In these lectures I'll try to give a whirlwind tour of this fascinating subject, stopping to smell at least some of the roses. I will also state a number of open problems along the way.

Lecture III will be a crash course on the Torelli group. The Torelli group is the subgroup of the mapping class group consisting of elements acting trivially on H_1 of the surface. The study of this group goes back to Nielsen (1918) and Magnus (1934), but it was for the most part ignored for the next 50 years. This topic is remarkably rich, and connects to 3-manifold theory, symplectic representation theory, combinatorial group theory, and algebraic geometry. In this lecture I will explain some of the main themes in this beautiful subject, including the important role it plays in the theory of mapping class groups.

Lecture #12555

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