CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / FundaÃÃo de Amparo à Pesquisa do Estado do Cearà / Estabelece desigualdades isoperimÃtricas e estimativas do tempo mÃdio de saÃda para subvariedades mÃnimas de N x R, onde N à uma variedade riemanniana completa com curvatura seccional nÃo-positiva. Prova desigualdades isoperimÃtricas para subvariedades com segunda forma fundamental dominada em espaÃos de Hadamard com curvatura seccional limitada. / It establishes isoperimetric inequalities and exit mean time estimates for minimal submanifolds of N x R, where N is a complete Riemannian manifold with sectional curvature non-positive. It proves isoperimetric inequalities for submanifolds with tamed second fundamental form in Hadamard spaces with bounded sectional curvature.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Este trabalho consiste em duas partes. Na primeira parte, estudaremos hipersuperfÃcies compactas sem bordo imersas no espaÃo Euclidiano com o quociente das curvaturas mÃdias anisotrÃpicas constante. Provaremos que tais hipersuperfÃcies sÃo pontos crÃticos para um problema variacional de preservar uma combinaÃÃo linear da (k; F)-Ãrea e do (n+1)-volume determinado por M. Demostraremos que a hipersuperfÃcie à (r; k; a; b)-estÃvel se, e somente se, a menos de translaÃÃo e homotetia, ela à a Wulff shape de F (veja SeÃÃo 2.1), sob algumas condiÃÃes acerca de a; b â R. Na segunda parte desse trabalho, obtemos outras caracterizaÃÃes para a Wulff shape envolvendo as curvaturas mÃdias anisotrÃpicas de ordem superior de uma hipersuperfÃ- cie M em Rn+1 e o conjunto W = Rn+1 -UpâM Tp. Os resultados sÃo obtidos para hipersuperfÃcies compactas nÃo convexas satisfazendo W ╪ Ã. / This work consists of two parts. In the first part we deal with a compact hypersurface without boundary immersed in to the Euclidean space with the quotient of anisotropic mean curvatures constant. Such a hypersurface is a critical point for the variational problem preserving a linear combination of the (k; F)-area and (n + 1)-volume enclosed by M. We show that it is (r; k; a; b)-stable if, and only if, up to translations and homotheties, it is the Wulff shape, under some assumptions on a; b â R. In the second part we obtain further characterizations for the Wulff shape involving the anisotropic mean curvatures of higher order of a hypersurface M in Rn+1 and the set W = Rn+1-UpâM Tp. Results are obtained for non-convex compact hypersurfaces satisfying W ╪ Ã.

This thesis consists of two parts. In the first part, we study certain Ricci flow invariant nonnegative curvature conditions as given by B. Wilking. We begin by proving that any such nonnegative curvature implies nonnegative isotropic curvature in the Riemannian case and nonnegative orthogonal bisectional curvature in the K¨ahler case. For any closed AdSO(n,C) invariant subset S so(n, C) we consider the notion of positive curvature on S, which we call positive S- curvature. We show that the class of all such subsets can be naturally divided into two subclasses: The first subclass consists of those sets S for which the following holds: If two Riemannian manifolds have positive S- curvature then their connected sum also admits a Riemannian metric of positive S- curvature. The other subclass consists of those sets for which the normalized Ricci flow on a closed Riemannian manifold with positive S-curvature converges to a metric of constant positive sectional curvature. In the second part of the thesis, we study the behavior of Ricci flow for a manifold having positive S - curvature, where S is in the first subclass. More specifically, we study the Ricci flow for a special class of metrics on Sp+1 x S1 , p ≥ 4, which have positive isotropic curvature.

The abstract boundary construction of Scott and Szekeres has proven a practical classification scheme for boundary points of pseudo-Riemannian manifolds. It has also proved its utility in problems associated with the re-embedding of exact solutions containing directional singularities in space-time. Moreover it provides a model for singularities in space-time - essential singularities. However the literature has been devoid of abstract boundary results which have results of direct physical applicability.¶ This thesis presents several theorems on the existence of essential singularities in space-time and on how the abstract boundary allows definition of optimal em- beddings for depicting space-time. Firstly, a review of other boundary constructions for space-time is made with particular emphasis on the deficiencies they possess for describing singularities. The abstract boundary construction is then pedagogically defined and an overview of previous research provided.¶ We prove that strongly causal, maximally extended space-times possess essential singularities if and only if they possess incomplete causal geodesics. This result creates a link between the Hawking-Penrose incompleteness theorems and the existence of essential singularities. Using this result again together with the work of Beem on the stability of geodesic incompleteness it is possible to prove the stability of existence for essential singularities.¶ Invariant topological contact properties of abstract boundary points are presented for the first time and used to define partial cross sections, which are an generalization of the notion of embedding for boundary points. Partial cross sections are then used to define a model for an optimal embedding of space-time.¶ Finally we end with a presentation of the current research into the relationship between curvature singularities and the abstract boundary. This work proposes that the abstract boundary may provide the correct framework to prove curvature singularity theorems for General Relativity. This exciting development would culminate over 30 years of research into the physical conditions required for curvature singularities in space-time.

