WiSe 26/27: Motivic homotopy theory

Lectures: Wednesday and Thursday 8-10 M102
Exercises: Wednesday 10-12 M103

Motivic homotopy theory was invented by F. Morel and V. Voevodsky in the late 90s in order to “do homotopy theory” with schemes. This course will introduce motivic homotopy theory from a modern perspective. Topics will include:

• 1) Categorical preliminaries (presentable categories, Bousfield localization, sheaves)
• 2) Algebro-geometric preliminaries (smoothness, blowups, the tubular neighborhood theorem)
• 3) The purity and localization theorems
• 4) Geometric models for classifying spaces
• 5) The A¹-homotopical classification of vector bundle
• 6) Motivic spectra and the stable A¹-connectivity theorem
• 7) Examples: algebraic K-theory, hermitian K-theory, motivic cohomology
• 8) Oriented cohomology theories and algebraic cobordism
• 9) The six-functor formalism

References

The original article of Morel and Voevodsky (the foundational material is now largely outdated) and their respective ICM addresses::

• F. Morel and V. Voevodsky, A¹-homotopy theory of schemes, 1999
• V. Voevodsky, A¹-homotopy theory, 1998
• F. Morel, A¹-algebraic topology, 2006

A useful “primer” covering the basic theory, examples and some applications:

• B. Antieau and E. Elmanto, A primer for unstable motivic homotopy theory, 2016