MATH266 Motivic homotopy theory (Fall 2024)

• Lectures: TTh, 10:30-11:45, SC507
• Office hours: Th, 2-3pm, SC242a
• My email: brazelton@math.harvard.edu
• Notes: GitHub
• Canvas

Other references

• A diagram relating various topologies on varieties to one another
• A brief discussion of torsors by Alex Youcis
• $\infty$-categories, a first course, lecture notes by Martin Gallauer

Overview

This fall we'll be learning unstable motivic homotopy theory with an eye towards computations in the unstable setting, and the development of motivic obstruction theory. Some topics we hope to cover (subject to change) include:

• Chow groups, Chow-Witt groups, $I^j$-cohomology
• Gersten and Rost-Schmid complexes
• The Bloch-Ogus spectral sequence
• Euler classes in the motivic setting
• Homotopy purity
• Affine representability

Background

At the very least, we will assume a strong handle of homological algebra, commutative algebra, category theory, and basic algebraic geometry. Knowledge of homotopy theory will be helpful mostly for intuition but we won't assume everyone is a homotopy theorist. We might use some of the language of infinity-categories, so familiarity with this is a plus.

Grading

Students interested in taking this class for credit will be expected to write a small expository paper on an extra topic, due before the end of the semester.