Algebraic Geometry and Combinatorics Seminar: Homotopical combinatorics and equivariant derived algebraic geometry

Abstract: Equivariant derived algebraic geometry (EDAG) seeks a “spectral algebraic geometry with norms,” where the basic functions are genuine G-E_∞ rings (from equivariant stable homotopy theory) and descent must remember restriction, transfer, and norm data. A central obstacle is conceptual and technical: how do we control which norms are present so that geometry is flexible enough to glue, but rigid enough to compute? This talk proposes transfer systems as the correct combinatorial lens. A transfer system on a finite group G is a conjugation-invariant refinement of subgroup inclusion that records exactly which norm maps are allowed; it governs the operadic realizability of so-called N_∞ structures and, by extension, the kinds of “affine pieces” EDAG can hope to assemble.
After briefly motivating EDAG in as accessible a fashion as possible, I will turn to the combinatorics of transfer systems, which has deep connections with Catalan numbers, the Tamari lattice, and the theory of factorization systems on posets. Much of this work is joint with Angélica Osorno and our Reed College undergraduate collaborators.

Host: Martha Precup