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The primary aim of this thesis is to construct explicit examples of four dimensional hyperbolic link complements. Using the theory of Kirby diagrams and Kirby calculus we set up a general framework that one can use to attack such a problem. We use this framework to construct explicit examples in a smooth standard S⁴ and a smooth standard S² x S². We then characterise which homeomorphism types of smooth simply connected closed 4-manifolds can admit a hyperbolic link complement, along the way giving constructions of explicit examples.

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La diffusion rapide du taoïsme dans le monde chinois dont il est issu, mais aussi bien au-delà, pose la question du modèle économique qui permet à ces organisations religieuses de se développer. Cette thèse étudie les aspects économiques de ce taoïsme contemporain à travers l'analyse des instances économiques élémentaires et leurs interactions entre elles et avec le monde laïc. Les résultats dessinent une économie du taoïsme monastique contemporain fondée sur un système de lignées de pseudo-parenté religieuse où les maîtres se transmettent une approche spécialisée de leur rôle d’échange entre les mondes humains et divins. Ces lignées s'inscrivent dans un ordre organisé pour favoriser la mobilité et le tissage de liens entre lignées, qui s’opposent autour de leurs activités religieuses spécialisées. Les apports de ressources laïcs qui financent les temples et lignées, dépendent de l'adéquation de la réponse des maitres aux besoins des laïcs, à travers leurs services et enseignements. / The rapid growth of Daoism in the Chinese world and beyond questions the economic model at play in these religious organisations. This thesis analyses the economic aspects of this contemporary monastic daoism through the study of basic economic elements, their interactions with each other and with the laity. Results point to an economy based on a religious lineage system, where is transmitted a specialised approach to the master's role of medium between worlds. Those lineages are affiliated to an order organised to promote mobility and networking between them, which oppose each other on their specialised religious activity. Lay ressources on which temples and lignages are funded, depend on the adequacy of the masters’ response to the laity's needs through their services and teachings.

We propose a new type of state sum model for two-dimensional surfaces that takes into account topology and spin. The definition used - new to the literature - provides a rich class of extended models called spin models. Both examples and general properties are studied. Most prominently, we find this type of model can depend on a surface spin structure through parity alone and we explore explicit cases that feature this behaviour. Further directions for the two dimensional world are analysed: we introduce a source of new information - defects - and show how they can enlarge the class of spin models available.

