RESUM: Aquesta tesi s’organitza segons l’esquema per capítols següent: El capítol primer va encaminat a presentar el teorema de descomposició de de Rham-Wu, que estén, al cas semiriemannià, el conegut teorema de descomposició de de Rham. La demostració que donem és de Wu (W.1) i és inspirada en el fet que les transformacions de curvatura i les seves derivados caracteritzen una v a r i e t a t. El començament del capítol segon és un recull de resultats que necessitarem posteriorment. Així s'introdueix la forma de Cartan-Killing (algú pot pensar que ens hem pres una Ilicéncia massa agosarada anomenant-la així), i s'exposen els rudiments sobre isometries infinitesimals. Tot aixó permet de donar una generalització del teorema de Kostant i d'estudiar el comportament de l'ope rador A(X), associat a un camp de Killing X, en varietats de curvatura constant, tant si la constant val zero com si no val zero. En el capítol primer ja comentem que les varietats irreduïbles resulten insuficients en el cas semiriemanniá. Fan falta, a més, varietats que tinguin subespais de l'espai tangent, degenerats, invariants per l'acció del grup d'holonomia. D'aquestes en diem varietats gairebé irreduïbles seguint la notació de Wu. El capítol tercer és dedicat a estudiar d'entre aquestes varietats, aquelles que, a mes, siguin de Lorentz. Cal destacar que aqüestes darreres son proveïdes d'una foliació de dimensió 1 (conseqüentment, també una de codimensió 1) paral.lela, en la direcció d'un camp vectorial de norma zero. El capítol quart és dedicat a estudiar les possibles àlgebres d'holonomia de varietats de dimensió menor o igual que cinc, que siguin de Lorentz i gairebé irreduibles localment. Tot aixó va encaminat a escatir el caracter holónom o no holónom deis camps de Killing sobre aquestes varietats. Tant en el capítol anterior com en aquest, donem exemples de varietats de Lorentz gairebé irreduïbles. Un d'ells és, a mes, una varietat compacta i és proveit d'un camp de Killing no holónom. Finalment, el capítol cinqué conté generalitzacions d'aquests exemples, així com del teorema de Kostant, ja comentada en el capítol segon. També conté aplicacions d'aquests teoremes quan el tensor de Ricci de la varietat satisfà condicions prou bones.

Aquest treball es refereix a foliacions i camps de Killing. Les foliacions, com a part individualitzada dins la Geometria Diferencial, pot considerarse que neixen a partir de la teoria dels sistemes dinàmics en varietats i de la teoria de connexions en fibrats desenvolupada per Ch. Ehresmann i G. Reeb entre 1940 i 1960. Resultats d'aquesta disciplina s'utilitzen en camps com ara sistemes d'equacions diferencials, termodinàmica, teoria del control…

