Nesta tese apresentamos uma contribuição para o estudo da transição do retrato de fase de uma equação diferencial descontínua específica ao longo de uma linha de descontinuidade. A equação diferencial que tratamos neste trabalho é a das linhas de curvatura principal de uma superfície S contendo uma curva distinguida B e imersa em R^3. A linha de descontinuidade é a curva B, a qual é o bordo comum de duas superfícies suaves justapostas que formam S. Na primeira parte do trabalho consideramos a superfície seccionalmente suave, S = S+ U B U S-, obtida pela justaposição de S+ e S- ao longo do bordo comum B. O estudo da configuração principal de S nos casos em que as linhas de curvatura principal das superfícies S+ e S- tem contato quadrático ou cruzam transversalmente B foi feito por comparação com a configuração principal de uma superfície suave, obtida de S pelo processo da \"regularização\" ao longo da curva de descontinuidade B. Na segunda parte do trabalho estudamos as linhas de curvatura principal de uma superfície S em R^3 com bordo B e da superfície suave obtida de S através dos processos de engrossamento e regularização definidos por Garcia e Sotomayor em [5], onde os autores consideraram o caso genérico, sem pontos umbílicos e contato quadrático de uma linha de curvatura principal com B. Damos aqui continuidade ao estudo feito em [5] analisando o caso de contato cúbico com o bordo B. Obtivemos que dos pontos da curva bordo comum B de contato quadrático e de cruzamento transversal emergem, sobre a superfície regularizada, pontos umbílicos Darbouxianos dos tipos D1 e D3, enquanto que, para o ponto sobre B de contato cúbico obtivemos pontos umbílicos Darbouxianos dos tipos D1, D2 e D3 e também pontos umbílicos não Darbouxianos dos tipos D12 e D23. [5] Garcia, R., and Sotomayor, J. Umbilic and tangential singularities on configurations of principal curvature lines. Anais da Academia Brasileira de Ciências 74, 1 (2002), 117. / In this work we present a contribution to the study of the transition of the phase portrait of a specific discontinuous differential equation along a line of discontinuity. The differential equations under consideration will be that of the principal curvature lines of a surface S with a distinguished curve B immersed in R^3, where the line of discontinuity is the curve B which is the common border of two smooth surfaces attached to make up S. In the first part of the work we consider a piecewise smooth surface S = S+ U B U S-, obtained by the juxtaposition of two smooth surfaces S+ and S- along their common border B. The analysis of the principal configuration of S in the cases where the principal curvature lines of the surfaces S+ and S- have quadratic contact or cross transversally B was carried out by comparison with a smooth surface, obtained from S by the \"regularization\" along the discontinuity curve B. In the second part of the work we study the principal curvature lines of a surface S in R^3 with boundary B and of the smooth surface obtained from S by thickening and smoothing introduced by Garcia and Sotomayor in [5], where they considered the generic case of no umbilic points and at most quadratic contact of principal lines with B. Here we pursue the study in [5] and analyze the case of cubic contact with the border B. We established that while from quadratic contact points with B emerge on the smoothed surface Darbouxian umbilics of D1 and D3 types, from the cubic contact points appear Darbouxian umbilics of types D1, D2 and D3 as well as non Darbouxian points of types D12 and D23. [5] Garcia, R., and Sotomayor, J. Umbilic and tangential singularities on configurations of principal curvature lines. Anais da Academia Brasileira de Ciências 74, 1 (2002), 117.

Nesta tese estudamos a geometria extrínseca de superfícies suave em R4 via seu contato com retas e hiperplanos. Uribe-Vargas introduziu um cr-invariante (crossratio) em uma cúspide de Gauss de uma superfície em R3. Para uma superfície em R4, o ponto P3(c) tem comportamento similar a uma cúspide de Gauss de uma superfície em R3. Estabelecemos nesta tese cross-ratio invariantes para superfícies em R4 de uma maneira análoga ao trabalho de Uribe-Vargas para superfícies em R3. Estudamos os lugares geométricos das singularidades locais e multi-locais das projeções ortogonais da superfície e classificamos os k-jatos de parametrizações de germes de superfícies no espaço projetivo P4 dadas na forma de Monge por mudanças projetivas. Os cross-ratio invariantes nos pontos P3(c) são usadas para recuperar os dois módulos no 4-jato da parametrização projetiva da superfície. / In this thesis we study the extrinsic geometry of smooth surfaces in R4 via their contact with lines and hyperplanes. Uribe-Vargas introduced a cr-invariant (crossratio) at a cusp of Gauss of a surface in R3. For a surface in R4, the point P3(c) has similar behavior to that of a cusp of Gauss of a surface in R3. We establish in this thesis cross-ratio invariants for surfaces in R4 in an analogous way to Uribe- Vargass work for surfaces in R3. We study the geometric locii of local and multilocal singularities of ortogonal projections of the surface and classify the k-jets of parametrizations of germs of surfaces in the projection space P4 given in Monge form by projective transformations. The cross-ratio invariants at P3(c) points are used to recover two moduli in the 4-jet of the projective parametrization of the surfaces.

