## Persi Diaconis

A classical problem of enumerative combinatorics counts the proportion of permutations with j fixed points (Monmort 1708). In joint work with Jason Fulman and Bob Guralnick, we study the same question for other actions of the symmetric group. We prove that, except for the action on k-sets, almost all permutations have no fixed points in any action. The underlying mathematics involves the O'nann-Scott classification of maximal subgroups and a variety of generating function identities.

## Jameel Al-Aidroos

There are spaces $\mathcal{M}_{g,n}$ whose points parameterize smooth genus $g$ algebraic curves with $n$ distinct marked points. These spaces admit certain beautiful compactifications $\overline{\mathcal{M}}_{g,n}$, to which we associate a system of rings $R^*(\overline{\mathcal{M}}_{g,n})$ called the tautological rings of the moduli spaces of pointed curves. The algebraic structure of the tautological rings encodes some of the geometric data of the moduli spaces, and thus geometric information about algebraic curves. For example, it encodes in a remarkably combinatorial way the intersection data for some important subspaces of moduli space. Most of this talk will be an introduction to the moduli spaces of curves and their tautological rings. I'll talk mainly about the combinatorics of these structures in some very basic but enlightening examples. I'll also mention the overarching conjectures about the tautological rings and some recent progress in that direction.