Understanding humans from photographs has always been a fundamental goal of computer vision. Early works focused on simple tasks such as detecting the location of individuals by means of bounding boxes. As the field progressed, harder and more higher level tasks have been undertaken. For example, from human detection came the 2D and 3D human pose estimation in which the task consisted of identifying the location in the image or space of all different body parts, e.g., head, torso, knees, arms, etc. Human attributes also became a great source of interest as they allow recognizing individuals and other properties such as gender or age. Later, the attention turned to the recognition of the action being performed. This, in general, relies on the previous works on pose estimation and attribute classification. Currently, even higher level tasks are being conducted such as predicting the motivations of human behavior or identifying the fashionability of an individual from a photograph. In this thesis we have developed a hierarchy of tools that cover all these range of problems, from low level feature point descriptors to high level fashion-aware conditional random fields models, all with the objective of understanding humans from monocular, RGB images. In order to build these high level models it is paramount to have a battery of robust and reliable low and mid level cues. Along these lines, we have proposed two low-level keypoint descriptors: one based on the theory of the heat diffusion on images, and the other that uses a convolutional neural network to learn discriminative image patch representations. We also introduce distinct low-level generative models for representing human pose: in particular we present a discrete model based on a directed acyclic graph and a continuous model that consists of poses clustered on a Riemannian manifold. As mid level cues we propose two 3D human pose estimation algorithms: one that estimates the 3D pose given a noisy 2D estimation, and an approach that simultaneously estimates both the 2D and 3D pose. Finally, we formulate higher level models built upon low and mid level cues for human understanding. Concretely, we focus on two different tasks in the context of fashion: semantic segmentation of clothing, and predicting the fashionability from images with metadata to ultimately provide fashion advice to the user. In summary, to robustly extract knowledge from images with the presence of humans it is necessary to build high level models that integrate low and mid level cues. In general, using and understanding strong features is critical for obtaining reliable performance. The main contribution of this thesis is in proposing a variety of low, mid and high level algorithms for human-centric images that can be integrated into higher level models for comprehending humans from photographs, as well as tackling novel fashion-oriented problems. / Siempre ha sido una meta fundamental de la visión por computador la comprensión de los seres humanos. Los primeros trabajos se fijaron en objetivos sencillos tales como la detección en imágenes de la posición de los individuos. A medida que la investigación progresó se emprendieron tareas mucho más complejas. Por ejemplo, a partir de la detección de los humanos se pasó a la estimación en dos y tres dimensiones de su postura por lo que la tarea consistía en identificar la localización en la imagen o el espacio de las diferentes partes del cuerpo, por ejemplo cabeza, torso, rodillas, brazos, etc...También los atributos humanos se convirtieron en una gran fuente de interés ya que permiten el reconocimiento de los individuos y de sus propiedades como el género o la edad. Más tarde, la atención se centró en el reconocimiento de la acción realizada. Todos estos trabajos reposan en las investigaciones previas sobre la estimación de las posturas y la clasificación de los atributos. En la actualidad, se llevan a cabo investigaciones de un nivel aún superior sobre cuestiones tales como la predicción de las motivaciones del comportamiento humano o la identificación del tallaje de un individuo a partir de una fotografía. En esta tesis desarrollamos una jerarquía de herramientas que cubre toda esta gama de problemas, desde descriptores de rasgos de bajo nivel a modelos probabilísticos de campos condicionales de alto nivel reconocedores de la moda, todos ellos con el objetivo de mejorar la comprensión de los humanos a partir de imágenes RGB monoculares. Para construir estos modelos de alto nivel es decisivo disponer de una batería de datos robustos y fiables de nivel bajo y medio. En este sentido, proponemos dos descriptores novedosos de bajo nivel: uno se basa en la teoría de la difusión de calor en las imágenes y otro utiliza una red neural convolucional para aprender representaciones discriminativas de trozos de imagen. También introducimos diferentes modelos de bajo nivel generativos para representar la postura humana: en particular presentamos un modelo discreto basado en un gráfico acíclico dirigido y un modelo continuo que consiste en agrupaciones de posturas en una variedad de Riemann. Como señales de nivel medio proponemos dos algoritmos estimadores de la postura humana: uno que estima la postura en tres dimensiones a partir de una estimación imprecisa en el plano de la imagen y otro que estima simultáneamente la postura en dos y tres dimensiones. Finalmente construimos modelos de alto nivel a partir de señales de nivel bajo y medio para la comprensión de la persona a partir de imágenes. En concreto, nos centramos en dos diferentes tareas en el ámbito de la moda: la segmentación semántica del vestido y la predicción del buen ajuste de la prenda a partir de imágenes con meta-datos con la finalidad de aconsejar al usuario sobre moda. En resumen, para extraer conocimiento a partir de imágenes con presencia de seres humanos es preciso construir modelos de alto nivel que integren señales de nivel medio y bajo. En general, el punto crítico para obtener resultados fiables es el empleo y la comprensión de rasgos fuertes. La aportación fundamental de esta tesis es la propuesta de una variedad de algoritmos de nivel bajo, medio y alto para el tratamiento de imágenes centradas en seres humanos que pueden integrarse en modelos de alto nivel, para mejor comprensión de los seres humanos a partir de fotografías, así como abordar problemas planteados por el buen ajuste de las prendas.

Systems of quasilinear partial differential equations of the first order, known as hydrodynamic type systems, are one of the most important classes of nonlinear partial differential equations in the modern theory of integrable systems. They naturally arise in continuum mechanics and in a wide range of applications, both in pure and applied mathematics. Deep connections between the mathematical theory of hydrodynamic type systems with differential geometry, firstly revealed by Riemann in the nineteenth century, have been thoroughly investigated in the eighties by Dubrovin and Novikov. They introduced and studied a class of Poisson structures generated by a flat pseudo-Riemannian metric, called first-order Poisson brackets of hydrodynamic type. Subsequently, these structures have been generalised in a whole variety of different ways: degenerate, non-homogeneous, higher order, multi-dimensional, and non-local. The first part of this thesis is devoted to the classification of such structures in two dimensions, both non-degenerate and degenerate. Complete lists of such structures are provided for a small number of components, as well as partial results in the multi-component non-degenerate case. In the second part of the thesis we deal with deformations of Poisson structures of hydrodynamic type. The deformation theory of Poisson structures is of great interest in the theory of integrable systems, and also plays a key role in the theory of Frobenius manifolds. In particular, we investigate deformations of two classes of structures of hydrodynamic type: degenerate one-dimensional Poisson brackets and non-semisimple bi-Hamiltonian structures associated with Balinskii-Novikov algebras. Complete classification of second-order deformations are presented for two-component structures.

