Galois 2023 – Covering Theory in Algebra and Geometry
Overview:
This is an IAP 2023 topics course in the theory of Galois correspondences, a framework for studying number theory, algebraic topology, complex analysis, and algebraic geometry.
The course is over, but contact us if you have any questions!
We will first discuss the classical theory of (infinite) Galois extensions. Then, we will explore how the main theorem of Galois theory has an analog in the theory of covering spaces and fundamental groups. Finally, we will introduce some algebraic geometry to formulate modern versions of these correspondences via étale fundamental groups.
Notes:
Click here to access the lecture notes! These are still being updated!
Roughly, we followed Szamuely’s Galois Groups and Fundamental Groups, but with an algebraic focus.
Syllabus:
| Date | Topic | References |
|---|---|---|
| 01/09 | Rushil: Algebraic and Galois extensions | Szamuely 1.1-1.2 |
| 01/10 | Yuyuan: Intro to category theory | Szamuely 1.4, Vakil 2.2 Yuyuan’s notes |
| 01/11 | Rushil: Infinite Galois theory and fiber functors | Szamuely 1.3,1.5, Keith Conrad’s notes |
| 01/12 | Yuyuan: Covering spaces and the monodromy action | Szamuely 2.1-2.3 Yuyuan’s notes |
| 01/17 | Rushil: Construction of the universal cover | Szamuely 2.4-2.5 |
| 01/18 | Yuyuan: Introduction to schemes | Szamuely 5.1 Yuyuan’s notes |
| 01/19 | Guest lecture by Hari Iyer: Étale morphisms of schemes |
Szamuely 5.2 |
| 01/20 | Yuyuan: Galois theory for étale covers | Szamuely 5.3 Yuyuan’s notes |
| 01/24 | Rushil: Étale fundamental groups | Szamuely 5.4-5.5 |
| 01/25 | Guest lecture by Daniel Santiago: The Riemann-Hilbert correspondance |
Szamuely 2.6-2.7 |
| 01/26 | Rushil: The Weil conjectures and Étale Cohomology | Milne 3-9, 26-27 |
Questions:
If you have any questions, feel free to contact either of us!
- Rushil Mallarapu — rushil_mallarapu@college.harvard.edu
- Yuyuan Luo — lyuyuan@mit.edu