The equivalence of differential graded modules and HZ-module spectra, applications, and generalizations

This document consists of lecture notes for four lectures given by Brooke Shipley at the 2017 Young Topologists Meeting at the Swedish Royal Institute of Technology (KTH) in Stockholm, Sweden. The original conference abstract for the lecture series is below. This sequence of lectures will explore the connections between the differential graded world and the spectral world. There will first be a brief introduction to model categories, stable homotopy, and symmetric spectra. Then we will discuss the equivalence between the homotopy theories of HZ-module (respectively, algebra) spectra and of differential graded modules (respectively, algebras or DGAs), where HZ here is the Eilenberg-Mac Lane spectrum associated with ordinary homology. We will then use this comparison to develop algebraic models of rational (equivariant) stable homotopy theories and to define topological equivalences of DGAs. In both of these applications we will discuss current on-going work; in the latter case, this uses a variant of Goerss-Hopkins obstruction theory to compute topological equivalences. As time permits we will also discuss extensions of the original comparison to the commutative (or E-infinity) case and to co-modules and co-algebras. These notes were typed post-mortem from my original handwritten notes, and I am certain that the transcription introduced errors. Additionally, I have attempted to cite original sources for ease of reference, but I may have accidentally left some out. Please let me know if you find any errors or omissions by sending me an email at [email protected].