The objective of this dissertation is to develop robust and accurate numerical methods for solving the compressible, non-linear Euler equations of gas dynamics in one and two space dimensions. In theory, solutions of the Euler equations can display various characteristics including shock waves, rarefaction waves and contact discontinuities. To capture these features correctly, highly accurate numerical schemes are designed. In this thesis, two different projects have been studied to show the accuracy and utility of these numerical schemes. Firstly, the compressible, non-linear Euler equations of gas dynamics in one space dimension are considered. Since the non-linear partial differential equations (PDEs) can develop discontinuities (shock waves), the numerical code is designed to obtain stable numerical solutions of the Euler equations in the presence of shocks. Discontinuous solutions are defined in a weak sense, which means that there are many different solutions of the initial value problems of PDEs. To choose the physically relevant solution among the others, the entropy condition was applied to the problem. This condition is then used to derive a bound on the solution in order to satisfy L2-stability. Also, it provides information on how to add an adequate amount of diffusion to smooth the numerical shock waves. Furthermore, numerical solutions are obtained using far-field and no penetration (wall) boundary conditions. Grid interfaces were also included in these numerical computations. Secondly, the two dimensional compressible, non-linear Euler equations are considered. These equations are used to obtain numerical solutions for compressible ow in a shock tube with a 90° circular bend for two channels of different curvatures. The cell centered finite volume numerical scheme is employed to achieve these numerical solutions. The accuracy of this numerical scheme is tested using two different methods. In the first method, manufactured solutions are used to the test the convergence rate of the code. Then, Sod's shock tube test case is implemented into the numerical code to show the correctness of the code in both ow directions. The numerical method is then used to obtain numerical solutions which are compared with experimental data available in the literature. It is found that the numerical solutions are in a good agreement with these experimental results.

Given a sequence of n topological pairs (X_i,A_i) for i=1,...,n, and a simplicial complex K, on n vertices, there is a topological space (X,A)^K by a construction of Buchstaber and Panov. Such spaces are called polyhedral products and they generalize the central notion of the moment-angle complex in toric topology. We study certain classes of polyhedral products from a homotopy theoretic point of view. The boundary of the 2-dimensional n-sided polygon, where n is greater than or equal to 3, may be viewed as a 1-dimensional simplicial complex with n vertices and n faces which we call the n-gon. When K is an n-gon for n at least 5, (D^2,S^1)^K is a hyperbolic space, by a theorem of Debongnie. We show that there is an infinite basis of the rational homotopy of the based loop space of (D^2,S^1)^K represented by iterated Samelson products. When K is an n-gon, for n at least 3, and P^m(p^r) is a mod p^r Moore space with m at least 3 and r at least 1, we show that the order of the elements in the p-primary torsion component in the homotopy groups of (Cone X, X)^K, where X is the loop space of P^m(p^r), is bounded above by p^{r+1}. This result provides new evidence in support of a conjecture of Moore. Moreover, this bound is the best possible and in fact, if a certain conjecture of M.J Barratt is assumed to be true, then this bound is also valid, and is the best possible, when K is an arbitrary simplicial complex.

Kawasaki has calculated the integral homology groups of weighted projective space, and his results imply the existence of a homotopy equivalence between weighted projective space and a CW-complex, with a single cell in each even dimension less than or equal to that of weighted projective space. When the weights satisfy certain divisibility conditions then the associated weighted projective space is actually homeomorphic to such an minimal CW-complex and such decompositions are well-known in these cases. Otherwise this minimal CW-complex is not a weighted projective space. Our aim is to give an explicit CW-structure on any weighted projective space, using an invariant decomposition of complex projective space with respect to the action of a product of finite cyclic groups. The result has many cells, in both odd and even dimensions; nevertheless, we identify it with a subdivision of the minimal decomposition whenever the weights are divisive. We then extend the decomposition to stunted weighted projective space, defined as the quotient of one weighted projective space by another. Finally, we compute the integral homology groups of stunted weighted projective space, identify generators in terms of cellular cycles, and describe cup products in the corresponding cohomology ring.

