How homotopy theory helps algebraic geometry
In spring 2025, I'm teaching a Master's course (M2) at Sorbonne.
- Chapter 1 - Introduction
- Chapter 2 - Vector bundles
- Chapter 3 - Obstruction theory
- Chapter 4.1 - K-theory intro
- Chapter 4.2 - Algebraic K-theory
- Chapter 5 - Motivic spaces
- Chapter 6.1 - Milnor K-theory and its friends
- Chapter 6.2 - Homotopy modules & Co
- Chapter 7 - Motivic obstruction theory
- Chapter 8 - Rigidity theorem
- Chapter 9 - Motivic cohomology (extended version)
Helpful sources:
- A. Asok "Algebraic geometry from an A1-homotopic viewpoint"
- T. Brazelton "Unstable motivic homotopy theory"
- A. Asok, J. Fasel "Vector bundles on algebraic varieties"
- M. Gallauer "Infinity-categories: a first course"
- J. Lurie "On infinity-topoi"
- A. Suslin "On the K-theory of algebraically closed fields"
- C. Mazza, V. Voevodsky, C. Weibel
"Lecture notes on motivic cohomology"
"A wise man loves to learn, and a fool loves to teach."
Anton Chekhov, From the diaries
Algebraic K-theory and Motivic cohomology
In spring 2023, I taught a Master's course (M2) at Sorbonne.
- Chapter 0 - Introduction
- Chapter 1 - K_0 of rings
- Chapter 2 - K_1 of rings
- Chapter 3 - K-theory space
- Chapter 4 - K_0 vs Chow groups
- Chapter 5 - K-theory of schemes
- Chapter 6 - Rigidity theorem
- Chapter 7 - Motivic cohomology
- Extra lecture by Elden Elmanto
Helpful sources:
- M. Hoyois’ lecture course on algebraic K-theory
- A. Khan’s lecture course on algebraic K-theory and intersection theory
- F. Binda and A. Khan’s lecture course on Grothendieck-Riemann-Roch theorem
- D. Nardin's lecture course on Stable homotopy theory
- S. Kelly "User's guide to Voevodsky's correspondences"
- D. Grayson "Motivic spectral sequence"
- T. Bachmann "Algebraic K-theory from the viewpoint of motivic homotopy theory"
Essays & blogposts
Send me your favourite ones!
- G. Sanderson "Math poetry"
- L. Pierce "Paths to math"
- R. Vakil "For potential PhD students"
- W. Thurston "On proof and progress in mathematics"
- W. Thurston "What's a mathematician to do?"
- J. Littlewood "The mathematician's art of work"
- W. Gowers "The two cultures of mathematics"
- P. Lockhart "A mathematician's lament"
- T. Tao "Does one have to be a genius to do math?"
- T. Tao "On time management"
- T. Tao "Enjoy your work"
- "Steps to solving a math problem"