In Lecture 11 and Lecture 12, we discussed examples of Riemann Surfaces and their relation with the Riemann Sphere . Today, we put things into perspective by considering the larger picture: we discuss tesselations of the upper half plane, covering spaces, and deck transformations.
Last time, we introduced the Fundamental Domain in terms of the extended complex plane . Felix Klein showed the existence of a function , invariant under , giving these isomorphisms. Today, we use Richard Dedekind‘s function to create similar functions .
Over the past several lectures, we have focused on triangle groups and ways to tesselate both the plane and the sphere. In order to generalize this, we will focus on ways to tesselate compact, connected Riemann surfaces. Today we’ll begin our discussion on how to do this by focusing on complex functions with lots of symmetries: we give a historical approach using elliptic modular functions.
Recall that the collection of Möbius transformations is the group
while the (ordinary) triangle group is the set
This homework set is meant to discuss the relationship with these two sets. This assignment is due Friday, September 20, 2013 at the start of class.
In the late 1700’s and early 1800’s, the French mathematician Augustin-Louis Cauchy (1789 – 1857) and German mathematician Bernhard Riemann (1826 – 1866) worked independently to create a theory of differentiating complex-valued functions . In order to define such functions, they realized they would need a way to define their domains of definition. In this lecture, we review some of the ideas surrounding Riemann Surfaces.
In the previous lecture, we discussed how to draw triangles in the plane and on the unit sphere such that the triangles tile these surfaces. In this lecture, we discuss triangle groups in more detail by focusing on discrete symmetries of the unit sphere.
Over the next couple of lectures, we will generalize cyclic groups and dihedral groups . Given positive integers , , and , formally define the abstract group
Such a group is called a Triangle Group, although it is also known as a von Dyck Group after the German mathematician Walther Franz Anton von Dyck (1856 — 1934). Today, we focus on why these are called “triangle groups.”