Title: 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 André-Quillen cohomology, Koszul duality, and Goodwillie
calculus.