UBC Topology Seminar: Dev Sinha

We present mod-two cohomology of both symmetric and alternating groups as (almost) Hopf rings.

As is being seen in subjects such as representation stability, in these settings one gains substantial insight by considering all cases together, binding them with some structure. For these group cohomologies, that structure is a transfer or induction product, akin to taking external tensor product of S_n and S_m representations and inducing up to S_{n+m}. This product along with cup product and a standard coproduct together define a ring object in the category of coalgebras in the setting of symmetric groups, and are close enough to such an object for practical purposes in the setting of symmetric groups.

Settings such as general linear groups over finite fields now beg for investigation, as do a number of questions internal to topology (e.g. Margolis homology, working towards Morava K-theory) and at the interface with algebra (e.g. how do analogous structures interface with current understanding of modular representation theory of symmetric groups).

We will aim for the talk to be accessible to algebraists. While topology will be mentioned throughout, it will be presented in a supporting role, familiar to topologists but treatable as a black box when necessary to algebraists.