L’objectif à moyen terme de ce travail est d’explorer quelques formulations des problèmes d’identification de forme et de reconnaissance de surface à partir de mesures ponctuelles. Ces problèmes ont plusieurs applications importantes dans les domaines de l’imagerie médicale, de la biométrie, de la sécurité des accès automatiques et dans l’identification de structures cohérentes lagrangiennes en mécanique des fluides. Par exemple, le problème d’identification des différentes caractéristiques de la main droite ou du visage d’une population à l’autre ou le suivi d’une chirurgie à partir des données générées par un numériseur. L’objectif de ce mémoire est de préparer le terrain en passant en revue les différents outils mathématiques disponibles pour appréhender la géométrie comme variable d’optimisation ou d’identification. Pour l’identification des surfaces, on explore l’utilisation de fonctions distance ou distance orientée, et d’ensembles de niveau comme chez S. Osher et R. Fedkiw ; pour la comparaison de surfaces, on présente les constructions des métriques de Courant par A. M. Micheletti en 1972 et le point de vue de R. Azencott et A. Trouvé en 1995 qui consistent à générer des déformations d’une surface de référence via une famille de difféomorphismes. L’accent est mis sur les fondations mathématiques sous-jacentes que l’on a essayé de clarifier lorsque nécessaire, et, le cas échéant, sur l’exploration d’autres avenues. / The mid-term objective of this work is to explore some formulations of shape identification and surface recognition problems from point measurements. Those problems have important applications in medical imaging, biometrics, security of the automatic access, and in the identification of Lagrangian Coherent Structures in Fluid Mechanics. For instance, the problem of identifying the different characteristics of the right hand or the face from a population to another or the follow-up after surgery from data generated by a scanner. The objective of this mémoire is to prepare the ground by reviewing the different mathematical tools available to apprehend the geometry as an identification or optimization variable. For surface identification it explores the use of distance functions, oriented distance functions, and level sets as in S. Osher and R. Fedkiw ; for surface recognition it emphasizes the construction of Courant metrics by A. M. Micheletti in 1972 and the point of view of R. Azencott and A. Trouvé in 1995 which consists in generating deformations of a reference surface via a family of diffeomorphisms. The accent will be put on the underlying mathematical foundations that it will attempt to clarify as necessary, and, if need be, on exploring new avenues.

In an attempt to prove Amitsur's Conjecture for cyclic subgroups of the Brauer group, we look at rational embeddings of the Severi Brauer variety of an algebra into its norm hypersurface. We enlarge the collection of such embeddings, and generalize them to embeddings of generalized Severi Brauer varieties into determinantal varieties. / text

This  thesis  consists  of two papers  and  a summary.  The  papers  both  deal with  affine algebraic complex  varieties,  and  in particular such  varieties  in dimension  three  that have a non-trivial action  of one of the  one-dimensional  algebraic  groups  Ga   :=  (C, +) and  Gm  :=  (C*, ·).  The methods  used  involve  blowing up  of subvarieties, the correspondances between  Ga - and  Gm - actions  on an affine variety  X with locally nilpotent derivations  and Z-gradings  respectively  on O(X) and passing from a filtered algebra  A to its associated graded  algebra  gr(A). In Paper  I, we study  Russell’s hypersurface  X , i.e. the affine variety  in the affine space A4 given by the equation  x + x2y + z3 + t2 = 0. We reprove by geometric means Makar-Limanov’s result which states  that X is not isomorphic to A3 – a result which was crucial to Koras-Russell’s proof of the linearization conjecture  for Gm -actions on A3. Our method consist in realizing X as an open part  of a blowup M  −→ A3 and to show that each Ga -action on X descends to A3 . This follows from considerations of the graded  algebra  associated to O(X ) with respect  to a certain filtration. In Paper  II, we study  Ga-threefolds X  which have  as their  algebraic  quotient  the  affine plane  A2  = Sp(C[x, y]) and  are a principal  bundle  above the  punctured plane  A2  :=  A2 \ {0}. Equivalently, we study  affine Ga -varieties  Pˆ  that extend  a principal  bundle  P over A2, being P together  with an extra  fiber over the origin in A2. First  the trivial  bundle  is studied,  and some examples of extensions  are given (including  smooth  ones which are not isomorphic  to A2 × A). The  most  basic among  the  non-trivial  principal  bundles  over A2 is SL2 (C)  −→ A2, A  1→  Ae1 where e1  denotes  the first unit  vector,  and we show that any non-trivial  bundle  can be realized as a pullback  of this  bundle  with  respect  to  a morphism  A2  −→ A2. Therefore  the  attention is then  restricted to extensions  of SL2(C)  and  find two families of such extensions  via a study of the  graded  algebras  associated  with  the  coordinate  rings  O(Pˆ)  '→ O(P ) with  respect  to  a filtration  which is defined in terms  of the Ga -actions  on P and Pˆ  respectively.

