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Friday, November 13, 2015

**Abstract:** The mapping class group of a closed surface $S$, denoted by $Mod(S)$, is the collection of all orientation-preserving homeomorphisms of $S$ up to homotopy. We will begin by explicitly computing the mapping class group of the torus $T^2$ and see that there are three types of elements in $Mod(T^2)$: finite order, powers of Dehn twists, and Anosov. We will then look at closed surfaces of genus $g \geq 2$ and learn that a similar classification statement (known as the Nielsen-Thurston classification) holds. I hope to convince you that the Nielsen-Thurston classification is meaningful. After all, any small child can arbitrarily place objects into bins and call the result a classification. Were Nielsen and Thurston just a pair of toddlers? Come to my talk and find out.