The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Title from PDF of title page (University of Missouri--Columbia, viewed on January 14, 2010). Thesis advisor: Dr. Alex Iosevich. Includes bibliographical references.

Not available / text

Thesis (Ph.D.)--Tufts University, 2004. / Adviser: Eric Todd Quinto. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 282-290). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 42).

Thesis (M.S.) -- Central Connecticut State University, 1998. / Thesis advisor: Dr. Jeffrey McGowan. "...in partial fulfillment of the requirements for the degree of Master of Science." Includes bibliographical references (leaves [78-79]).

This dissertation studies quantum invariants of knots and links, particularly the colored Jones polynomials, and their relationships with classical invariants like the hyperbolic volume and the A-polynomial. We consider the volume conjecture that relates the Kashaev invariant, a specialization of the colored Jones polynomial at a specific root of unity, and the hyperbolic volume of a link; and the AJ conjecture that relates the colored Jones polynomial and the A-polynomial of a knot. We establish the AJ conjecture for some big classes of two-bridge knots and pretzel knots, and confirm the volume conjecture for some cables of knots.

Le groupe de Cremona de rang 2 est le groupe des transformations birationnelles du plan projectif. Le but de cette thèse est d'étudier et de construire des espaces hyperboliques sur lesquels le groupe de Cremona agit et qui permettent de mettre en œuvre des méthodes provenant de la théorie géométrique des groupes. Il est connu depuis une dizaine d'année que le groupe de Cremona agit sur un espace hyperbolique H analogue au plan hyperbolique classique mais de dimension infinie. Dans un premier temps, nous montrons que le groupe de Cremona défini sur un corps quelconque n'est pas simple en le faisant agir sur cet espace hyperbolique. Ceci prolonge un résultat déjà connu dans le cas d'un corps de base algébriquement clos. Nous nous intéressons ensuite à un graphe construit par D. Wright sur lequel agit le groupe de Cremona. Nous montrons qu'il ne possède pas la propriété que nous souhaitions, à savoir qu'il n'est pas hyperbolique au sens de Gromov. Nous construisons également un domaine fondamental pour l'action du groupe de Cremona sur H via la méthode des cellules de Voronoï. Nous caractérisons les applications du groupe de Cremona qui correspondent à un domaine adjacent au domaine fondamental. Cela nous permet de prouver que le graphe de Wright est quasi-isométrique au graphe dual à ce pavage. Nous obtenons ainsi une manière de retrouver le graphe de Wright dans H. Nous montrons enfin qu'en modifiant ce graphe dual, nous obtenons un graphe hyperbolique au sens de Gromov. Dans une dernière partie, nous nous intéressons à une autre propriété naturelle qui est la propriété CAT(0). Nous construisons un complexe cubique CAT(0) de dimension infinie muni d'une action naturelle du groupe de Cremona. / The Cremona group of rank 2 is the group of birational transformations of the projective plane. The aim of this thesis is to study and build some hyperbolic spaces with a natural action of the Cremona group. We want these spaces to have good geometric properties in order to use methods coming from geometric group theory. It is known that the Cremona group acts on a hyperbolic space H which is similiar to the classical hyperbolic plane but in infinite dimension. First, using this action, we show that the Cremona group is not simple over any field. This extends previous results over an algrebraic closed field. Then we study the Wrigth's graph. We show that it doesn't have the property we are looking for, in the sense that it is not Gromov hyperbolic. We build a fundamental domain for the action of the Cremona group on H 8 via Voronoï's cells. We characterize birational tranformations that correspond to adjacent domains of the fundamental domain. This allows us to prove that the Wright's graph is quasi-isometric to the dual graph of this tessellation. It's give us a way of realizing the Wright's graph inside H. Finally, we show that by modifying the dual graph we obtain a Gromov hyperbolic graph. In the last part, we are interested in another classical property which is the CAT(0) property. We build an infinite dimensional CAT(0) cubical complex which comes with a natural action of the Cremona group.