This work motivates the study of open dynamical systems corresponding to the doubling map. In particular, the dynamical properties of the attractor of the doubling map when a symmetric, centred open interval is removed are studied. Using the arithmetical properties of the binary expansion of the points on the boundary of the removed interval, we study properties such as topological transitivity, the specification property and intrinsic ergodicity. The properties of the function that associates to each hole $(a,b)$ the topological entropy of the attractor of the considered dynamical system are also shown. For these purposes, a subshift corresponding to an element of the lexicographic world is associated to each attractor and the mentioned properties are studied symbolically.

In dieser Arbeit untersuchen wir Varietäten singulärer, quadratischer Hyperflächen, die eine projektive Kurve enthalten, und effektive Divisoren im Modulraum von Kurven, die mittels verschiedener Eigenschaften von quadratischen Hyperflächen definiert werden. In Kapitel 2 berechnen wir die Klasse des effektiven Divisors im Modulraum von Kurven mit Geschlecht g und n markierten Punkten, der als der Ort von solchen markierten Kurven definiert ist, dass das Projektion der kanonischen Abbildung der Kurve von den markierten Punkten auf einer quadratischen Hyperfläche liegt. Mithilfe dieser Klasse zeigen wir, dass die Modulräume mit Geschlecht 16, 17 und 8 markierten Punkten Varietäten von allgemeinem Typ sind. In Kapitel 3 stratifizieren wir den Raum von quadratischen Hyperflächen, die eine projektive Kurve enthalten, mithilfe des Rangs dieser Hyperflächen. Wir zeigen, dass jedes Stratum die erwartete Dimension hat, falls die Kurve ein allgemeines Element des Hilbertschemas ist. Mit Rücksicht auf Rang von quadratischen Hyperflächen, eine ähnliche Konstruktion wie in Kapitel 2 ergibt neue Divisoren im Modulraum von Kurven. Wir berechnen die Klasse von diesen Divisoren und zeigen, dass der Modulraum von Kurven mit Geschlecht 15 und 9 markierten Punkten eine Varietät von allgemeinem Typ ist. In Kapitel 4 präsentieren wir unterschiedliche Resultate, die mit Themen von vorigen Kapiteln im Zusammenhang stehen. Zum Ersten berechnen wir die Klasse von Divisoren im Modulraum von Kurven, die als die Orte von Kurven definiert sind, wo die maximale Rang Vermutung nicht gilt. Zweitens zeigen wir, dass jedes Geradenbündel als Tensorprodukt von zwei Geradenbündeln mit zwei Schnitten geschrieben werden kann, falls die Kurve allgemein ist und eine gewisse numerische Bedingung erfüllt ist. Zuletzt benutzen wir bekannte Divisorklassen zu zeigen, dass der Modulraum von Kurven mit Geschlecht 12 und 10 markierten Punkten eine Varietät von allgemeinem Typ ist. / In this thesis, we study varieties of singular quadrics containing a projective curve and effective divisors in the moduli space of pointed curves defined via various constructions involving quadric hypersurfaces. In Chapter 2, we compute the class of the effective divisor in the moduli space of n-pointed genus g curves, which is defined as the locus of pointed curves such that the projection of the canonical model of the curve from the marked points lies on a quadric hypersurface. Using this class, we show that the moduli spaces of 8-pointed genus 16 and 17 curves are varieties of general type. In Chapter 3, we stratify the space of quadrics that contain a given curve in the projective space, using the ranks of the quadrics. We show, in a certain numerical range, that each stratum has the expected dimension if the curve is general in its Hilbert scheme. By incorporating the datum of the rank of quadrics, a similar construction as the one in Chapter 2 yields new divisors in the moduli space of pointed curves. We compute the class of these divisors and show that the moduli space of 9-pointed genus 15 curves is a variety of general type. In Chapter 4, we present miscellaneous results, which are related with our main work in the previous chapters. Firstly, we consider divisors in the moduli space of genus g curves, which are defined as the failure locus of maximal rank conjecture for hypersurfaces of degree greater than two. We illustrate three examples of such divisors and compute their classes. Secondly, using the classical correspondence between rank 4 quadrics and pencils on curves, we show that the map that associates to a pair of pencils their tensor product in the Picard variety is surjective, when the curve is general and obvious numerical assumptions are satisfied. Finally, we use divisor classes, that are already known in the literature, to show that the moduli space of 10-pointed genus 12 curves is a variety of general type.

