## Homework Assignment 3

Recall that the upper half plane $\mathbb H^2$ consists of complex numbers $z = x + i \, y$ with $y > 0$. Let $\overline{\mathbb D} = \left \{ x + i \, y \in \mathbb C \, \bigl| \, x^2 + y^2 \leq 1 \right \} = \mathbb D \cup \partial \mathbb D$ denote the closed unit disk as the union of the unit disk $\mathbb D$ with its boundary $\partial \mathbb D = \left \{ x + i \, y \in \mathbb C \, \bigl| \, x^2 + y^2 = 1 \right \} \simeq S^1(\mathbb R)$. This assignment is meant to explain the pictures at the Wolfram Demonstrations site as well as on Wikipedia:

This assignment is due Friday, October 18, 2013 at the start of class.

## “Separated Belyĭ Maps” by Zachary Scherr and Michael E. Zieve

Zachary Scherr and his graduate advisor Michael E. Zieve have a new paper on the ArXiv entitled “Separated Belyĭ Maps”.

## Lecture 15: Monday, September 23, 2013

Today we discuss how some familiar objects — namely the circle, the sphere, the torus, and even elliptic curves — are each examples of non-singular algebraic curves which are also compact, connected Riemann surfaces.

## Lecture 14: Friday, September 20, 2013

Eventually, we wish to show that every compact, connected Riemann surface $X$ is a nonsingular algebraic curve. Today, we discuss what is means to be a “nonsingular algebraic curve” using Dedekind Domains.

## “An Elementary Approach to Dessins d’enfants and the Grothendieck-Teichmüller Group” by Pierre Guillot

Pierre Guillot has a new paper on the ArXiv entitled “An Elementary Approach to Dessins d’enfants and the Grothendieck-Teichmüller Group”.