En aquesta tesi interdisciplinar desenvolupem eines algebraiques per a problemes en filogenètica i genòmica. Per estudiar l'evolució molecular de les espècies sovint s'usen models evolutius estocàstics. L'evolució es representa en un arbre (anomenat filogenètic) on les espècies actuals corresponen a fulles de l'arbre i els nodes interiors corresponen a ancestres comuns a elles. La longitud d'una branca de l'arbre representa la quantitat de mutacions que han ocorregut entre les dues espècies adjacents a la branca. Llavors l'evolució de seqüències d'ADN en aquestes espècies es modelitza amb un procés Markov ocult al llarg de l'arbre. Si el procés de Markov se suposa a temps continu, normalment s'assumeix que també és homogeni i, en tal cas, els paràmetres del model són les entrades d'una raó de mutació instantània i les longituds de les branques. Si el procés de Markov és a temps discret, llavors els paràmetres del model són les probabilitats condicionades de substitució de nucleòtids al llarg de l'arbre i no hi ha cap hipòtesi d'homogeneïtat. Aquests últims són els tipus de models que considerem en aquesta tesi i són, per tant, més generals que els de temps continu. Des d'aquesta perspectiva s'estudien els problemes més bàsics de la filogenètica: donat un conjunt de seqüències d'ADN, com decidim quin és el model evolutiu més adequat? com inferim de forma eficient els paràmetres del model? I fins i tot, tal i com també hem provat en aquesta tesi, és possible que les espècies no hagin evolucionat seguint un sol arbre sinó una mescla d'arbres i llavors cal abordar aquestes preguntes en aquest cas més general. Per a models evolutius a temps continu i homogenis, s'ha proposat solucions diverses a aquestes preguntes al llarg de les últimes dècades. En aquesta tesi resolem aquests dos problemes per a models evolutius a temps discret usant tècniques algebraiques provinents d'àlgebra lineal, teoria de grups, geometria algebraica i estadística algebraica. A més a més, la nostra solució per al primer problema és vàlida també per a mescles filogenètiques. Hem fet tests dels mètodes proposats en aquesta tesi sobre dades simulades i dades reals del projectes ENCODE (Encyclopedia Of DNA Elements). Per tal de provar els nostres mètodes hem donat algoritmes per a generar seqüències evolucionant sota un model a temps discret amb un nombre esperat de mutacions prefixat. I així mateix, hem demostrat que aquests algorismes generen totes les seqüències possibles (per la majoria de models). Els tests sobre dades simulades mostren que els mètodes proposats són molt acurats i els resultats sobre dades reals permeten corroborar hipòtesis prèviament formulades. Tots els mètodes proposats en aquesta tesi han estat implementats per a un nombre arbitrari d'espècies i estan disponibles públicament. / In this thesis we develop interdisciplinary algebraic tools for genomic and phylogenetic problems. To study the molecular evolution of species one often uses stochastic evolutionary models. The evolution is represented in a tree (called phylogenetic tree) whose leaves represent current species and whose internal nodes correspond to their common ancestors. The length of a branch of the tree represents the number of mutations that have occurred between the two species adjacent to the branch. Then ,the evolution of DNA sequences in these species is modeled with a hidden Markov process along the tree. If the Markov process is assumed to be continuous in time, it is usually assumed homogeneous as well and, if so, the model parameters are the instantaneous rate of mutation and the lengths of the branches. If the Markov process is discrete in time, then the model parameters are the conditional probabilities of nucleotide substitution along the tree and there is no assumption of homogeneity. The latter are the types of models we consider in this thesis and are therefore more general than the homogeneous continuous ones. From this perspective we study the basic problems of phylogenetics: Given a set of DNA sequences, what is the evolutionary model that best fits the data? how can we efficiently infer the model parameters? Also, as we also checked in this thesis, it is possible that species have not evolved along a single tree but a mixture of trees so that we need to address these questions in this more general case. For continuous-time, homogeneous, evolutionary models, several solutions to these questions have been proposed during the last decades. In this thesis we solve these two problems for discrete-time evolutionary models, using algebraic techniques from linear algebra, group theory, algebraic geometry and algebraic statistics. In addition, our solution to the first problem is also valid for phylogenetic mixtures. We have made tests of the methods proposed in this thesis on simulated and real data from ENCODE Project (Encyclopedia Of DNA Elements). To test our methods, we also provide algorithms to generate sequences evolving under discrete-time models with a given expected number of mutations. Even more, we have proved that these algorithms generate all possible sequences (for most models). Tests on simulated data show that the methods are very accurate and our results on real data confirm hypotheses previously formulated. All the methods in this thesis have been implemented for an arbitrary number of species and are publicly available.

