Syllabus and Course Structure

Basic Structure

• Lectures will be in person in MOOR 212 and simultaneously on Zoom (link available by request)
• For those attending in person, Masks are required during lecture
• Grading will be based on attendance and presentations (10-20 minutes long, towards the end of the semester)

An aspirational outline of the course is below (please don’t take this too seriously!)

Intro, basics of Chow groups

• Lecture 1: Intro, summary of the interaction between algebraic cycles and K-theory, higher analogs (motivic cohomology, higher K-theory), and applications
• Lecture 2: definition of Chow groups, pullbacks, pushforwards, fundamental classes, present the problem of reasonable pullbacks over closed imbeddings (intersections! how to pullback a class to itself?)

Vector bundles, chern classes, pullbacks

• Lecture 3: vector bundles and Chern classes / Segre classes
• Lecture 4: pullbacks via the normal bundle, refined Gysin homomorphisms, definition of Chow ring

Riemann-Roch

• Lecture 5: Intro to Riemann-Roch problem
• Lecture 6: Statement of Riemann-Roch, some hints at proofs (or special cases)

A Glimpse of motives

• Lecture 7: Correspondences, cohomology theories
• Lecture 8: Introduction to Chow motives

Classical Algebraic K-theory

• Lecture 9: Classical K-theory, $K_0, K_1$ and Meyer-Vietoris
• Lecture 10: Classical $K_2$, Milnor K-theory

The $K_0$ functor in Algebraic Geometry

• Lecture 11: Functorial properties of the $K_0$ functor, resolutions
• Lecture 12: Topological and Gamma filtrations, towards Chern classes
• Lecture 13: Riemann-Roch
• Lecture 14: Some relationships to Chow groups

Quillen K-theory

• Lecture 15: Quillen’s Q-construction, nerves, definition of higher K-theory
• Lecture 16: Localization, review of spectral sequences
• Lecture 17: Coniveau filtration and the BGQ spectral sequence
• Lecture 18: Working with the BGQ spectral sequence

The Merkurjev-Suslin Theorem

• Lecture 19: Review of aspects of Brauer groups, étale cohomology
• Lecture 20: Statement of the Merkurjev-Suslin theorem and its significance, relationship to Bloch-Kato, Milnor conjectures
• Lecture 21: Severi-Brauer varieties and their $K$-theory and geometry
• Lecture 22: Overview of the proof of the Merkurjev-Suslin theorem

Student Presentations

• Lectures 23-25