Summary

Serre’s classical work in 1955 established a dictionary between finitely generated projective modules over a ring and algebraic vector bundles over the affine scheme associated to the ring. This motivated transport of questions and techniques for topological vector bundles to the algebraic setting. For example: How do algebraic bundles decompose into smaller bundles? Are they representable by homotopy classes of maps into an infinite Grassmannian, as they are in the topological setting?

One of the first instances of such a question is Serre’s problem – over affine space over a field, is every algebraic vector bundle trivial? This question motivated techniques at the genesis of algebraic K-theory, and hints at the idea of A¹-invariance, the key idea underlying motivic homotopy theory.

Recent work of Asok, Fasel, Hopkins, Hoyois, Wendt, and others demonstrates that vector bundles can be studied using motivic methods and that the motivic setting is intermediate between algebraic and topological vector bundles. Methods from motivic obstruction theory can be used to resolve longstanding questions about classifying algebraic vector bundles.

In this seminar, we will work through classical results in this area, including Quillen’s solution to Serre’s problem, aspects of the Bass-Quillen conjecture, and Suslin cancellation. We will build towards modern results, including the affine representability theorem of Asok-Hoyois-Wendt extending earlier work of Morel, and obstructions for lifting topological bundles to algebraic vector bundles following Asok-Fasel-Hopkins.

Here is the seminar syllabus.

Participant expectations

Participants will take turns giving presentations. The seminar organizers will make sure that presenters have written source material upon which to base their presentations. Participants are expected to attend regularly.

Participants are encouraged to skim the source material being presented each week. They are also encouraged to participate in the private shared online workspace.

As a general rule, students are expected to obtain 1-2 independent study credits from their home universities for their participation in the seminar. This requirement is waived on an individual basis for students whose home universities do not have formal independent study opportunities.

Organizers

The seminar is organized by Morgan Opie (UCLA) and Thomas Brazelton (Harvard University). They are assisted by graduate student assistants Scotty Tilton (UC San Diego) and Millie Deaton (University of Kentucky).

Application

Please submit the application form to apply. Applications received by August 26 will receive full consideration. Contact Dan Isaksen for inquiries about applications.

Schedule

The seminar meets regularly at 1:00pm Eastern time on Thursdays, starting September 7 and ending December 14. There will be no meeting on November 23 during the week of Thanksgiving (US). Office hours/informal discussion/problem sessions will meet regularly at 1:00pm Eastern time on Tuesdays.