Nesta tese apresentamos uma contribuição para o estudo da transição do retrato de fase de uma equação diferencial descontínua específica ao longo de uma linha de descontinuidade. A equação diferencial que tratamos neste trabalho é a das linhas de curvatura principal de uma superfície S contendo uma curva distinguida B e imersa em R^3. A linha de descontinuidade é a curva B, a qual é o bordo comum de duas superfícies suaves justapostas que formam S. Na primeira parte do trabalho consideramos a superfície seccionalmente suave, S = S+ U B U S-, obtida pela justaposição de S+ e S- ao longo do bordo comum B. O estudo da configuração principal de S nos casos em que as linhas de curvatura principal das superfícies S+ e S- tem contato quadrático ou cruzam transversalmente B foi feito por comparação com a configuração principal de uma superfície suave, obtida de S pelo processo da \"regularização\" ao longo da curva de descontinuidade B. Na segunda parte do trabalho estudamos as linhas de curvatura principal de uma superfície S em R^3 com bordo B e da superfície suave obtida de S através dos processos de engrossamento e regularização definidos por Garcia e Sotomayor em [5], onde os autores consideraram o caso genérico, sem pontos umbílicos e contato quadrático de uma linha de curvatura principal com B. Damos aqui continuidade ao estudo feito em [5] analisando o caso de contato cúbico com o bordo B. Obtivemos que dos pontos da curva bordo comum B de contato quadrático e de cruzamento transversal emergem, sobre a superfície regularizada, pontos umbílicos Darbouxianos dos tipos D1 e D3, enquanto que, para o ponto sobre B de contato cúbico obtivemos pontos umbílicos Darbouxianos dos tipos D1, D2 e D3 e também pontos umbílicos não Darbouxianos dos tipos D12 e D23. [5] Garcia, R., and Sotomayor, J. Umbilic and tangential singularities on configurations of principal curvature lines. Anais da Academia Brasileira de Ciências 74, 1 (2002), 117. / In this work we present a contribution to the study of the transition of the phase portrait of a specific discontinuous differential equation along a line of discontinuity. The differential equations under consideration will be that of the principal curvature lines of a surface S with a distinguished curve B immersed in R^3, where the line of discontinuity is the curve B which is the common border of two smooth surfaces attached to make up S. In the first part of the work we consider a piecewise smooth surface S = S+ U B U S-, obtained by the juxtaposition of two smooth surfaces S+ and S- along their common border B. The analysis of the principal configuration of S in the cases where the principal curvature lines of the surfaces S+ and S- have quadratic contact or cross transversally B was carried out by comparison with a smooth surface, obtained from S by the \"regularization\" along the discontinuity curve B. In the second part of the work we study the principal curvature lines of a surface S in R^3 with boundary B and of the smooth surface obtained from S by thickening and smoothing introduced by Garcia and Sotomayor in [5], where they considered the generic case of no umbilic points and at most quadratic contact of principal lines with B. Here we pursue the study in [5] and analyze the case of cubic contact with the border B. We established that while from quadratic contact points with B emerge on the smoothed surface Darbouxian umbilics of D1 and D3 types, from the cubic contact points appear Darbouxian umbilics of types D1, D2 and D3 as well as non Darbouxian points of types D12 and D23. [5] Garcia, R., and Sotomayor, J. Umbilic and tangential singularities on configurations of principal curvature lines. Anais da Academia Brasileira de Ciências 74, 1 (2002), 117.

