Plan for this course

I plan to lecture about foundations and applications of stable homotopy theory.

As motivation, we may study examples of generalized (co)homology theories, such as bordism [Pontryagin, Thom] and $K$-theory [Bott, Atiyah-Hirzebruch].  Such functors become representable in the stable homotopy category [Boardman, Adams], which is triangulated and closed symmetric monoidal.  In the 1990s, this structure was found to arise as the homotopy category of a model category, in several ways, including $S$-modules, symmetric spectra and orthogonal spectra [Elmendorf-Kriz-Mandell-May, Hovey-Shipley-Smith, Lydakis, May-Mandell-Schwede-Shipley, ...].  We will follow a book project by Stefan Schwede, titled "Symmetric Spectra", as an introduction to these results.  Thereafter we will turn to the equivariant theory [Adams, Lewis-May-Steinberger, Mandell-May, Stolz, Hausmann, ...], with possible applications to the Kervaire invariant one problem [Hill-Hopkins-Ravenel].