The aim of this thesis is to expand on parts of the work of Björn–Björn–Shanmugalingam [2] and in particular on the construction and properties of hyperbolic fillings of nonempty bounded metric spaces. In light of [2], we introduce two new parameters λ and ξ to the construction while relaxing a specific maximal-condition. With these modifications we obtain a slightly more flexible model that generates a larger family of hyperbolic fillings. We then show that every hyperbolic filling in this family possess the desired property of being Gromov hyperbolic. Next, we uniformize an arbitrary hyperbolic filling of this type and show that, under fairly weak conditions, the boundary of the uniformization is snowflake-equivalent to the completion of the metric space it corresponds to. Finally, we show that this unifomized hyperbolic filling is a uniform space. In summary, our construction generates hyperbolic fillings which satisfy the necessary conditions for it to serve its intended purpose of an analytical tool for further studies in [2, Chapters 9-13 ] or similar. As such, it can be regarded as an improvement to the reference model.

The ping-pong lemma was introduced by Klein in the late 1800s to show that certain subgroups of isometries of hyperbolic 3-space are free and remains one of very few tools that certify when a pair of group elements generate a free subgroup or semigroup. Quantitatively applying the ping-pong lemma to more general group actions on metric spaces requires a blend of understanding the large-scale global geometry of the underlying space with local combinatorial and dynamical behavior of the action. In the 1980s, Gromov publish a sequence of seminal works introducing several metric notions of non-positive curvature in group theory where he asked which finitely generated groups have uniform exponential growth. We give an overview of various developments of non-positive curvature in group theory and past results related to building free semigroups in the setting of non-positive curvature. We highlight joint work with Radhika Gupta and Kasia Jankiewicz and with Carolyn Abbott and Davide Spriano that extends these tools and techniques to show several groups with that act on cube complexes and many hierarchically hyperbolic groups have uniform exponential growth. / Mathematics

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface -we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs. A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility. Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).

The first part of this thesis deals with detonations in gaseous reactive mixtures. Various technological applications have been proposed involving detonations, particularly in the field of propulsion. However, it has been confirmed experimentally that detonations generally exhibit an unstable behaviour, leading to complicated flow structures. A thorough understanding of the evolution of detonation waves is needed before they can be used for propulsion purposes. Herein, we present the first detailed numerical study of three-dimensional structures in gaseous detonations. This study is based on a parallelized, unsplit, shock-capturing algorithm. We show that we can reproduce all types of detonations that have been observed experimentally. The advancements in the field of gaseous compressible reactive flows paved the way for the study of the significantly more complex phenomena that occur in the flow of two-phase, heterogeneous compressible reactive mixtures. In the second part of this thesis, we develop a new shock-capturing algorithm for the study of these flows. We first present a new numerical procedure for solving exactly the Riemann problem of compressible two-phase flow models containing non-conservative products. We then examine the accuracy and robustness of three known methods for the integration of the non-conservative products. The issue of existence and uniqueness of solutions to the Riemann problem is also discussed. Due to the ill-posedness of the Riemann problem of standard two-phase models, we present and analyze, in the third and last part of this work, a conservative approximation to reduced one-pressure one-velocity models for compressible two-phase flows that contain non-conservative products. Herein, we develop an exact Riemann solver for the proposed reduced model. Further, we investigate the structure of the steady two-phase detonation waves admitted by this model. Finally, we report on numerical simulations of the transmission of a purely gaseous detonation to heterogeneous mixtures. The effect of the solid particles on the structure of the resulting two-phase detonation is discussed in detail.

Spherical, Euclidean, and hyperbolic simplices can be characterized by the dihedral angles on their codimension-two faces. These characterizations analyze the Gram matrix, a matrix with entries given by cosines of dihedral angles. Hyperideal hyperbolic simplices are non-compact generalizations of hyperbolic simplices wherein the vertices lie outside hyperbolic space. We extend recent characterization results to include fully general hyperideal simplices. Our analysis utilizes the Gram matrix, however we use inversive distances instead of dihedral angles to accommodate fully general hyperideal simplices.For two-dimensional triangulations, an angle structure is an assignment of three face angles to each triangle. An angle structure permits a globally consistent scaling provided the faces can be simultaneously scaled so that any two contiguous faces assign the same length to their common edge. We show that a class of symmetric Euclidean angle structures permits globally consistent scalings. We develop a notion of virtual scaling to accommodate spherical and hyperbolic triangles of differing curvatures and show that a class of symmetric spherical and hyperbolic angle structures permit globally consistent virtual scalings.The double tetrahedron is a triangulation of the three-sphere obtained by gluing two congruent tetrahedra along their boundaries. The pentachoron is a triangulation of the three-sphere obtained from the boundary of the 4-simplex. As piecewise flat manifolds, the geometries of the double tetrahedron and pentachoron are determined by edge lengths that gives rise to a notion of a metric. We study notions of Einstein metrics on the double tetrahedron and pentachoron. Our analysis utilizes Regge's Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds.A notion of conformal structure on a two dimensional piecewise flat manifold is given by a set of edge constants wherein edge lengths are calculated from the edge constants and vertex based parameters. A conformal variation is a smooth one parameter family of the vertex parameters. The analysis of conformal variations often involves the study of degenerating triangles, where a face angle approaches zero. We show for a conformal variation that remains weighted Delaunay, if the conformal parameters are bounded then no triangle degenerations can occur.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / Let A[subscript]n(S) be a coefficient free commutative cluster algebra over a field K. A cluster automorphism is an element of Aut.[subscript]KK(t[subscript]1,[dot, dot, dot],t[subscript]n) which leaves the set of all cluster variables, [chi][subscript]s invariant. In Chapter 2, the group of all such automorphisms is studied in terms of the orbits of the symmetric group action on the set of all seeds of the field K(t[subscript]1,[dot,dot, dot],t[subscript]n). In Chapter 3, we set up for a new class of non-commutative algebras that carry a non-commutative cluster structure. This structure is related naturally to some hyperbolic algebras such as, Weyl Algebras, classical and quantized universal enveloping algebras of sl[subscript]2 and the quantum coordinate algebra of SL(2). The cluster structure gives rise to some combinatorial data, called cluster strings, which are used to introduce a class of representations of Weyl algebras. Irreducible and indecomposable representations are also introduced from the same data. The last section of Chapter 3 is devoted to introduce a class of categories that carry a hyperbolic cluster structure. Examples of these categories are the categories of representations of certain algebras such as Weyl algebras, the coordinate algebra of the Lie algebra sl[subscript]2, and the quantum coordinate algebra of SL(2).