<p>In the last few decades, data analysis techniques have had to expand to handle large sets of data with complicated structure. This includes identifying low dimensional structure in high dimensional data, analyzing shape and image data, and learning from or classifying large corpora of text documents. Common Bayesian and Machine Learning techniques rely on using the unique geometry of these data types, however departing from Euclidean geometry can result in both theoretical and practical complications. Bayesian nonparametric approaches can be particularly challenging in these areas. </p><p> </p><p>This dissertation proposes a novel approach to these challenges by working with convenient embeddings of the manifold valued parameters of interest, commonly making use of an extrinsic distance or measure on the manifold. Carefully selected extrinsic distances are shown to reduce the computational cost and to increase accuracy of inference. The embeddings are also used to yield straight forward derivations for nonparametric techniques. The methods developed are applied to subspace learning in dimension reduction problems, planar shapes, shape constrained regression, and text analysis.</p> / Dissertation

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Apresentamos um mÃtodo para a obtenÃÃo de limites inferiores para o primeiro autovalor de Dirichlet em termos de campos vetoriais com divergÃncia positiva. Aplicando-o ao gradiente de uma funÃÃo distante, obtemos estimativas de de autovalor em bolas geodÃsicas em cut locus e dos domÃnios de subvariedades com curvatura mÃdia localmente limitada.Para subvariedades das variedade de Hadamard com limites mÃdios de curvaturas, estes limites inferiores dependem da dimensÃo das subvariedades e limite sobre sua curvatura mÃdia. / We present a method to obtain lower bounds for first Dirichlet eigenvalue in terms of vector fields with positive divergence. Applying this to the gradient of a distance function we obtain estimates of eigenvalue of geodesic balls inside the cut locus and of domains in submanifolds with locally bounded mean curvature. For submanifolds of Hadamard manifolds with bounded mean curvature these lower bounds depend only on the dimension of the submanifold and the bound on its mean curvature.

Orientador: Pedro José Catuogno / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-23T19:39:42Z (GMT). No. of bitstreams: 1 Neves_EduardodeAmorim_D.pdf: 2575933 bytes, checksum: c52665fabd4cb103ecdff3106990155e (MD5) Previous issue date: 2013 / Resumo: Esta tese está dividida em duas partes. Na primeira parte, faremos uma abordagem probabilística para a teoria de aplicações L-harmônicas em variedades diferenciáveis, passaremos para esse contexto os Teoremas de Liouville, Picard, Elworthy e Dirichlet. Na segunda parte do trabalho, o objetivo é generalizar e caracterizar o conceito de difusão, martingale e movimento Browniano em variedades que estejam munidas por uma família de métricas e conexões que variam diferenciavelmente com o tempo / Abstract: This thesis is divided into two parts. In the first part, we will make a probabilistic approach to the theory of L-harmonic applications on manifolds; we generalize to this context Theorems of Liouville, Picard, Elworthy and Dirichlet. In the second part of the work, the goal is to generalize and characterize the concept of diffusion, martingale and Brownian motion on manifolds that are provided by a family of metrics and connections which depends smoothly on time / Doutorado / Matematica / Doutor em Matemática

Orientadores: Alcibiades Rigas, Carlos Eduardo Duran Fernandez / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T13:56:12Z (GMT). No. of bitstreams: 1 Speranca_LlohamDallagnol_D.pdf: 994466 bytes, checksum: 472919d7eec0f563b673a0307450dc49 (MD5) Previous issue date: 2012 / Resumo: Nesse trabalho estudaremos a geometria e a topologia de algumas variedades homeomorfas, porém não difeomorfas, à esfera padrão Sn, chamadas esferas exóticas. Realizaremos duas dessas variedades como quocientes isométricos de fibrados principais com métricas de conexão sobre esferas de curvatura constante. Através disso, apresentaremos simetrias desses espaços e exemplos explícitos de difeomorfismos não isotópicos a identidade, usando-os para o cálculo de grupos de homotopia equivariante. Como mais uma aplicação dessa construção, provaremos que, se uma esfera homotópica de dimensão 15 é realizável como um fibrado linear sobre S8, então a mesma esfera é realizável como um fibrado linear sobre a esfera exótica de dimensão 8 com as mesmas funções de transição. No ultimo capítulo lidaremos com a geometria de fibrados induzidos, deduzindo uma condição necessária sobre a função indutora para que a métrica da conexão induzida tenha curvatura seccional não-negativa / Abstract: In this work we study the geometry and topology of manifolds homemorphic, but not diffeomorphic, to the standard sphere Sn, the so called exotic spheres. We realize two of these manifolds as isometric quotients of principal bundles with connection metrics over the constant curved sphere. Through this, we present symmetries in these spaces and explicit examples of diffeomorphisms not isotopic to the identity, using them for the calculation of equivariant homotopy groups. As another application, we prove that, if a homotopy 15-sphere is realizeble as the total space of a linear bundle over the standard 8-sphere, then, it is realizeble as the total space of a linear bundle over the exotic 8-sphere with the same transition maps. In the last chapter we deal with the geometry of pull-back bundles, deducing a necessary condition on the pull-back map for nonnegative curvature of the induced connection metric / Doutorado / Matematica / Doutor em Matemática