La principale motivation de cette thèse est de mettre en relation les aspects extrinsèque et intrinsèque des hypersurfaces d'espaces modèles au moyen de résultats de rigidité. Dans un premier temps, nous donnons des résultats de pincment pour des minorations du rayon extrinsèqueen fonction des r-courbures moyennes dans les trois espaces modèles. Nous obtenons ensuite des résultats de pincement comparables pour des majorations de la première valeur propre du laplacien dans l'espace euclidien, ce qui nous permet d'obtenir des résultats concernant les hypersurfaces presque Einstein. Dans un second temps, nous donnons une caractérisation spinorielle des surfaces dans les 3-variétés homogènes à groupe d'isométries de dimension 4.

在這篇論文中，我們從K energy的角度探討緊緻法諾超平面上的K穩定性。首先，我們給K energy一個較明確的型式，接著再透過分析的手法求解其導函數。後續，我們引進熱帶幾何的結構來重新分析主要的結果，最後給一些法諾超平面的實例，驗證我們所得到的公式。 / In this thesis, we analyze K stability on compact Fano hypersurfaces from K energy. We first represent the K energy into an explicitly formula. Then we compute the derivative by using some analytic techniques. Furthermore, we introduce some structures of tropical geometry to analyze the main result. Finally, we give some examples of compact Fano hypersurface to test and verify the formula we get.

Made available in DSpace on 2016-06-02T20:28:25Z (GMT). No. of bitstreams: 1 3184.pdf: 615801 bytes, checksum: 5d8654ee242eff8f78e530be4b12eaf5 (MD5) Previous issue date: 2010-06-24 / Financiadora de Estudos e Projetos / In his retiring Presidential address to the American Mathematical Society in 1971, Zariski proposed some questions in the Theory of Singularities. One of them concerns the topological invariance of the multiplicity of complex hypersurfaces. In more accurate terms, Zariski asked: if two complex hypersurfaces are homeomorphic as embedded varieties, then are their multiplicities at the origin the same? The multiplicity of a complex hypersurface at the origin is the number of points of intersection of the hypersurface with a generic complex line passing close to the origin, but not through it. The problem still remains unsolved. However, there are some special cases which were answered affirmatively, such as the case of homeomorphic hypersurfaces by a bilipschitz homeomorphism. This work aims at understanding the main results settled for the problem. In the present dissertation, we will make a precise concept of multiplicity of a complex hypersurface and we will give special emphasis to C1-invariance of the multiplicity, bilipschitz invariance and quasihomogeneous hypersurfaces. Besides having great importance by themselves, these cases bring their own interpretations of multiplicity helping us to understand better such an object. / Em seu discurso de saída da presidência da Sociedade Americana de Matemática em 1971, Zariski propôs algumas questões na Teoria de Singularidades. Uma delas diz respeito `a invariância topológica da multiplicidade de hipersuperfícies complexas. Em termos mais precisos, Zariski perguntou: se duas hipersuperfícies complexas são homeomorfas como variedades imersas, então suas multiplicidades na origem são as mesmas? A multiplicidade de uma hipersuperfície complexa na origem é o número de pontos de interseção da hipersuperfície com uma reta complexa genérica passando próximo da origem, mas não por ela. O problema permanece ainda sem solução. Entretanto, existem alguns casos especiais que foram respondidos afirmativamente, tais como o caso de hipersuperfícies homeomorfas por um homeomorfismo bilipschitz. Este trabalho tem por objetivo compreender os principais resultados estabelecidos para o problema. Na presente dissertação, faremos um conceito preciso de multiplicidade de uma hipersuperfície complexa e daremos ênfase especial `a C1-invariância da multiplicidade, `a invariância bilipschitz e `as hipersuperfícies quase homogêneas. Além de terem grande importância por si só, estes casos trazem suas próprias interpretações de multiplicidade, ajudando-nos a compreender melhor tal objeto.

