Motivic Homotopy Learning Seminar (Summer 2025)

Welcome to our seminar on Motivic Homotopy Theory, organized by Xingting Wang, Guanyu Li and me, based on the seminal works of Voevodsky, Morel, and others. This 10-week program introduces participants to the foundations and advanced developments of this exciting intersection of algebraic geometry and homotopy theory. We will be meeting over Zoom for all of our meetings.

Time: Tuesdays and Fridays, 3-4.15 pm (CST)

Zoom Information:   

Zoom link: https://lsu.zoom.us/j/97530641887?pwd=obPBM0BOV7DOiKcLxfqFGrqBRAO1VL.1

Meeting ID: 975 3064 1887

Passcode: 532008

Resources:

• Motivic Homotopy Theory Lectures at a Summer School in Nordfjordeid, Norway, August 2002. (The main resource for this seminar. These notes are very detailed and elaborated.)

• Notes on Homotopy and A1 homotopy,  Ben Williams, 2020. (Excellent notes, easy to follow through and very close to our main reference)

• A primer for unstable motivic homotopy theory, Benjamin Antieau, Elden Elmanto, 2016. (Expository article for foundations.)

• Lecture Notes on Motivic Cohomology, Carlo Mazza, Vladimir Voevodsky, Charles Weibel. (Classic notes.)

Structure:

We’re getting together twice a week to work through Voevodsky’s motivic homotopy theory lectures, aiming for a mix of serious math and relaxed, open conversation. Each session is about 1.5 hours, and one person will lead with a rough outline to keep things on track, but the real goal is group understanding, not perfect presentations. We’ll start with the basics from homotopy theory and algebraic geometry, then build up to motivic spaces, spectra, and cohomology. Expect lots of questions, examples, and moments where we pause to figure things out together. It's meant to be rigorous, but friendly, more of a chalkboard jam session than a formal course.

Here is the overall plan for the seminar.

Week 1-3 (Foundations in Topology)

Topological spaces, simplicial sets, Dold-Kan correspondence, Quillen model categories, Motivic spaces.

Week 4-6 (Foundations in Algebraic Geometry)

Sheaves, Schemes, Morphisms of schemes, Grothendieck topology.

Week 7-10 (Motivic Homotopy Theory Core)

Construction of motivic spaces, The Nisnevich Topology & Homotopy Sheaves, Eilenberg–MacLane Spaces in Motivic Context, Motivic Functors & Mapping Spaces, Stable Homotopy Category SH(k), Motivic Cohomology.