I will give an overview of the development of symmetric spectra, orthogonal spectra, and other monoidal model categories of spectra as well as some generalizations and applications. I then expect small groups to each focus on one of the various references listed below.

Reading List:

Foundations:

• M. Hovey, B. Shipley, J. Smith, Symmetric spectra, J. Amer. Math. Soc., 13 (2000), 149–208. (Only the first three sections are essential.) 

• M. Mandell, J. P. May, S. Schwede, B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001), 441–512. 
Generalizations and applications: 


• M. Hovey, Spectra and symmetric spectra in general model categories, Journal of Pure and Applied Algebra 165 (2001) 63–127. 

• T. Geisser and L. Hesselholt, Topological cyclic homology of schemes. Algebraic K-Theory (Seattle, WA, 1997), 41–87, Proc. Sympos. Pure Math., 67, Amer. Math. Soc., Providence, RI, 1999. (In particular, section 6.1) Available at: http://www-math.mit.edu/∼larsh/papers/008/gh.pdf

• B. Shipley, HZ-algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007) 351-379.

General reference:

• S. Schwede, Symmetric spectra, untitled book in progress. Available at: www.math.uni- bonn.de/people/schwede/SymSpec