Let $G$ be a very nice $p$-adic analytic group; I have in mind examples such as $\mathsf{Gl}_n(\mathbb{Z}_p)$. The category of continuous $G$-modules has a very elegant theory of duality reflecting Poincare duality for $G$. We would very much like to extend this to stable homotopy theory where, in various contexts, it would help explain some deep structure we have seen so far only through computations. It is easy enough to define the dualizing objects, but then we are left with understanding them. It turns out that if we are only interested in finite subgroups of $G$ (which would be a serious start) we can get away with classical computations with Stiefel-Whitney classes. This is an on-going project with Agnes Beaudry, Mike Hopkins, and Vesna Stojanoska.

In the first talk, we introduced Tits' projection map for real hyperplane arrangements, and gave a purely combinatorial description of this map in the special case of braid arrangements.

In this talk, we will see how this projection map appears naturally in the Hopf theory for Joyal's category of combinatorial species, motivating us to generalize the definition of species from braid arrangements to Coxeter arrangements and further to hyperplane arrangements. This talk is based on the work of Aguiar and Mahajan.