Given a compact smooth manifold Mn without boundary and n ≥ 3, the Lp-norm of the curvature tensor, defines a Riemannian functional on the space of Riemannian metrics with unit volume M1. Consider C2,α-topology on M1 Rp remains invariant under the action of the group of diffeomorphisms D of M. So, Rp is defined on M1/ D. Our first result is that Rp restricted to the space M1/D has strict local minima at Riemannian metrics with constant sectional curvature for certain values of p. The product of spherical space forms and the product of compact hyperbolic manifolds are also critical point for Rp if they are product of same dimensional manifolds. We prove that these spaces are strict local minima for Rp restricted to M1/D. Compact locally symmetric isotropy irreducible metrics are critical points for Rp. We give a criteria for the local minima of Rp restricted to the conformal class of metrics of a given irreducible symmetric metric. We also prove that the metrics with constant bisectional curvature are strict local minima for Rp restricted to the space of Kahlar metrics with unite volume quotient by D. Next we consider the Riemannian functional given by In [GV], M. J. Gursky and J. A. Viaclovsky studied the local properties of the moduli space of critical metrics for the functional Ric2.We generalize their results for any p > 0.

The representer theorem from the reproducing kernel Hilbert space theory is the origin of many kernel-based machine learning and signal modelling techniques that are popular today. Most kernel functions used in practical applications behave in a homogeneous manner across the domain of the signal of interest, and they are called stationary kernels. One open problem in the literature is the specification of a non-stationary kernel that is computationally tractable. Some recent works solve large-scale optimization problems to obtain such kernels, and they often suffer from non-identifiability issues in their optimization problem formulation. Many practical problems can benefit from using application-specific prior knowledge on the signal of interest. For example, if one can adequately encode the prior assumption that edge contours are smooth, one does not need to learn a finite-dimensional dictionary from a database of sampled image patches that each contains a circular object in order to up-convert images that contain circular edges. In the first portion of this thesis, we present a novel method for constructing non-stationary kernels that incorporates prior knowledge. A theorem is presented that ensures the result of this construction yields a symmetric and positive-definite kernel function. This construction does not require one to solve any non-identifiable optimization problems. It does require one to manually design some portions of the kernel while deferring the specification of the remaining portions to when an observation of the signal is available. In this sense, the resultant kernel is adaptive to the data observed. We give two examples of this construction technique via the grayscale image up-conversion task where we chose to incorporate the prior assumption that edge contours are smooth. Both examples use a novel local analysis algorithm that summarizes the p-most dominant directions for a given grayscale image patch. The non-stationary properties of these two types of kernels are empirically demonstrated on the Kodak image database that is popular within the image processing research community. Tensors and tensor decomposition methods are gaining popularity in the signal processing and machine learning literature, and most of the recently proposed tensor decomposition methods are based on the tensor power and alternating least-squares algorithms, which were both originally devised over a decade ago. The algebraic approach for the canonical polyadic (CP) symmetric tensor decomposition problem is an exception. This approach exploits the bijective relationship between symmetric tensors and homogeneous polynomials. The solution of a CP symmetric tensor decomposition problem is a set of p rank-one tensors, where p is fixed. In this thesis, we refer to such a set of tensors as a rank-one decomposition with cardinality p. Existing works show that the CP symmetric tensor decomposition problem is non-unique in the general case, so there is no bijective mapping between a rank-one decomposition and a symmetric tensor. However, a proposition in this thesis shows that a particular space of rank-one decompositions, SE, is isomorphic to a space of moment matrices that are called quasi-Hankel matrices in the literature. Optimization over Riemannian manifolds is an area of optimization literature that is also gaining popularity within the signal processing and machine learning community. Under some settings, one can formulate optimization problems over differentiable manifolds where each point is an equivalence class. Such manifolds are called quotient manifolds. This type of formulation can reduce or eliminate some of the sources of non-identifiability issues for certain optimization problems. An example is the learning of a basis for a subspace by formulating the solution space as a type of quotient manifold called the Grassmann manifold, while the conventional formulation is to optimize over a space of full column rank matrices. The second portion of this thesis is about the development of a general-purpose numerical optimization framework over SE. A general-purpose numerical optimizer can solve different approximations or regularized versions of the CP decomposition problem, and they can be applied to tensor-related applications that do not use a tensor decomposition formulation. The proposed optimizer uses many concepts from the Riemannian optimization literature. We present a novel formulation of SE as an embedded differentiable submanifold of the space of real-valued matrices with full column rank, and as a quotient manifold. Riemannian manifold structures and tangent space projectors are derived as well. The CP symmetric tensor decomposition problem is used to empirically demonstrate that the proposed scheme is indeed a numerical optimization framework over SE. Future investigations will concentrate on extending the proposed optimization framework to handle decompositions that correspond to non-symmetric tensors.