Tuesday 11 October 2022, 16:00 - 17:00 in HG03.054
Paul Goerss (Northwestern)
Refining duality from algebra to topology
Poincaré Duality for compact manifolds is a model for duality in a variety of contexts. For example, Serre wrote down a very similar looking duality in the study of compact p-adic analytic groups. These groups, the analog of Lie groups over the p-adic numbers, arise naturally in representation theory and arithmetic geometry. Atiyah had noticed that one way to prove Poincaré Duality is to use ideas from vector bundles and a much more basic duality theory applicable to all finite simplicial complexes. Some years ago, Hopkins and, independently, Clausen noticed it would be very useful to import Atiyah’s ideas into the context of p-adic analytic groups. The problem was that there were two candidates for Atiyah duality in this context. It was natural to hypothesize that they were the same, as this would give very conceptual explanations for results only known by very hard calculation. The point of this talk is to make a story out of these ideas, give further context, and to explain the state of the art.