Nowadays, the concept of homogeneity is one of the fundamental notions in geometry although its meaning must be always specified for the concrete situations. In this thesis, we consider the homogeneity of Riemannian manifolds and the homogeneity of manifolds equipped with affine connections. The first kind of homogeneity means that, for every smooth Riemannian manifold (M, g), its group of isometries I(M) is acting transitively on M. Part I of this thesis fits into this philosophy. Afterwards in Part II, we treat the homogeneity concept of affine connections. This homogeneity means that, for every two points of a manifold, there is an affine diffeomorphism which sends one point into another. In particular, we consider a local version of the homogeneity, that is, we accept that the affine diffeomorphisms are given only locally, i.e., from a neighborhood onto a neighborhood. More specifically, we devote the first Chapter of Part I to make a brief overview of some special kinds of homogeneous Riemannian manifolds which will be of special relevance in Part I and to show how the software MATHEMATICA© becomes useful. For that, we prove that "the three-parameter families of flag manifolds constructed by N. R. Wallach in "Compact homogeneous Riemannian manifols with strictly positive curvature, Ann. of Math. 96 (1972), p. 276-293" are D'Atri spaces if and only if they are naturally reductive spaces. Thus, we improve the previous results given by D'Atri, Nickerson and by Arias-Marco, Naveira.Moreover, in Chapter 2 we obtain the complete 4-dimensional classification of homogeneous spaces of type A. This allows us to prove correctly that every 4-dimensional homogeneous D'Atri space is naturally reductive. Therefore, we correct, complete and improve the results presented by Podestà, Spiro, Bueken and Vanhecke. Chapter 3 is devoted to prove that the curvature operator has constant osculating rank over g.o. spaces. It is mean that a real number 'r' exists such that under some assumptions, the higher order derivatives of the curvature operator from 1 to r are linear independent and from 1 to r + 1 are linear dependent. As a consequence, we also present a method valid on every g.o. space to solve the Jacobi equation. This method extends the method given by Naveira and Tarrío for naturally reductive spaces. In particular, we prove that the Jacobi operator on Kaplan's example (the first known g.o. space that it is not naturally reductive) has constant osculating rank 4. Moreover, we solve the Jacobi equation along a geodesic on Kaplan's example using the new method and the well-known method used by Chavel, Ziller and Berndt,Tricerri, Vanhecke. Therefore, we are able to present the main differences between both methods.In Part II, we classify (locally) all locally homogeneous affine connections with arbitrary torsion on two-dimensional manifolds. Therefore, we generalize the result given by Opozda for torsion-less case. Moreover, from our computations we obtain interesting consequences as the relation between the classifications given for the torsion less-case by Kowalski, Opozda and Vlá ek. In addition, we obtain interesting consequences about flat connections with torsion.In general, the study of these problems sometimes requires a big number of straightforward symbolic computations. In such cases, it is a quite difficult task and a lot of time consuming, try to make correctly this kind of computations by hand. Thus, we try to organize our computations in (possibly) most systematic way so that the whole procedure is not excessively long. Also, because these topics are an ideal subject for a computer-aided research, we are using the software MATHEMATICA©, throughout this work. But we put stress on the full transparency of this procedure. / En esta tesis, se consideran dos tipos bien diferenciados de homogeneidad: la de las variedades riemannianas y la de las variedades afines. El primer tipo de homogeneidad se define como aquel que tiene la propiedad de que el grupo de isometrías actúa transitivamente sobre la variedad. La Parte I, recoge todos los resultados que hemos obtenido en esta dirección. Sin embargo, en la Parte II se presentan los resultados obtenidos sobre conexiones afines homogéneas. Una conexión afín se dice homogénea si para cada par de puntos de la variedad existe un difeomorfismo afín que envía un punto en otro. En este caso, se considera una versión local de homogeneidad. Más específicamente, la Parte I de esta tesis está dedicada a probar que "las familias 3-paramétricas de variedades bandera construidas por Wallach son espacios de D'Atri si y sólo si son espacios naturalmente reductivos". Más aún, en el segundo Capítulo, se obtiene la clasificación completa de los espacios homogéneos de tipo A cuatro dimensionales que permite probar correctamente que todo espacio de D'Atri homogéneo de dimensión cuatro es naturalmente reductivo.Finalmente, en el tercer Capítulo se prueba que en cualquier g.o. espacio el operador curvatura tiene rango osculador constante y, como consecuencia, se presenta un método para resolver la ecuación de Jacobi sobre cualquier g.o. espacio. La Parte II se destina a clasificar (localmente) todas las conexiones afines localmente homogéneas con torsión arbitraria sobre variedades 2-dimensionales. Para finalizar el cuarto Capítulo, se prueban algunos resultados interesantes sobre conexiones llanas con torsión.En general, el estudio de estos problemas requiere a veces, un gran número de cálculos simbólicos aunque sencillos. En dichas ocasiones, realizarlos correctamente a mano es una tarea ardua que requiere mucho tiempo. Por ello, se intenta organizar todos estos cálculos de la manera más sistemática posible de forma que el procedimiento no resulte excesivamente largo. Este tipo de investigación es ideal para utilizar la ayuda del ordenador; así, cuando resulta conveniente, utilizamos la ayuda del software MATHEMATICA para desarrollar con total transparencia el método de resolución que más se adecua a cada uno de los problemas a resolver.

In this thesis we answer questions in two related areas of combinatorics: Ramsey theory and asymptotic enumeration. In Ramsey theory we introduce a new method for finding desired structures. We find a new upper bound on the Ramsey number of a path against a kth power of a path. Using our new method and this result we obtain a new upper bound on the Ramsey number of the kth power of a long cycle. As a corollary we show that, while graphs on n vertices with maximum degree k may in general have Ramsey numbers as large as ckn, if the stronger restriction that the bandwidth should be at most k is given, then the Ramsey numbers are bounded by the much smaller value. We go on to attack an old conjecture of Lehel: by using our new method we can improve on a result of Luczak, Rodl and Szemeredi [60]. Our new method replaces their use of the Regularity Lemma, and allows us to prove that for any n > 218000, whenever the edges of the complete graph on n vertices are two-coloured there exist disjoint monochromatic cycles covering all n vertices. In asymptotic enumeration we examine first the class of bipartite graphs with some forbidden induced subgraph H. We obtain some results for every H, with special focus on the cases where the growth speed of the class is factorial, and make some comments on a connection to clique-width. We then move on to a detailed discussion of 2-SAT functions. We find the correct asymptotic formula for the number of 2-SAT functions on n variables (an improvement on a result of Bollob´as, Brightwell and Leader [13], who found the dominant term in the exponent), the first error term for this formula, and some bounds on smaller error terms. Finally we obtain various expected values in the uniform model of random 2-SAT functions.