Tong, Pun Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 77-78). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Basic Terminologies --- p.8 / Chapter 3 --- Causality in space-time --- p.12 / Chapter 3.1 --- Preliminaries in space-time --- p.12 / Chapter 3.2 --- Global causality condition --- p.14 / Chapter 3.3 --- Domains of Dependence --- p.23 / Chapter 4 --- Conjugate Points --- p.29 / Chapter 4.1 --- Space of causal curves --- p.29 / Chapter 4.2 --- "Jacobi field, conjugate point and length of geodesic" --- p.35 / Chapter 4.3 --- Congruence of causal geodesics --- p.47 / Chapter 5 --- Singularity Theorems --- p.57 / Chapter 5.1 --- Definition of singularities in space-time --- p.57 / Chapter 5.2 --- A singularity theorem of R. Penrose --- p.60 / Chapter 5.3 --- A singularity theorem of S.W. Hawking and R. Penrose --- p.64 / Appendix --- p.73 / Bibliography --- p.77

Soit f un germe de fonction holomorphe dans deux variables qui s'annule à l'origine. L'ensemble zéro de cette fonction définit un germe de courbe analytique. Bien que la classification topologique d'un tel germe est bien connue depuis les travaux de Zariski, la classification analytique est encore largement ouverte. En 2012, Hefez et Hernandes ont résolu le cas irréductible et ont annoncé le cas de deux components. En 2015, Genzmer et Paul ont résolu le cas des fonctions topologiquement quasi-homogènes. L'objectif principal de cette thèse est d'étudier la première classe topologique de fonctions non quasi-homogènes. Dans le deuxième chapitre, nous décrivons l'espace local des modules des feuillages de cette classe et nous donnons une famille universelle de formes normales analytiques. Dans le même chapitre, nous prouvons l'unicité globale de ces formes normales. Dans le troisième chapitre, nous étudions l'espace des modules de courbes, qui est l'espace des modules des feuillages à une équivalence analytique des séparatrices associées près. En particulier, nous présentons un algorithme pour calculer sa dimension générique. Le quatrième chapitre présente une autre famille universelle de formes normales analytiques, qui est globalement unique aussi. En effet, il n'ya pas de modèle canonique pour la distribution de l'ensemble des paramètres sur les branches. Ainsi, avec cette famille, nous pouvons voir que la famille précédente n'est pas la seule et qu'il est possible de construire des formes normales en considérant une autre distribution des paramètres. Enfin, pour la globalisation, nous discutons dans le cinquième chapitre une stratégie basée sur la théorie géométrique des invariants et nous expliquons pourquoi elle ne fonctionne pas jusqu'à présent. / Let f be a germ of holomorphic function in two variables which vanishes at the origin. The zero set of this function defines a germ of analytic curve. Although the topological classification of such a germ is well known since the work of Zariski, the analytical classification is still widely open. In 2012, Hefez and Hernandes solved the irreducible case and announced the two components case. In 2015, Genzmer and Paul solved the case of topologically quasi-homogeneous functions. The main purpose of this thesis is to study the first topological class of non quasi-homogeneous functions. In chapter 2, we describe the local moduli space of the foliations in this class and give a universal family of analytic normal forms. In the same chapter, we prove the global uniqueness of these normal forms. In chapter 3, we study the moduli space of curves which is the moduli space of foliations up to the analytic equivalence of the associated separatrices. In particular, we present an algorithm to compute its generic dimension. Chapter 4 presents another universal family of analytic normal forms which is globally unique as well. Indeed, there is no canonical model for the distribution of the set of parameters on the branches. So, with this family, we can see that the previous family is not the only one and that it is possible to construct normal forms by considering another distribution of the parameters. Finally, concerning the globalization, we discuss in chapter 5 a strategy based on geometric invariant theory and explain why it does not work so far.

Les fonctions analytiques généralisées sont définies par des séries convergentes de monômes à coeficients réels et exposants réels positifs. Nous étudions l'extension de la géométrie analytique réelle associée à ces algèbres de fonctions. Nous introduisons pour cela la notion de variété analytique réelle généralisée. Il s'agit de variétés topologiques à bord munies de la structure du faisceau des fonctions analytiques réelles généralisées. Notre résultat principal est un théorème de monomialisation locale de ces fonctions.

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For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero.

We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.

On a compact closed manifold with edges live pseudodifferential operators which are block matrices of operators with additional edge conditions like boundary conditions in boundary value problems. They include Green, trace and potential operators along the edges, act in a kind of Sobolev spaces and form an algebra with a wealthy symbolic structure. We consider complexes of Fréchet spaces whose differentials are given by operators in this algebra. Since the algebra in question is a microlocalization of the Lie algebra of typical vector fields on a manifold with edges, such complexes are of great geometric interest. In particular, the de Rham and Dolbeault complexes on manifolds with edges fit into this framework. To each complex there correspond two sequences of symbols, one of the two controls the interior ellipticity while the other sequence controls the ellipticity at the edges. The elliptic complexes prove to be Fredholm, i.e., have a finite-dimensional cohomology. Using specific tools in the algebra of pseudodifferential operators we develop a Hodge theory for elliptic complexes and outline a few applications thereof.