Nicolas Guès: A homotopy-theoretic approach to representation stability

Abstract: In a series of papers from the 2010s, Church, Ellenberg and Farb developed the notion of representation stability, aimed at understanding the asymptotic behavior of natural sequences of representations of the symmetric groups S_n, particularly those arising from homological invariants. Such phenomena typically occur for FI-modules, i.e., functors from FI (the category of finite sets and injections) to abelian groups.

In this talk, I will propose a homotopical refinement of representation stability: a derived version adapted to (co)FI-spaces. This viewpoint sheds light on how the phenomenon of representation stability can arise from the high (co)cartesianity of certain cubical diagrams. With this point of view, I will show how one can deduce polynomial growth results for the homotopy groups of ordered configuration spaces on closed manifolds, recover the best known bounds on their cohomology, and generalize stability results of Palmer concerning moduli spaces of submanifolds.