A Hyperbolic Take of Non-Euclidean Geometry

Abstract: A discussion of the history of non-Euclidean geometry broadly answering the question: "Why hyperbolic geometry?" First, Riemann surfaces and their canonical geometries will be introduced. In particular, the model geometries for surfaces will come out of the uniformization theorem. Then, the history of non-Euclidean geometry will be discussed through model theory. Different models of spherical and hyperbolic geometry will be presented and results between the different geometries will be illustrated. In particular, the differences in isometries of Euclidean and hyperbolic geometry will be demonstrated. This leads to a discussion of the classification of surfaces via "pants decomposition" and its 3-manifold analogue, Thurston's geometrization conjecture. Finally, the unique problems and questions of hyperbolic 3-manifold theory will be touched on.