Abstract

 In 1999, Mark Mahowald and Charles Rezk introduced a class of spectra which are particularly amenable to understanding using the classical Adams spectral sequence, called fp-spectra. As first described by Rognes, these play a pivotal role in generalizing Quillen-Lichtenbaum conjectures to the setting of ring spectra.The Quillen-Lichtenbaum conjectures were proven for truncated Brown-Peterson spectra by Dylan Wilson and Jeremy Hahn in 2021, who in this way discovered the first highly non-obvious example of an fp-spectrum in the form of algebraic K-theory. This led them to ask about a general structure result for fp-spectra, and to conjecture that they can all be built out of particularly simple ones.I will talk about recent joint work with David Lee where we prove a monochromatic analogue of the Hahn-Wilson conjecture, and deduce the original conjecture at height one.