Math 80430: Topics in topology: chromatic homotopy theory

Math 80430: Topics in topology: chromatic homotopy theory

Info:

Lecturer: Mark Behrens
Time and place: MW 11:00-12:15, Riley 200
Office: Hurley 287
E-mail address: mbehren1@nd.edu

Office hours:

By appointment (email me or talk after class)

References

Although I will not be following any textbook in particular, there are some good source materials that you can refer to.
  1. H. Margolis: Spectra and the Steenrod Algebra. djvu scan
  2. J.F. Adams, Stable Homotopy and Generalised Homology.
  3. A.K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257-281.
  4. D.C. Ravenel: Complex Cobordism and the Stable Homotopy Groups of Spheres ("Green Book"). Online Edition
  5. D.C. Ravenel: Nilpotence and Periodicity in Stable Homotopy Theory ("Orange Book"). Online Edition

Topics to be covered (subject to change!)

The stable homotopy groups of any finite complex admits a filtration, called the chromatic filtration, where the height n stratum consists of periodic families of elements. This filtration is intimately tied to the algebraic geometry of formal group laws, and via this connection computations in stable homotopy theory can be tied to certain computations in arithmetic geometry. Topics I plan to cover include:
  1. Brief review of stable homotopy theory
  2. Quillen's theorem, Complex cobordism, and BP
  3. Morava K-theory, E-theory, and stabilizer group
  4. Chromatic spectral sequence
  5. Nilpotence and Periodicity theorems of Devinatz-Hopkins-Smith
  6. TMF and EO_n
  7. telescope conjecture
  8. unstable v_n-periodic homotopy (time permitting)
  9. Lurie ambidexterity (time permitting)
Click here for notes and exercises.