Dans ce travail de thèse , je présente comment extraire les géométries discrètes de l'espace-temps de la formulation covariante de la gravitaté quantique à boucles, qui est appelé le formalisme de la mousse de spin. LQG est une théorie quantique de la gravité qui non-perturbativement quantifie la relativité générale indépendante d'un fond fixe. Il prédit que la géométrie de l'espace est quantifiée, dans lequel l'aire et le volume ne peuvent prendre que la valeur discrète. L'espace de Hilbert cinématique est engendré par les fonctions du réseau de spin. L'excitation de la géométrie peut être parfaitement visualisée comme des polyèdres floue qui collées à travers leurs facettes. La mousse de spin définit la dynamique de la LQG par une amplitude de la mousse de spin sur un complexe cellulaire avec un état du réseau de spin comme la frontiére. Cette thèse présente deux résultats principaux. Premièrement, la limite semi-classique de l'amplitude de la mousse de spin sur un complexe simplicial arbitraire avec une frontière est complètement étudiée. La géométrie discrète classique de l'espace-temps est reconstruite et classée par les configurations critiques de l'amplitude de la mousse de spin. Deuxièmement, la fonction de trois-point de LQG est calculé. Il coïncide avec le résultat de la gravité discrète. Troisièmement, la description des géométries discrètes de hypersurfaces nulles est explorée dans le cadre de la LQG. En particulier, la géométrie nulle est décrit par une structure singulière euclidienne sur la surface de type espace à deux dimensions définie par un feuilletage de l'espace-temps par hypersurfaces nulles. / In this thesis, I will present how to extract discrete geometries of space-time fromthe covariant formulation of loop quantum gravity (LQG), which is called the spinfoam formalism. LQG is a quantum theory of gravity that non-perturbative quantizesgeneral relativity independent from a fix background. It predicts that the geometryof space is quantized, in which area and volume can only take discrete value. Thekinematical Hilbert space is spanned by Penrose's spin network functions. The excita-tion of geometry can be neatly visualized as fuzzy polyhedra that glued through theirfacets. The spin foam defines the dynamics of LQG by a spin foam amplitude on acellular complex, bounded by the spin network states. There are three main results inthis thesis. First, the semiclassical limit of the spin foam amplitude on an arbitrarysimplicial cellular complex with boundary is studied completely. The classical discretegeometry of space-time is reconstructed and classified by the critical configurations ofthe spin foam amplitude. Second, the three-point function from LQG is calculated.It coincides with the results from discrete gravity. Third, the description of discretegeometries of null hypersurfaces is explored in the context of LQG. In particular, thenull geometry is described by a Euclidean singular structure on the two-dimensionalspacelike surface defined by a foliation of space-time by null hypersurfaces. Its quan-tization is U(1) spin network states which are embedded nontrivially in the unitaryirreducible representations of the Lorentz group.

Cette thèse est constituée de quatre chapitres. Le premier contient les notions de base qui permettent d'aborder les divers thèmes qui y sont étudiés. Le second est consacré à l'étude des sous-variétés lagrangiennes d'une variété presque kählérienne. J'y présente les résultats obtenus en collaboration avec Burcu Bektas, Joeri Van der Veken et Luc Vrancken. Dans le troisième, je m'intéresse à un problème de géométrie différentielle affine et je donne une classification des hypersphères affines qui sont isotropiques. Ce résultat a été obtenu en collaboration avec Luc Vrancken. Et enfin dans le dernier chapitre, je présente quelques résultats sur les surfaces de translation et les surfaces homothétiques, objet d'un travail en commun avec Rafael López. / Abstract in English not available

This dissertation has as base Jocelino Sato and Vicente de Souza Neto's paper called Complete and Stable O(p + 1) x O(q + 1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space Rp+q+2, published on the Annals of Global Analysis and Geometry - 29 in 2006. The main result of this dissertation is the Classi_cation Theorem, which states: The O(p+1) x O(q+1)-Invariant Hypersurfaces in Rp+q+2, p; q > 1, with zero scalar curvature belong to one of the following classes: (1) Cones with a singularity at the orign of Rp+q+2; (2) Hypersurfaces having one orbit of singularity and asymptoting both of the cones Cα and Cβ; (3) Regular hypersurfaces asymptoting the cone Cα; (4) Regular hypersurfaces asymptoting the cone Cβ; (5) Regular hypersurfaces asymptoting both of the cones Cα and Cβ. It was reached by the studies of the ordinary differential equation on R2, involving the coordenate curves that generate these hypersurfaces. Such differential equation, in its turn, is associated with a vector field X : R22 → R2 on the plan. The study of the orbits space in this field is essential; after all, because of it, it was possible to translate the X orbits' behavior into information concerning the profile curves and, finally, reach the theorem. / Fundação de Amparo a Pesquisa do Estado de Alagoas / Esta dissertação está baseada no artigo de Jocelino Sato e Vicente de Souza Neto intitulado Complete and Stable O(p+1) x O(q+1) - Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space Rp+q+2, publicado na revista Annals of Global Analysis and Geometry, volume 29, em 2006. O principal resultado desta dissertação é o Teorema de Classicação, que afirma o seguinte: Uma hipersuperfície Mp+q+1 que é invariante pela açãoao do grupo O(p + 1) x O(q + 1), p; q > 1, com curvatura escalar identicamente nula deve pertencer a uma das seguintes classes: (1) Cones com uma singularidade na origem de Rp+q+2; (2) Hipersuperfícies possuindo uma órbita de singularidades e assintotando ambos os cones Cα e Cβ; (3) Hipersuperfícies regulares que assintotam o cone Cα; (4) Hipersuperfícies regulares que assintotam o cone Cβ; (5) Hipersuperfícies regulares que assintotam ambos os cones Cα e Cβ. A demonstração do teorema requer um estudo de uma equação diferencial ordinária envolvendo as coordenadas das curvas, no plano, que geram estas hipersuperfícies. Esta equação diferencial, por sua vez, está associada a um campo de vetores X : R2 → R2 no plano. O estudo do retrato de fase deste campo é fundamental. Através dele, foi possível traduzir o comportamento das trajetórias de X em informações com respeito às curvas geratrizes e desta maneira obter o teorema.

Die Frage nach dem Verstäandnis der Topologie symplektischer Mannigfaltigkeiten erhielt immer größere Aufmerksamkeit, insbesondere seit den Arbeiten von A. Weinstein und V. Arnold. Ein bewährtes Mittel ist dabei die Theorie der Gromov-Witten-Invarianten. Eine Gromov-Witten-Invariante zählt Schnitte von rationalen Zyklen mit Modulräumen J-holomorpher Kurven, die eine fixierte Homologieklasse repräsentieren, für eine zahme fast komplexe Struktur. Allerdings ist es im Allgemeinen schwierig, solche Schnittzahlen zu definieren, ohne zusätzliche Annahmen an die symplektische Mannigfaltigkeit zu treffen, da mehrfach überlagerte J-holomorphe Kurven mit negativer Chernzahl vorkommen können. Die vorliegende Dissertation folgt einem alternativen Ansatz zur Definition von Gromov-Witten-Invarianten, der von K. Cieliebak und K. Mohnke eingeführt wurde. Dieser Ansatz liefert für jede fixierte Homologieklasse einen Pseudozykel für jede geschlossene glatte Mannigfaltigkeit mit einer rationalen symplektischen Form. Wir erweitern diesen Ansatz in der vorliegenden Arbeit für eine beliebige symplektische Form. Wie bereits in der ursprünglichen Arbeit betrachten wir nur den Fall holomorpher Sphären. Wir zeigen, dass für jede Klasse der zweiten Koholomogie eine offene Umgebung existiert, so dass für zwei beliebige rationale symplektische Formen, desser Klassen in der gewählten Umgebung liegen, die dazugehörigen Pseudozykel rational kobordant sind. Der Beweis basiert auf einer Modifikation der Argumente des Ansatzes von Cieliebak und Mohnke für den Fall von zwei sich transversal schneidenden Hyperflächen, die jeweils zu verschiedenen symplektischen Formen gehören. / The question of understanding the topology of symplectic manifolds has received great attention since the work of A. Weinstein and V. Arnold. One of the established tools is the theory of Gromov-Witten invariants. A Gromov-Witten invariant counts intersections of rational cycles with the moduli space of J-holomorphic curves representing a fixed class for a tame almost complex structure. However, without imposing additional assumptions on the symplectic manifold such counts are difficult to define in general due to the occurence of multiply covered J-holomorphic curves with negative Chern numbers. This thesis deals with an alternative approach to Gromov-Witten invariants introduced by K. Cieliebak and K. Mohnke. Their approach delivers a pseudocycle for any closed symplectic manfold equipped with a rational symplectic form. Here this approach is extended to the case of an arbitrary symplectic form on a closed symplectic manifold.As in the original work we consider only the case of holomorphic spheres. We show that for any second cohomology class there exists an open neighbourhood, such that for any two rational symplectic forms, whose cohomolgy classes are contained in this neighbourhood, the corresponding pseudocycles are rationally cobordant. The proof is based on an adaptation of the arguments from the original Cieliebak-Mohnke approach to a more general situation - presence of two transversely intersecting hypersurfaces coming from two different symplectic forms.