## Lecture 13: Wednesday, September 18, 2013

In Lecture 11 and Lecture 12, we discussed examples of Riemann Surfaces $X$ and their relation with the Riemann Sphere $X(1) = SL_2(\mathbb Z) \backslash \mathbb H^\ast \simeq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R)$. Today, we put things into perspective by considering the larger picture: we discuss tesselations of the upper half plane, covering spaces, and deck transformations.

## Lecture 12: Monday, September 16, 2013

Last time, we introduced the Fundamental Domain $X(1) = SL_2(\mathbb Z) \backslash \mathbb H^\ast \simeq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R)$ in terms of the extended complex plane $\mathbb H^\ast = \mathbb H^2 \cup \mathbb P^1(\mathbb C)$. Felix Klein showed the existence of a function $J: \mathbb H^\ast \to \mathbb P^1(\mathbb C)$, invariant under $SL_2(\mathbb Z)$, giving these isomorphisms. Today, we use Richard Dedekind‘s function $\eta: \mathbb H^\ast \to \mathbb P^1(\mathbb C)$ to create similar functions $\mathbb H^\ast \to \mathbb P^1(\mathbb C)$.

## Lecture 11: Friday, September 13, 2013

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.

## Homework Assignment 2

Recall that the collection of Möbius transformations is the group
$\text{Aut} \bigl( \mathbb P^1(\mathbb C) \bigr) = \left \{ \gamma: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) \, \biggl| \, \gamma(z) = \dfrac {a \, z + b}{c \, z + d} \ \text{for some} \ \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \in GL_2(\mathbb C) \right \},$
while the (ordinary) triangle group is the set
$D(m,n,k) = \left \langle \gamma_0, \, \gamma_1, \, \gamma_\infty \, \bigl| \, {\gamma_0}^m = {\gamma_1}^n = {\gamma_\infty}^k = \gamma_0 \, \gamma_1 \, \gamma_\infty = 1 \right \rangle.$
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.

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## Lecture 10: Wednesday, September 11, 2013

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 $f: \mathbb C \to \mathbb C$. 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.

## Lecture 9: Monday, September 9, 2013

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.

## Lecture 8: Friday, September 6, 2013

Over the next couple of lectures, we will generalize cyclic groups $Z_n$ and dihedral groups $D_n$. Given positive integers $m$, $n$, and $k$, formally define the abstract group
$D(m,n,k) = \left \langle \gamma_0, \, \gamma_1, \, \gamma_\infty \ \biggl| \ {\gamma_0}^m = {\gamma_1}^n = {\gamma_\infty}^k = \gamma_0 \, \gamma_1 \, \gamma_\infty = 1 \right \rangle.$
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.”

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