notes

Link to 2015 notes

Lecture notes for Math 549, Spring 2016, in chronological order

• January 14. Statement of the Kervaire invariant theorem and outline of the proof, I

• January 20. Some definitions and examples

• January 21. Representations rings, RO(G)-graded homotopy groups, the construction on the spectrum Ω and the slice filtration.

• January 25. Das on adjoint functors and related topics.

• January 27. Das on monads, limits and colimits.

• January 28. Das on ends/coends and Qu on Kan extensions.

• February 1. Qu on Kan extensions and monoidal categories.

• February 3. Yigit on enriched categories.

• February 4. Doug on the definition of spectra.

• February 22. Zhu on model categories.

• February 24. Zeng on the small object argument and cofibrantly generated model categories.

• February 25. Zeng on the homotopy in a model category, followed by Doug on MC structures in diagram categories.

• February 29. McTague on derived functors and homotopical categories.

• March 2. McTague on homotopical categories and flat maps.

• March 3. McTague on flat maps followed by Doug on a model category structure for G-spectra.

• March 21. Doug on Bousfield localization.

• March 23. Doug on localizing subcategories and the slice tower.

• March 24. Zou on the positive complete model category structure on G-spectra.

• March 28. Zeng on the periodicity theorem I.

• March 30. Zeng on the periodicity theorem II.

• March 31. Doug on the gap theorem.

• April 11. Doug on the geometric fixed points.

• April 13. Zhu on the homotopy limits and colimits.

• April 14. Doug on the slice filtration revisited.

• April 18. Zou on the detection theorem.

• April 20. Zou finishes the detection theorem and Doug talks about commutative ring spectra.

• April 21. Doug talks more about Thom spectra and commutative ring spectra.

• April 25. Doug talks more about commutative ring spectra and the Smith positivity condition.

• April 27. Doug summarizes the course.