A Stable Homotopy Invariant for Legendrians in Jet Bundles

Symplectic geometry is anticipating a gold rush of data to arise by lifting "abelian group"- or "chain complex"-valued invariants to the level of "spectra" (also known as stable homotopy types). But the community has not yet struck gold in all the places it wants. In this talk, I'll talk about (very) recent success arising from "generating family" invariants, where we have actually managed to lift rich chain-complex-valued invariants to the level of spectra. I will try to give an overview of central concepts such as Legendrians, jet bundles, spectra, et cetera, and if time allows (which it will not) I will try to discuss a spectrally enriched cocategory of generating families, conjecturally expected to encode various Koszul duals to a spectral lift of a famous invariant of Legendrians. This is joint work with Lisa Traynor.