Junior Algebra and Representation Theory Seminar

For any manifold, we can assign its mapping class group, that is, the group of its diffeomorphisms modulo isotopies. Although such a group can be studied for manifolds of any dimension, the mapping class groups of surfaces draw special attention. They are isomorphic to the outer automorphism groups of $\pi_1(S)$ and have many properties similar to lattices in semisimple Lie groups, as well as connections with the theory of moduli of curves.

One of the most important parts of the research on mapping class groups is the study of their representation. In particular, in the general situation, we still don't know if they have a faithful representation into $\operatorname{GL}_n(\mathbb{C})$.

In my talk, I will show basic facts about mapping class groups and briefly describe a few known methods for constructing their representations and discuss their properties. In particular, I will present recent results classifying low-dimensional representations of the mapping class group.