Cette thèse porte sur le développement d’un dispositif laser à 1,54 µm, triplé en fréquence et stabilisé sur une transition hyperfine de l’iode moléculaire au voisinage de 514 nm.Une partie importante de ce travail est consacrée au triplage de la fréquence d’une diode laser à 1,54 µm, en utilisant deux cristaux non linéaires de Niobate de lithium en structure guide d’onde (PPLN), fibrés. Une efficacité de conversion non linéaire P3w/Pw > 36 % a été obtenue, constituant le meilleur rendement jamais démontré pour un processus de triplage de fréquence en mode continu. Une puissance harmonique de 300 mW a été ainsi générée à 514 nm, à partir d’une puissance fondamentale de 800 mW à 1,54 µm. Le banc optique est totalement fibré, et la puissance électrique totale consommée, nécessaire pour réaliser le triplage de fréquence, n’est que de 20 W. Selon un mode opératoire spécifique, ce dispositif laser permet de fournir simultanément trois radiations intenses, stabilisées en fréquence, à 1.54 µm, 771 nm et 514 nm.Suite à ce développement, un banc de spectroscopie laser très compact a été mis en place, basé sur une courte cellule en quartz scellée, contenant une vapeur d’iode moléculaire. Une puissance optique < 10 mW dans le vert est suffisante pour détecter les transitions hyperfines de l’iode, de grand facteur de qualité au voisinage de 514 nm (Q > 2x109).Une stabilité de fréquence de 4,5 x 10-14 τ-1/2 avec un minimum de 6 x 10-15 de 50 s à 100 s a été démontrée dans le cadre de cette étude. Cette stabilité de fréquence constitue la meilleure performance jamais conférée à une source laser à 1,5 µm à l’aide d’une vapeur atomique, en utilisant une technique simple d’interrogation sub-Doppler.Cette étude a permis d’identifier les points clés permettant de mettre en place dans le futur proche, un dispositif laser stabilisé, totalement fibré, d’un volume < 10 litres.Ce développement pourrait répondre aux besoins de nombreux projets spatiaux nécessitant des liens optiques ultrastables en fréquence, inter-satellites ou bord-sol, pour la géodésie spatiale (GRICE), la mesure du champ gravitationnel terrestre (GRACE FO, NGGM), la détection d’ondes gravitationnelles (LISA), etc. … / This thesis describes the frequency stabilization of a 1.54 µm laser diode on an iodine hyperfine line at 514 nm, after a frequency tripling process.An important part of this work is dedicated to the development of the frequency tripling process of a 1.54 µm laser diode, using two periodically polled wave guided Lithium Niobate nonlinear crystals. A nonlinear conversion efficiency P3w/Pw > 36 % is obtained. This result is the best efficiency ever demonstrated for a CW frequency tripling process. 300 mW of harmonic power is generated at 514 nm from a fundamental optical power of 800 mW at 1.54 µm. The optical setup is fully fibered. The total power consumption of this frequency tripling process is 20 W only. Using a specific operation mode, this laser setup emits simultaneously three frequency-stabilized and intense radiations at 1.54 µm, 771 nm and 514 nm.Following this development, a very compact laser spectroscopy setup was built, based on a short sealed quartz cell, which contains the molecular iodine vapor. An optical power lower than 10 mW in the green is sufficient to carry out the iodine vapor interrogation, and to detect the hyperfine saturation transitions, which have a high quality factor around 514 nm (Q > 2x109).A frequency stability at the level of 4.5 x 10-14 τ-1/2 with a minimum value of 6 x 10-15 from 50 s to 100 s is demonstrated in this study. This frequency stability is the best result ever conferred to a laser diode at 1.54 µm, using in a simple way a Doppler-free iodine spectroscopy technique.This work has allowed to identify the major key components, in order to develop in the near future, a fully fibered and compact stabilized laser prototype occupying a total optical volume < 10 liters.Such a laser source could cover the needs of numerous space projects that require ultra-stable frequency optical links, inter-satellite or ground to space, for space geodesy (GRICE), Earth gravitational field measurement (GRACE-FO, NGGM), gravitational waves detection (LISA) , etc. …