Suppose you want to stir a pot of soup with several spoons. What is the most efficient way to do this? Thurston's theory of surface homeomorphisms gives us a concrete way to analyze this question.

First, we will explain how the different mixing patterns can be encoded via mathematical braids. Then, to each mixing pattern we can associate a real number called the entropy. This number gives a first approximation to the amount of mixing that is happening.

We will start from scratch with a simple example, state the Nielsen-Thurston classification of surface homeomorphisms, and give some open questions about entropies of general surface homeomorphisms.