MthSc 481 (Topics in Geometry and Topology), Spring 2012

MthSc 481 (Topics in Geometry and Topology), Spring 2012

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty." --Bertrand Russell

About the class

This course will be an introduction to selected mainstream topics in geometry and topology. Special focus will be given to understanding hyperbolic geometry and its symmetries. Hyperbolic geometry is similar to standard high-school geometry, but with a few subtle differences (notably as the absence of the "parallel postulate") that lead to deep consequences and beauty. At first, it may seem exotic, but it is just as prevalent and important to mathematics and physics as its familiar cousin, Euclidean geometry. We will begin with a quick review of basic Euclidean geometry, and then move into few advanced aspects including its isometries, conformal maps, and circle inversions. These will provide a foundation for an introduction to hyperbolic geometry. After defining it formally, we will examine some of its basic constituents such as its isometries and trigonometry. This will lead us to the general linear group GL(n,R), the projective linear group PGL(n,R), and Möbius transformations. We will study the isometries of the hyperbolic plane in more detail, and classify them by type: elliptic, parabolic, and hyperbolic. Next, we will turn our attention to both orientable and non-orientable surfaces, such as spheres, tori, and projective planes. This will be done in the setting of simplicial complexes, and in particular, quotient maps of side-paired polygons. Finally, we will learn about the invariance of the Euler characteristic, and use this to give the classification of two-dimensional surfaces. If time permits, we will learn about the celebrated Gauss-Bonnet theorem which provides a remarkable link between the geometry and the topology of a surface.



Class lecture notes