principalmente en la teoría de las representaciones lineales de los planes de grupo, y en "teorías de dualidad". En este trabajo se estudian determinados funtores reflexivos de los módulos y ofrecemos muchas propiedades de funtores reflexivos de los módulos, de álgebra y de biálgebras.Reflexive functors of modules naturally appear in Algebraic Geometry, mainly in the theory of linear representations of group schemes, and in “duality theories”. In this paper we study and determine reflexive functors of modules and we give many properties of reflexive functors of modules, of algebras and of bialgebras

Affine functors and duality

Un funtor de conjuntos X más de la categoría de álgebras-K conmutativa se dice que es un funtor afín si su funtor de funciones, Aₓ, es reflexiva y X = SpecAₓ. Probamos que funtores afines son igual a un límite directo de los esquemas afines y que los sistemas afines, esquemas formales, la finalización de los planes afines a lo largo de un subesquema cerrado, etc., son funtores afines. Dotar a un funtor afín X con un funtor de estructura monoide equivale a dotar Aₓ con un funtor de estructura biálgebras. Si G es un funtor afín de monoides, entonces A*G es el funtor envolvente de álgebras de G y la categoría de G-módulos es equivalente a la categoría de A*G-módulos. Las aplicaciones de estos resultados incluyen la dualidad Cartier, dualidad Tannakian neutral para los esquemas del grupo afín, la equivalencia entre los grupos formales y álgebras de Lie de característica cero, etc.A functor of sets X over the category of K-commutative algebras is said to be an affine functor if its functor of functions, Aₓ, is reflexive and X = SpecAₓ. We prove that affine functors are equal to a direct limit of affine schemes and that affine schemes, formal schemes, the completion of affine schemes along a closed subscheme, etc., are affine functors. Endowing an affine functor X with a functor of monoids structure is equivalent to endowing Aₓ with a functor of bialgebras structure. If G is an affine functor of monoids, then A∗G is the enveloping functor of algebras of G and the category of G-modules is equivalent to the category of A∗G-modules. Applications of these results include Cartier duality, neutral Tannakian duality for affine group schemes, the equivalence between formal groups and Lie algebras in characteristic zero, etc

Apuntes de Topología

Apuntes destinados a servir de base para la asignatura de Topología (Grado en Matemáticas de la Universidad de Extremadura). El objetivo fundamental de estas notas es que el estudiante asimile la naturaleza topológica de varios conceptos estudiados en asignaturas previas (abierto, cerrado, interior, clausura, límite de una sucesión, aplicación continua, conjuntos conexos, compactos,...) y que los distinga de aquellos que no tienen tal naturaleza (como la acotación, la noción de sucesión de Cauchy o de aplicación uniformemente continua, etc.). Para facilitar la asimilación de estas ideas en el texto hay multitud de ejemplos, amplias baterías de ejercicios y problemas. Cada uno de los temas contiene una colección de veinte ejercicios de autoevaluación, que posibilitan al estudiante comprobar si ha asimilado correctamente los conceptos estudiados. A modo de apéndice, se incluyen las respectivas soluciones.Notes intended to serve as a basis for the subject of Topology (Degree in Mathematics from the University of Extremadura). The fundamental objective of these notes is that the student assimilates the topological nature of various concepts studied in previous subjects (open, closed, interior, closure, limit of a succession, continuous application, related sets, compacts,...) and that they distinguish them from those that do not have such a nature (such as dimensioning, the notion of Cauchy succession or uniformly continuous application, etc.). To facilitate the assimilation of these ideas in the text there are many examples, extensive sets of exercises and problems. Each of the topics contains a collection of twenty self-evaluation exercises, which enable the student to check whether he or she has assimilated the concepts studied correctly. The respective solutions are included as an appendix

Difusión del software libre en la universidad : la experiencia piloto Moodle en la Universidad del País Vasco

La experiencia piloto Moodle UPV/EHU que aquí se presenta, comenzó en 2003 impulsada por un pequeño grupo de profesores apoyados por el centro informático de la universidad (CIDIR). Este grupo ha colaborado de manera desinteresada con los profesores investigadores de la Universidad que han solicitado su ayuda en el uso de Moodle, ofreciéndoles soporte técnico y formación. A partir del curso 2004-2005, este colectivo de profesores ha conformado el Grupo iKide. En este artículo se muestra cómo la filosofía del software libre puede ayudar a impulsar la innovación docente.The UPV-EHU Moodle pilot experience presented in this paper was set up in 2003, driven by a small group of lecturers with the help of the Computer Centre of the UPV/EHU (CIDIR). This group has voluntarily collaborated with those university teachers who were interested in working with Moodle and offered them technical support and training. From the course 2004-2005 onwards, the foremencioned group of lecturers became the iKide Group. This paper shows how Open Source philosophy can help to foster teaching innovation

Morphoagronomic characterization and wholegenome resequencing of eight highly diverse wild and weedy S. pimpinellifolium and S. lycopersicum var. cerasiforme accessions used for the first interespecific tomato MAGIC population

[EN] The wild Solanum pimpinellifolium (SP) and the weedy S. lycopersicum var. cerasiforme (SLC) are largely unexploited genetic reservoirs easily accessible to breeders, as they are fully cross-compatible with cultivated tomato (S. lycopersicum var. lycopersicum). We performed a comprehensive morphological and genomic characterization of four wild SP and four weedy SLC accessions, selected to maximize the range of variation of both taxa. These eight accessions are the founders of the first tomato interspecific multi-parent advanced generation inter-cross (MAGIC) population. The morphoagronomic characterization was carried out with 39 descriptors to assess plant, inflorescence, fruit and agronomic traits, revealing the broad range of diversity captured. Part of the morphological variation observed in SP was likely associated to the adaptation of the accessions to different environments, while in the case of SLC to both human activity and adaptation to the environment. Whole-genome resequencing of the eight accessions revealed over 12 million variants, ranging from 1.2 to 1.9 million variants in SLC and from 3.1 to 4.8 million in SP, being 46.3% of them (4,897,803) private variants. The genetic principal component analysis also confirmed the high diversity of SP and the complex evolutionary history of SLC. This was also reflected in the analysis of the potential footprint of common ancestors or old introgressions identified within and between the two taxa. The functional characterization of the variants revealed a significative enrichment of GO terms related to changes in cell walls that would have been negatively selected during domestication and breeding. The comprehensive morphoagronomic and genetic characterization of these accessions will be of great relevance for the genetic analysis of the first interspecific MAGIC population of tomato and provides valuable knowledge and tools to the tomato community for genetic and genomic studies and for breeding purposes.This work has been funded by the Spanish Ministerio de Economia y Competitividad and the Fondo Europeo de Desarrollo Regional/European Regional Development Fund, Grant AGL2015-71011-R. Authors also thank the G2P-SOL (linking genetic resources, genomes, and phenotypes of solanaceous crops) and BRESOV (breeding for resilient, efficient, and sustainable organic vegetable production) projects for support. G2P-SOL and BRESOV projects have received funding from the European Union's Horizon 2020 research and innovation programme under grant agreements 677379 (G2P-SOL), and 774244 (BRESOV). 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