# Fusion system, p-completed classifying spaces, loops, (co)singularity categories

GRA2 - Groups, representations and applications: new perspectives

Cochains on the $p$-completed classifying space of a finite group (or more generally, a fusion system), and chains on its loop space, are Koszul dual objects with interesting representation theory. My intention is to talk about the singularity and cosingularity categories of these objects. I shall discuss fusion systems, linking systems, Chermak’s theorem, classifying spaces, Bousfield—Kan $p$-completion, the loop space, $A_\infty$ structures, Koszul duality, the derived category, the singularity category, the cosingularity category, the role of the nucleus, and some conjectures. I shall concentrate on some examples, coming from finite groups with cyclic Sylow $p$-subgroups, and the tame representation type cases in characteristic two.

This talk is part of the Isaac Newton Institute Seminar Series series.