Math 264X: Algebraic Cobordism

This is the class webpage for my topics class at Harvard (Spring ’20). The goal is to give an overview of Levine-Morel’s algebraic cobordism via various perspectives.

  1. Lecture 1: antecedents and overview (last updated January 31)
  2. Lecture 2: derived commutative rings (last updated Feb 5; exercises modified)
  3. lecture 3: derived schemes (last updated Feb 6)
  4. lecture 4: Chow homology and oriented Borel-Moore functors (last updated Feb 6; with appendix)
  5. lectures 5+6: Koszul derived subschemes, the Hurewicz theorem, quasi-smoothness and classification of derived structures (last updated Feb 15) (thanks to Xu Kai for pointing out an earlier gap in the proof of 0.0.2)
  6. Lecture 7: the definition of algebraic cobordism
  7. Guest lecture by Martin Speirs.
  8. Lecture 8: derived double points
  9. Lecture 9: derived blow-ups and formal group laws
  10. Lecture 10: Lazard’s theorem and Quillens theorem, constructing an FGL I
  11. Lecture 11: constructing an FGL II, derived double point relations


A reading list:

  1. Derived algebraic geometry by Bertrand Toën
  2. Jacob Lurie’s thesis
  3. SGA 6