# Mathematics Colloquia and Seminars

The permutation representation of $S_n$ on $C^{2n}$, and the corresponding quotient space $C^{2n}/S_n$, are examples of great interest in combinatorics and geometry. The corresponding action of $S_n$ on either the polynomial ring $C[x_1,y_1,...,x_n,y_n]$ or a Weyl algebra, and the resulting invariant subrings and crossed product rings, are algebraically interesting. In either context, there are cohomology rings encoding important information that have combinatorial descriptions. In this talk, we will describe the structures of certain Hochschild and orbifold cohomology rings for these examples, and mention more general results involving arbitrary finite group actions on vector spaces.