Course content: This course will focus on the symmetry groups of manifolds and cell complexes. We will look at both the geometric and algebraic properties of the mapping class group, which can roughly be thought of as the group of symmetries of a surface. One of the ways in which we will study the group is by designing a certain cell complex on which it acts by isometries. This principle has wide generalizations: one can derive a great deal of information about a group by constructing a geometric object for the group to act on.

In the spring semester, we will focus our attention on the braid groups, which are a special class of mapping class groups. In particular, we will study the representation theory of these groups, with connections to invariants of knots and links.

Textbook:

• A Primer on Mapping Class Groups, by Benson Farb and Dan Margalit.

• Automorphisms of Surfaces after Nielsen and Thurston, by Andrew Casson and Steven Bleiler
• Braid Groups, by Christian Kassel and Vladimir Turaev

Grading: Grades will be assigned based on homework and a presentation toward the end of the semester.