Thompson’s groups F, T, and V represent crucial examples of groups in geometric group theory that bridge it with other areas of mathematics such as logic, computer science, analysis, and geometry. One of the ways to study these groups is by understanding the geometric meaning of their actions. In this thesis we deal with Thompson’s group T that acts naturally on the unit circle S1, that is identified with the segment [0, 1] with the end points glued together. The main result of this work is the explicit construction of the Schreier graph of T with respect to the action on the orbit of 1/2. This is done by careful examination of patterns in how the generators of T act on binary words. As a main application, the nonamenability of the action of T on S1 is proved by defining injections on the set of vertices of the constructed graph that satisfy Gromov’s doubling condition. This gives an alternative proof of the known fact that T is nonamenable.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico / Neste texto, à mostrado que conjuntos definÃveis bi-Lipschitz homeomorfos tem cones tangentes bi-Lipschitz homeomorfos. AlÃm disso, no caso de conjuntos analÃticos complexos, regularidade Lipschitz ou regularidade topolÃgica forte implica em regularidade analÃtica. TambÃm à feito um estudo regularidade de conjuntos analÃticos reais. Ademais, à dada uma classificaÃÃo completa para curvas analÃticas complexas no espaÃo e sÃo apresentados alguns resultados sobre invariÃncia da multiplicidade. Em especial, à mostrado que a multiplicidade mod 2 de conjuntos analÃticos reais à invariante por difeomorfismos. / In this paper, it is shown that definable sets bi-Lipschitz homeomorphic have tangent cones bi-Lipschitz homeomorphic. Furthermore, in the case of complex analytical sets, Lipschitz regularity or strong topological regularity implies analytical regularity. It is also done a complete study on regularity of real analytic sets. Furthermore, it is given a complete classification for complex analytical curves in space and are shown some results about invariance of the multiplicity. In particular, it is shown that the multiplicity of real analytical sets is invariant mod 2 under diffeomorphisms.

A mapping f from R^{n} to R^{n} is said to satisfy the Luzin condition N if f maps sets of measure zero to sets of measure zero. It is known to be valid for mappings in the Sobolev space W^{1,p} for p > n and for p <= n there are counterexamples. The aim of this thesis is to summarize known results and study the validity of Luzin condition N for mappings in the Sobolev space W^{2,p}.

Considerable study has been made concerning the properties of the iterations of a homeomorphism on a metric space. Much of this material is scattered throughout the literature and understood solely by a specialist. The main object of this paper is to put into readable form proofs of theorems found in G.T. Whyburn's "Analytic Topology" pertaining to this topic in topology. Properties of the decomposition space of point-orbits induced by the iterations of a homeomorphism will compose a major part of the study. Some theorems will be established through series of lemmas required to fill in much of the detail lacking in proofs found the book. Although an elementary knowledge of topology is assumed throughout the paper, references are given for basic definitions and theorems used in developing some of the proofs. The following symbols and notation will be used throughout the paper. X will denote a metric space with metric p, S a topological space, I the set of positive integers, A, B, C... sets of points or elements. Small letters, such as a, b, c, x, y, z... will designate elements or points of sets. U and V will denote open sets Sr(x) a spherical neighborhood of x with radius r. A' denotes the set of limit points of A. A- the set of closure points of A/ U, N, C will denote union, intersection, and set inclusion respectively. The symbol E will mean "is an element of". 0 denotes the empty set. S - A is the set of points in S which are not in A.

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Durant cette thèse, nous nous sommes intéressés à la problématique de la captation de mouvement sans marqueurs. Une approche classique est basée sur l'utilisation d'un modèle prédéfini du sujet, et est divisée en deux phases : celle d'initialisation, où la pose initiale du sujet est estimée, et celle de suivi, où la pose actuelle du sujet est estimée à partir des précédentes. Souvent, la phase d'initialisation est faite manuellement, rendant impossible l'utilisation en direct, ou nécessite des actions spécifiques du sujet. Nous proposons une phase d'initialisation automatique et temps-réel, utilisant l'information topologique extraite par squelettisation d'une reconstruction 3D du sujet. Cette information est représentée sous forme d'arbre (arbre de données), qui est mis en correspondance avec un arbre utilisé comme modèle, afin d'identifier les différentes parties du sujet. Pour obtenir une telle méthode, nous apportons des contributions dans les domaines de la topologie discrète et de la théorie des graphes. Comme notre méthode requiert le temps réel, nous nous intéressons d'abord à l'optimisation du temps de calcul des méthodes de squelettisation, ainsi qu'à l'élaboration de nouveaux algorithmes rapides fournissant de bons résultats. Nous nous intéressons ensuite à la définition d'une mise en correspondance efficace entre l'arbre de données et celui décrivant le modèle. Enfin, nous améliorons la robustesse de notre méthode en ajoutant des contraintes novatrices au modèle. Nous terminons par l'application de notre méthode sur différents jeux de données, démontrantses propriétés : rapidité robustesse et adaptabilité à différents types de sujet / This manuscript deals with the problem of markerless motion capture. An approach to thisproblem is model-based and is divided into two steps : an initialization step in which the initialpose is estimated, and a tracking which computes the current pose of the subject using infor-mation of previous ones. Classically, the initialization step is done manually, for bidding the possibility to be used online, or requires constraining actions of the subject. We propose an automatic real-time markerless initialization step, that relies on topological information provided by skeletonization of a 3D reconstruction of the subject. This topological information is then represented as a tree, which is matched with another tree used as modeldescription, in order to identify the different parts of the subject. In order to provide such a method, we propose some contributions in both digital topology and graph theory researchfields. As our method requires real-time computation, we first focus on the speed optimization of skeletonization methods, and on the design of new fast skeletonization schemes providing good results. In order to efficiently match the tree representing the topological information with the tree describing the model, we propose new matching definitions and associated algorithms. Finally, we study how to improve the robustness of our method by the use of innovative con-straints in the model. This manuscript ends by a study of the application of our method on several data sets, demon-strating its interesting properties : fast computation, robustness, and adaptability to any kindof subjects