The aim of this dissertation is to give explicit descriptions of the set of proper holomorphic mappings between two complex manifolds with reasonable restrictions on the domain and target spaces. Without any restrictions, this problem is intractable even when posed for do-mains in . We give partial results for special classes of manifolds. We study, broadly, two types of structure results: Descriptive. The first result of this thesis is a structure theorem for finite proper holomorphic mappings between products of connected, hyperbolic open subsets of compact Riemann surfaces. A special case of our result follows from the techniques used in a classical result due to Remmert and Stein, adapted to the above setting. However, the presence of factors that have no boundary or boundaries that consist of a discrete set of points necessitates the use of techniques that are quite divergent from those used by Remmert and Stein. We make use of a finiteness theorem of Imayoshi to deal with these factors. Rigidity. A famous theorem of H. Alexander proves the non-existence of non-injective proper holomorphic self-maps of the unit ball in . ,n >1. Several extensions of this result for various classes of domains have been established since the appearance of Alexander’s result, and it is conjectured that the result is true for all bounded domains in . , n > 1, whose boundary is C2-smooth. This conjecture is still very far from being settled. Our first rigidity result establishes the non-existence of non-injective proper holomorphic self-maps of bounded, balanced pseudo convex domains of finite type (in the sense of D’Angelo) in ,n >1. This generalizes a result in 2, by Coupet, Pan and Sukhov, to higher dimensions. As in Coupet–Pan–Sukhov, the aforementioned domains need not have real-analytic boundaries. However, in higher dimensions, several aspects of their argument do not work. Instead, we exploit the circular symmetry and a recent result in complex dynamics by Opshtein. Our next rigidity result is for bounded symmetric domains. We prove that a proper holomorphic map between two non-planar bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, all-encompassing argument that unifies the various special cases in which this result is known. Furthermore, our proof of this result does not rely on the fine structure (in the sense of Wolf et al.) of bounded symmetric domains. Thus, we are able to apply our techniques to more general classes of domains. We illustrate this by proving a rigidity result for certain convex balanced domains whose automorphism groups are assumed to only be non-compact. For bounded symmetric domains, our key tool is that of Jordan triple systems, which is used to describe the boundary geometry.

Orientador: Reginaldo Palazzo Júnior / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-18T16:43:26Z (GMT). No. of bitstreams: 1 Pilla_ElianeCristinaGeroli_M.pdf: 2007393 bytes, checksum: 2b95255e6d4fca123c23a039d1a083a5 (MD5) Previous issue date: 2005 / Resumo: O presente trabalho tem como meta principal construir constelações de sinais geometricamente uniformes no plano hiperbólico, visando considerá-las como alfabeto para geração de códigos de espaço de sinais, em particular os códigos de classes laterais generalizados. Para estabelecer estas constelações foi escolhido um conjunto de sinais geometricamente uniforme, constituído pelos centros dos octógonos da tesselação {8, 8}. Depois foi obtido um rotulamento para os elementos do grupo gerador dos conjuntos de sinais geometricamente uniformes em cada classe lateral. Finalmente, a partir do isomorfismo rótulo obtivemos um rotulamento isométrico para os elementos do conjunto de sinais / Abstract: Our goal in this work is to construct hyperbolic geometrically uniform signal constellations (more specifically g-torus) that are able to act as alphabets for ge neration of codes. To obtain these constellations we choose geometrically uniform signal sets consisting of the centers of the p-gons of tessellations of type {p, q}. From these constellations we obtain labelings for the elements of the generator group of the geometrically uniform signal sets in each coset. Finally, by the label isomorphism we obtain an isometric labeling for the elements of the signal set / Mestrado / Telecomunicações e Telemática / Mestre em Engenharia Elétrica

Dans une première partie de cette thèse, nous donnons des minorations universelles ne dépendant que de la dimension – explicites, de trois invariants globaux des quotients des espaces hyperboliques quaternioniques : leur rayon maximal, leur volume, ainsi que leur caractéristique d’Euler. Nous donnons également une majoration de leur constante de Margulis, montrant que celle-ci décroit au moins comme une puissance négative de la dimension. Dans une seconde partie, nous étudions un réseau remarquable des isométries du plan hyperbolique quaternionique, le groupe modulaire d’Hurwitz. Nous montrons en particulier qu’il est engendré par quatres éléments, et construisons un domaine fondamental pour le sous-groupe des isométries de ce réseau qui stabilisent un point à l’infini. / In the first part of this thesis, we derive explicit universal – that is, depending only on the dimension – lower bounds on three global invariants of quaternionic hyperbolic sapces : their maximal radius, their volume, and their Euler caracteristic. We also exhibit an upper bound on their Margulis constant, showing that this last quantity decreases at least like a negative power of the dimension. In the second part, we study a specific lattice of isometries of the quaternionic hyperbolic plane : the Hurwitz modular group. In particular, we show that this group is generated by four elements, and we construct a fundamental domain for the subgroup of isometries of this lattice stabilising a point on the boundary of the quaternionic hyperbolic plane.