Thesis advisor: Martin J. Bridgeman / Thesis advisor: Ian Biringer / The first part of this dissertation is on the quasiconformal homogeneity of surfaces. In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic surfaces in several cases, including the Torelli group, congruence subgroups, and pure cyclic subgroups. Further, we introduce a counting argument providing a possible path to exploring a uniform lower bound for the nonrestricted quasiconformal homogeneity constant across all closed hyperbolic surfaces. We then move on to identities on hyperbolic manifolds. We study the statistics of the unit geodesic flow normal to the boundary of a hyperbolic manifold with non-empty totally geodesic boundary. Viewing the time it takes this flow to hit the boundary as a random variable, we derive a formula for its moments in terms of the orthospectrum. The first moment gives the average time for the normal flow acting on the boundary to again reach the boundary, which we connect to Bridgeman's identity (in the surface case), and the zeroth moment recovers Basmajian's identity. Furthermore, we are able to give explicit formulae for the first moment in the surface case as well as for manifolds of odd dimension. In dimension two, the summation terms are dilogarithms. In dimension three, we are able to find the moment generating function for this length function. / Thesis (PhD) — Boston College, 2015. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

M.Sc., Faculty of Science, University of the Witwatersrand, 2011 / Continued fractions have been extensively studied in number-theoretic ways. In this text, we will illuminate some of the geometric properties of contin- ued fractions by considering them as compositions of MÄobius transformations which act as isometries of the hyperbolic plane H2. In particular, we examine the geometry of simple continued fractions by considering the action of the extended modular group on H2. Using these geometric techniques, we prove very important and well-known results about the convergence of simple con- tinued fractions. Further, we use the Farey tessellation F and the method of cutting sequences to illustrate the geometry of simple continued fractions as the action of the extended modular group on H2. We also show that F can be interpreted as a graph, and that the simple continued fraction expansion of any real number can be can be found by tracing a unique path on this graph. We also illustrate the relationship between Ford circles and the action of the extended modular group on H2. Finally, our work will culminate in the use of these geometric techniques to prove well-known results about the relationship between periodic simple continued fractions and quadratic irrationals.

Neste trabalho, mostramos que os bilhares hiperbólicos construídos originalmente por Bussolari- Lenci têm a propriedade de Bernoulli. Tais bilhares não satisfazem as técnicas standard de Wojtkowski-Markarian-Donnay-Bunimovich para bilhares focalizadores hiperbólicos, a qual requer que o diâmetro da mesa do bilhar seja de mesma ordem que o maior raio de curvatura ao longo da componente focalizadora. Nossa prova, utiliza um teorema ergódico local que nos diz que sob certas condições, existe um conjunto de medida total do espaço de fase do bilhar tal que cada ponto desse conjunto possui uma vizinhança contida (mod 0) em uma componente Bernoulli da aplicação do bilhar. / In this work, we show that hyperbolic billiards constructed originally by Bussolari-Lenci has the Bernoulli property. These billiards do not satisfy the standard Wojtkowski-Markarian-Donnay- Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane, which requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. Our proof employs a locally ergodic theorem which says that under a few conditions, there exists a full measure set of the billiard phase space such that each of its points has a neighborhood contained, up to a zero measure set, in one Bernoulli component of the billiard map.

Neste trabalho tem por objetivo a construção de conjugações suaves de pontos fixos hiperbólicos com condições de não ressonância. Por tanto, inicialmente são apresentados alguns conceitos básicos sobre espaços de Banach e alguns resultados de equações diferenciais ordinárias em espaços de Banach e sistemas dinâmicos, apresentamos o teorema de Hartman Grobman como motivação inicial de Linearização. Apresentamos também vários exemplos como motivação para estudar o Teorema de Sternberg para contrações hiperbólicas, o principal resultado estudado nesta dissertação para contrações hiperbólicas / This work has the objetive of building smooth conjugations of hyperbolic fixed points with non-resonance conditions. So, first we present some basics of Banach spaces and some results of ordinary differential equations in Banach spaces and dynamical systems, we present the theorem of Hartman Grobman as original motivation for linearization . We also present several examples as motivation to study the Sternberg theorem for hyperbolic contractions, as main result studied in this dissertation