Univerzita Karlova Abstract of the diploma thesis Analysis in Banach spaces Matěj Novotný, Praha 2013 In the thesis, connection between two certain types of equivalence on Ba- nach spaces is studied: Between Lipschitz and linear one. In general, linear equivalence of two Banach spaces implies their Lipschitz equivalence, but the converse need not be true, which is shown by some nonseparable examples. There are summarized several examples to this question in the thesis, both positive and negative ones. Moreover, it is shown that James' quasi-reflexive space and its dual space have unique Lipschitz structure. To prove this, theory of linearization of Lipschitz mappings and at the same time linear structure of the two mentioned spaces is used. 1

Finding new examples of compact simply connected spaces admitting a Riemannian metric of positive sectional curvature is a fundamental problem in differential geometry. Likewise, studying topological properties of families of manifolds is very interesting to topologists. The Eschenburg spaces combine both of those interests: they are positively curved Riemannian manifolds whose topological classification is known. There is a second family consisting of the Witten manifolds: they are the examples of compact simply connected spaces admitting Einstein metrics of positive Ricci curvature. Thirdly, there is a notion of generalized Witten manifold as well. Topologically, all three families share the same cohomology ring. This common ring structure motivates the definition of a manifold of type r, where r is the order of the fourth cohomology group. In 1991, M. Kreck and S. Stolz classified manifolds M of type r up to homeomorphism and dieomorphism using invariants s̄[subscript i](M) and s[subscript i](M), for i = 1, 2, 3. This gave rise to many new examples of nondieomorphic but homeomorphic manifolds. In this dissertation, new versions of the homeomorphism and dieomorphism classification of manifolds of type r are proven. In particular, we can replace s̄₁ and s̄₃ by the first Pontrjagin class and the self-linking number in the homeomorphism classification of spin manifolds of type r. As the formulas of the two latter invariants are in general much easier to compute, this simplifies the classification of these manifolds up to homeomorphism significantly. / Graduation date: 2012

In this thesis, we present solutions to several problems concerning one-dimensi- onal continua. We give an inductive description of graphs with a given disconnec- tion number, this answers a question of S. B. Nadler. Further, we state a topo- logical characterization of the Sierpi'nski triangle. In the study of shore sets in dendroids and λ-dendroids we obtain several positive results and we also provide some counterexamples. By doing this, we continue in the recent work of several authors. We are also dealing with the notion of 1 2 -homogeneity and we prove that up to homeomorphism there are only two 1 2 -homogeneous chainable continua with just two end points. We present also a new elegant proof of a classical theorem of Waraszkiewicz. 1

Seja K um conjunto de Cantor. Neste trabalho apresentamos dois teoremas relacionados a densidade do conjunto das dinâmicas simbólicas. No caso de endomorfismos provamos que, dado uma dinâmica T : K K, existe uma T : K K próxima a T, tal que toda órbita é finalmente periódica. Já no caso de homeomorfismos, mostramos que, dado uma dinâmica T : K K, existe uma T : K K próxima a T, tal que o w-limite de toda órbita de T é uma órbita periódica. Em particular, mostramos que, em ambos os casos, todas as medidas ergódicas estão suportadas em órbitas periódicas. / Let K be a Cantor set. In this work we present two theorems related to the density of symbolic dynamics. We prove that given an endomorphism T : K K then there exists an endomorphism ~ T : K K close to T such that every orbit is finally periodic. We also prove that given a homeomorphism T : K K then there exists a homeomorphism ~ T : K K close to T such that the w-limit of every orbit is a periodic orbit. In particular, we have shown, in both cases, that all ergodic measures have support on periodic orbits.