Based on the study and analysis of existing buildings, and using three-dimensional modeling software, we can understand and thus explain the main mechanisms for historic buildings design and construction, making them understandable to non-specialist audiences. Knowledge of CAD softwares devices allow us a deeper study of the development of mechanisms that describe the building before constructing it or when we have to rebuild them into their original state if they have been destroyed or partially sunk. The thesis explores these resources from various examples from Antoni Gaudí works, Catalonia's most influent architect, with special emphasis on the geometry and the form that defines it. The level of expertise achieved and the constant process of training gives us the capacity to transmit (to future generations) resources in order to achieve with judgment and utmost accurary the necessary documentation to be good professionals in the field of design,engineering and architecture. / A partir de l'estudi i anàlisi d'edificis existents ,i utilitzant el modelat tridimensional hom pot entendre i per tant explicar, els principals mecanismes de disseny i construcció d'edificis històrics fent-los comprensibles a públics no especialitzats. El coneixement de les eines que ens ofereixen avui els softwares de CAD permeten aprofundir molt en el desenvolupament de mecanismes descriptius de l'edifici, abans de construir-los o tornant-los al seu estat originari si han sigut enfonsats o parcialment enfonsats. La tesi aprofundeix en aquests recursos a partir de diversos exemples, obres d'Antoni Gaudí, arquitecte referent a Catalunya, incidint especialment en la geometria i en la forma que els defineix. El grau d'expertesa assolit i el constant procés de formació ens doten de capacitat per transmetre (a futures generacions) recursos per tal d'assolir amb criteri i rigor la documentació necessària pera ser bons professionals en el camp del disseny, l'enginyeria i l'arquitectura.

La Tesis Doctoral está dedicada, por un lado, al estudio de desigualdades de tipo Brunn-Minkowski, especialmente cuando se trabaja con hipótesis sobre proyecciones/secciones, y, por otro lado, al estudio de las raíces de polinomios geométricos que surgen de una generalización del denominado funcional de Wills. En medio, nos encontraríamos las salchichas, las cuales resultan ser, salvo cuerpos convexos degenerados, la familia de los ‘conjuntos extremales’ en relación a algunas mejoras lineales de desigualdades tales como la desigualdad de Brunn-Minkowski o la primera desigualdad de Minkowski (y por tanto también de las desigualdades isoperimétrica y de Uryshon). Además esta familia de cuerpos convexos está ampliamente relacionada con algunos problemas relativos al polinomio de Steiner. Comenzamos estableciendo las nociones básicas que se necesitarán en un desarrollo posterior. A continuación, estudiamos mejoras de la desigualdad de Brunn-Minkowski, en el sentido de ‘refinar’ el exponente 1/n, cuando se asume que los cuerpos comparten una proyección común sobre un (n-k)-plano, por un lado, y para familias de cuerpos particulares, por el otro. En el tercer capítulo, abordamos el caso de igualdad en la versión lineal de la desigualdad de Brunn-Minkowski; nuestro enfoque subyace en (el estudio de) una posible caracterización de la linealidad del volumen a través de salchichas. En el último capítulo, investigamos las raíces de una familia de polinomios geométricos de cuerpos convexos asociados a una medida dada en la semirrecta real no-negativa, que surgen de una generalización natural del funcional de Wills. / The Doctoral Dissertation is devoted, on the one hand, to the study of Brunn-Minkowski's type inequalities, especially when working with projections/sections assumptions, and, on the other hand, to the study of the roots of geometric polynomials which arise from a generalization of the so-called Wills functional. In the middle, we would find sausages, which turn out to be, up to degenerated convex bodies, the family of ‘extremal sets’ in relation to some linear improvements of inequalities such as Brunn-Minkowski's inequality or Minkowski's first inequality (and thus also the isoperimetric and Urysohn's inequalities). Furthermore, this family of convex bodies is strongly connected to some problems relative to the Steiner polynomial. We start establishing the basic notions that will be needed further on. Next, we study refinements of the Brunn-Minkowski inequality, in the sense of ‘enhancing’ the exponent 1/n, when assuming that the bodies share a common projection onto an (n-k)-plane on the one hand, and for particular families of bodies on the other hand. In the third chapter, we deal with the equality case in the linear version of Brunn-Minkowski’s inequality; our approach relies on (the study of) a possible characterization of the linearity of the volume through sausages. In the last chapter, we investigate the roots of a family of geometric polynomials of convex bodies associated to a given measure on the non-negative real line, which arise from a natural generalization of the Wills functional.