xxx INTRODUCTION

• Rezk, Charles. Notes on the Hopkins–Miller theorem. Homotopy theory via

algebraic geometry and group representations (Evanston, IL, 1997), 313–366,

Contemp. Math., 220, Amer. Math. Soc., Providence, RI, 1998.

• Schwede, Stefan. Book project about symmetric spectra. Book preprint. Avail-

able at

http://www.math.uni-bonn.de/people/schwede/SymSpec.pdf

Stacks:

• Complex oriented cohomology theories and the language of stacks. Course

notes for 18.917, taught by Mike Hopkins (1999), available at

http://www.math.

rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf

• Naumann, Niko. The stack of formal groups in stable homotopy theory. Adv.

Math. 215 (2007), no. 2, 569–600.

• The stacks project. Open source textbook, available at

http://stacks.math.columbia.

edu

• Vistoli, Angelo. Grothendieck topologies, fibered categories and descent theory.

Fundamental algebraic geometry, 1–104, Math. Surveys Monogr., 123, Amer.

Math. Soc., Providence, RI, 2005.

Spectral sequences:

• Hatcher, Allen. Spectral sequences in algebraic topology. Book preprint. Avail-

able at

http://www.math.cornell.edu/˜hatcher/SSAT/SSATpage.html

• McCleary, John. A user’s guide to spectral sequences. Second edition. Cam-

bridge Studies in Advanced Mathematics, 58. Cambridge University Press,

Cambridge, 2001. xvi+561 pp. ISBN: 0-521-56759-9

• Weibel, Charles A. An introduction to homological algebra. Cambridge Studies

in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994.

xiv+450 pp. ISBN: 0-521-43500-5; 0-521-55987-1

The contents of this book span four levels: the first five chapters (elliptic co-

homology, elliptic curves, the moduli stack, the exact functor theorem, sheaves in

homotopy theory) are more elementary, classical, and expository and we hope will

be tractable for all readers and instructive or at least entertaining for all but the

experts; the next three chapters (the Hasse square, the local structure of the moduli

stack, obstruction theory) are somewhat more sophisticated in both content and

tone, and especially for novice and intermediate readers will require more deter-

mination, patience, and willingness to repeatedly pause and read other references

before proceeding; the last five chapters (from spectra to stacks, string orientation,

the sheaf of ring spectra, the construction, the homotopy groups) are distinctly

yet more advanced, with Mike Hopkins’ reflective account of and perspective on

the subject, followed by an extensive technical treatment of the construction and

homotopy of tmf ; finally the three classic manuscripts (Hopkins–Miller, Hopkins–

Mahowald, Hopkins) illuminate the original viewpoint on tmf —a careful reading

of them will require serious dedication even from experts.

In addition to the references listed above, we encourage the reader to consult

the following sources about tmf more broadly: