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Thesis (Ph. D.)--University of Texas at Austin, 2005. / Vita. Includes bibliographical references.

Characterizing the long-time behavior of solutions to the Einstein field equations remains an active area of research today. In certain types of coordinates the Einstein equations form a coupled system of quasilinear wave equations. The investigation of the nature and properties of solutions to these equations lies in the field of geometric analysis. We make several contributions to the study of solution dynamics near singularities. While singularities are known to occur quite generally in solutions to the Einstein equations, the singular behavior of solutions is not well-understood. A valuable tool in this program has been to prove the existence of families of solutions which are so-called asymptotically velocity term dominated (AVTD). It turns out that a method, known as the Fuchsian method, is well-suited to proving the existence of families of such solutions. We formulate and prove a Fuchsian-type theorem for a class of quasilinear hyperbolic partial differential equations and show that the Einstein equations can be formulated as such a Fuchsian system in certain gauges, notably wave gauges. This formulation of Einstein equations provides a convenient general framework with which to study solutions within particular symmetry classes. The theorem mentioned above is applied to the class of solutions with two spatial symmetries -- both in the polarized and in the Gowdy cases -- in order to prove the existence of families of AVTD solutions. In the polarized case we find families of solutions in the smooth and Sobolev regularity classes in the areal gauge. In the Gowdy case we find a family of wave gauges, which contain the areal gauge, such that there exists a family of smooth AVTD solutions in each gauge.

In this paper, we present techniques for proving a group to be non-left-orderable or a unique product group. These methods involve the existence of a mapping from the group to R which obeys a left-multiplication criterion. By determining the existence or non-existence of such a mapping, the desired information about the group can be concluded. As examples, we apply this technique to groups of transformations in hyperbolic 2- and 3- space, and Fibonacci groups. / Master of Science

We construct Fuchsian groups [Gamma] of signature (0 : 2, ... ,2 ;1;0) so that the set of hyperbolic fixed points of [Gamma] will contain a given finite collection of elements in the boundary of the hyperbolic plane. We use this to establish that there are infinitely many non-commensurable non-cocompact Fuchsian groups [Delta] of finite covolume sitting in PSL₂(Q) so that the set of hyperbolic fixed points of [Delta] will contain a given finite collection of rational boundary points of the hyperbolic plane. We also give a parameterization of Fuchsian groups of signature (0:2,2,2;1;0) and investigate when particular hyperbolic elements have rational fixed points. Moreover, we include a detailed list of the group elements and their killer intervals for the known pseudomodular groups that Long and Reid found; in addition, the list contains a new list of killer intervals for a pseudomodular group not found by Long and Reid. / text

<p>A compact Riemann surface can be realized as a quotient space $\mathcal{U}/\Gamma$, where $\mathcal{U}$ is the sphere $\Sigma$, the euclidian plane $\mathbb{C}$ or the hyperbolic plane $\mathcal{H}$ and $\Gamma$ is a discrete group of automorphisms. This induces a covering $p:\mathcal{U}\rightarrow\mathcal{U}/\Gamma$.</p><p>For each $\Gamma$ acting on $\mathcal{H}$ we have a polygon $P$ such that $\mathcal{H}$ is tesselated by $P$ under the actions of the elements of $\Gamma$. On the other hand if $P$ is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group $\Gamma$ generated by the side pairing is discrete and $P$ tesselates $\mathcal{H}$ under $\Gamma$.</p>

A compact Riemann surface can be realized as a quotient space $\mathcal/\Gamma$, where $\mathcal$ is the sphere $\Sigma$, the euclidian plane $\mathbb$ or the hyperbolic plane $\mathcal$ and $\Gamma$ is a discrete group of automorphisms. This induces a covering $p:\mathcal\rightarrow\mathcal/\Gamma$. For each $\Gamma$ acting on $\mathcal$ we have a polygon $P$ such that $\mathcal$ is tesselated by $P$ under the actions of the elements of $\Gamma$. On the other hand if $P$ is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group $\Gamma$ generated by the side pairing is discrete and $P$ tesselates $\mathcal$ under $\Gamma$.

This is a thesis on general relativity. It analyzes dynamical properties of Einstein's field equations in cosmology and in the vicinity of spacetime singularities in a number of different situations. Different techniques are used depending on the particular problem under study; dynamical systems methods are applied to cosmological models with spatial homogeneity; Hamiltonian methods are used in connection with dynamical systems to find global monotone quantities determining the asymptotic states; Fuchsian methods are used to quantify the structure of singularities in spacetimes without symmetries. All these separate methods of analysis provide insights about different facets of the structure of the equations, while at the same time they show the relationships between those facets when the different methods are used to analyze overlapping areas. The thesis consists of two parts. Part I reviews the areas of mathematics and cosmology necessary to understand the material in part II, which consists of five papers. The first two of those papers uses dynamical systems methods to analyze the simplest possible homogeneous model with two tilted perfect fluids with a linear equation of state. The third paper investigates the past asymptotic dynamics of barotropic multi-fluid models that approach a `silent and local' space-like singularity to the past. The fourth paper uses Hamiltonian methods to derive new monotone functions for the tilted Bianchi type II model that can be used to completely characterize the future asymptotic states globally. The last paper proves that there exists a full set of solutions to Einstein's field equations coupled to an ultra-stiff perfect fluid that has an initial singularity that is very much like the singularity in Friedman models in a precisely defined way. / <p>Status of the paper "Perfect Fluids and Generic Spacelike Singularities" has changed from manuscript to published since the thesis defense.</p>

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [51]-52).

Es wird sowohl das klassischen Kreisproblem als auch dessen Verallgemeinerung auf geometrisch endliche Fuchssche Gruppen betrachten. Insbesondere werden obere und untere Schranken für die jeweiligen Zählfunktionen angeben. Mit einem Computerprogramm wird die Zählfunktion einer Heckegruppe bestimmt und damit eine Abschätzung ihres Konvergenzexponenten gegeben.