Spectral algebra models of unstable v_n-periodic homotopy theory

Abstract

We give a survey of a generalization of Quillen-Sullivan rational homotopy theory which gives spectral algebra models of unstable v_n-periodic homotopy types. In addition to describing and contextualizing our original approach, we sketch two other recent approaches which are of a more conceptual nature, due to Arone-Ching and Heuts. In the process, we also survey many relevant concepts which arise in the study of spectral algebra over operads, including topological Andr\'e-Quillen cohomology, Koszul duality, and Goodwillie calculus.