Continued fractions have been extensively studied in number theoretic ways. In this text we will consider continued fraction expansions with partial quotients that are in Z = f x : x 2 Zg and where = 2 cos( q ); q 3 and with 1 < < 2. These continued fractions are expressed as the composition of M obius maps in PSL(2;R), that act as isometries on H2, taken at 1. In particular the subgroups of PSL(2;R) that are studied are the Hecke groups G . The Modular group is the case for q = 3 and = 1. In the text we show that the Hecke groups are triangle groups and in this way derive their fundamental domains. From these fundamental domains we produce the v-cell (P0) that is an ideal q-gon and also tessellate H2 under G . This tessellation is called the -Farey tessellation. We investigate various known -continued fractions of a real number. In particular, we consider a geodesic in H2 cutting across the -Farey tessellation that produces a \cutting sequence" or path on a -Farey graph. These paths in turn give a rise to a derived -continued fraction expansion for the real endpoint of the geodesic. We explore the relationship between the derived -continued fraction expansion and the nearest - integer continued fraction expansion (reduced -continued fraction expansion given by Rosen, [25]). The geometric aspect of the derived -continued fraction expansion brings clarity and illuminates the algebraic process of the reduced -continued fraction expansion.

Let ΓM be the fundamental group of a knot or link complement M. The discrete faithful representation of ΓM into PSL2(C) has an associated quaternion algebra. We can extend this notation to other representations, which are encoded by the character variety X(ΓM). The generalization is the canonical quaternion algebra and can be used to find unifying features of irreducible representations, such as the splitting behavior of their associated quaternion algebras. Within this dissertation, we will determine properties of the canonical quaternion algebra for the Whitehead link complement and explore how the algebra can descend to quaternion algebras of the Dehn (d, m)-surgeries thereon. / Mathematics

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In 2004, Fisher constructed a map on a 2-disc that admitted a hyperbolic set not contained in any locally maximal hyperbolic set. Furthermore, it was shown that this was an open property, and that it was embeddable into any smooth manifold of dimension greater than one. In the present work we show that analogous results hold for flows. Specifically, on any smooth manifold with dimension greater than or equal to three there exists an open set of flows such that each flow in the open set contains a hyperbolic set that is not contained in a locally maximal one.

We study hyperbolic structures on the compression body C with genus 2 positive boundary and genus 1 negative boundary. We consider individual hyperbolic structures as well as special regions in the space of all such hyperbolic structures. We use some properties of the boundary hyperbolic structures on C to establish an interesting property of cusp shapes of tunnel number one manifolds. This extends a result of Nimershiem in [26] to the class of tunnel number one manifolds. We also establish convergence results on the geometry of compression bodies. This extends the work of Ito in [13] from the punctured-torus case to the compression body case.

Thesis advisor: Ian Biringer / We consider the space $\rootedH2$ of all complete hyperbolic surfaces without boundary with a basepoint equipped with the pointed Gromov-Hausdorff topology. Continuous paths within $\rootedH2$ arising from certain deformations on a hyperbolic surface and concrete geometric constructions are studied. These include changing some Fenchel-Nielsen parameters of a subsurface, pinching a simple closed geodesic to a cusp, and inserting an infinite strip along a proper bi-infinite geodesic. We then use these paths to show that $\rootedH2$ is path-connected and that it is locally weakly connected at points whose underlying surfaces are either the hyperbolic plane or hyperbolic surfaces of the first kind. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

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Certain fibered hyperbolic 3-manifolds admit a layered veering triangulation, which can be constructed algorithmically given the stable lamination of the monodromy. These triangulations were introduced by Agol in 2011, and have been further studied by several others in the years since. In the first part of this work, we obtain experimental results which shed light on the combinatorial structure of veering triangulations, and its relation to certain topological invariants of the underlying manifold. Among other things, our experimental results strongly suggest that typical veering triangulations are non-geometric, i.e., they cannot be realized as a union of isometrically embedded hyperbolic tetrahedra. In the second part, we prove that veering triangulations are in fact generically non-geometric. / Mathematics / Accompanied by two .py files. A Python interpreter is required to run a PY script in Windows.

This dissertation studies hyperbolic partial differential equations for Conservation Laws motivated by traffic control problems. New traffic models for multi-directional flow in two dimensions are derived and their properties studied. Control models are proposed where the control variable is a multiplicative term in the flux function. Control models are also proposed for relaxation type systems of hyperbolic PDEs. Existence of optimal control for the case of constant controls is presented. Unbounded and bounded feedback control designs are proposed. These include advective, diffusive, and advective-diffusive controls. Existence result for the bounded advective control is derived. Performance of the relaxation model using bounded advective control is analyzed. Finally simulations using Godunov scheme are performed on unbounded and bounded feedback advective controls. / Ph. D.