The structure of Euclidean, spherical, and hyperbolic geometries are compared, considering specifically postulates, curvature of the plane, and visual models. Implications for distance, the sum of the angles of triangles, and circumference to diameter ratios are investigated. / text

No description available.

The primary aim of this thesis is to construct explicit examples of four dimensional hyperbolic link complements. Using the theory of Kirby diagrams and Kirby calculus we set up a general framework that one can use to attack such a problem. We use this framework to construct explicit examples in a smooth standard S⁴ and a smooth standard S² x S². We then characterise which homeomorphism types of smooth simply connected closed 4-manifolds can admit a hyperbolic link complement, along the way giving constructions of explicit examples.

The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions. We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions. In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.

The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions. We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions. In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.

The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions. We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions. In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.

The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions. We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions. In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.

Recent work has demonstrated the impressive efficacy of computing representations in hyperbolic space rather than in Euclidean space. This is especially true for multi-relational data and for data containing latent hierarchical structures. In this work, we seek to understand why this is the case. We reflect on the intrinsic properties of hyperbolic geometry and then zero in on one of these as a possible explanation for the performance improvements --- projection. To validate this hypothesis, we propose our projected cone model, $\mathcal{PC}$. This model is designed to capture the effects of projection while not exhibiting other distinguishing properties of hyperbolic geometry. We define the $\mathcal{PC}$ model and determine all of the properties we need in order to conduct machine learning experiments with it. The model is defined as the stereographic projection of a cone into a unit disk. This is analogous to the construction of the Beltrami-Poincaré model of hyperbolic geometry by stereographic projection of one sheet of a two-sheet hyperboloid into the unit disk. We determine the mapping formulae between the cone and the unit disk, its Riemannian metric, and the distance formula between two points in the $\mathcal{PC}$ model. We investigate the learning capacity of our model. Finally, we generalize our model to higher dimensions so that we can perform representation learning in higher dimensions with our $\mathcal{PC}$ model. Because generalizing models into higher dimensions can be difficult, we also introduce a baseline model for comparison. This is a product space model, $\mathcal{PCP}$. It is built up from our rigourously developed, two-dimensional version of the $\mathcal{PC}$ model. We run experiments and compare our results with those obtained by others using the Beltrami-Poincaré model. We find that our model performs almost as well as their Beltrami-Poincaré model, far outperforming representation learning in Euclidean space. We thus conclude that projection indeed is key in explaining the success which hyperbolic geometry brings to representation learning.

No description available.

Thesis advisor: Ian Biringer / We are interested in proving the following statement: Given a 3-manifold M with boundary and a homeomorphism of the boundary f : ∂M → ∂M such that there is some power that extends to M, there is some k depending only on the genus g(∂M) and some l < k such that ƒᶩ extends to M. We will prove that the power needed to extend is not uniformly bounded with some examples, we will prove the statement is true if M is boundary incompressible and we will show that the general statement reduces to effectivising some technical results about pure homeomorphisms extending to compression bodies. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.