Course Information
This course will develop the theory of Riemann surfaces and algebraic curves. We will then study Belyi maps (three-point branched covers of the complex projective line) and their connection with the absolute Galois group and other parts of arithmetic geometry. We will use the following references.- Girondo, González-Diez - Introduction to Compact Riemann Surfaces and Dessins d’Enfants
- Miranda - Algebraic Curves and Riemann Surfaces
- Jones, Wolfart - Dessins d'Enfants on Riemann Surfaces
- Forster - Lectures on Riemann Surfaces
- Rotman - An Introduction to Algebraic Topology
Weekly homework assignments will be assigned and posted below. They will be "due" in one week, but late homework will be accepted. That said, these exercises are meant to help you learn the material, so it is strongly encouraged to do them in a timely manner!
Cooperation on homework is permitted (and encouraged!), but if you work together, do not take any paper away with you--in other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, you must write the solution up on your own. Please credit your collaborators on your assignments.
Daily Schedule
| Date | Topic | Notes | Homework |
|---|---|---|---|
| (M) Feb 22 | Review of complex analysis, Riemann surfaces | Lecture 1 | |
| (W) Feb 24 | Complex algebraic curves as Riemann surfaces | Lecture 2 |
HW 1: [TeX], [PDF] |
| (M) Mar 1 | Morphisms of Riemann surfaces; automorphisms of PP^1 | Lecture 3 | |
| (W) Mar 3 | Automorphisms of HH and DD; order of vanishing and ramification | Lecture 4 |
HW 2: [TeX], [PDF] |
| (T) Mar 9 | Local normal form; degree of a morphism | Lecture 5 | |
| (W) Mar 10 | Hyperelliptic curves; differentials | Lecture 6 |
HW 3: [TeX], [PDF] |
| (M) Mar 15 | Genus; canonical map; fundamental group | Lecture 7 | |
| (W) Mar 17 | Fundamental group, cont.; covering spaces | Lecture 8 |
HW 4: [TeX], [PDF] |
| (M) Mar 22 | Student holiday---no class | ||
| (W) Mar 24 | Coding session: Belyi maps in the LMFDB | ||
| (M) Mar 29 | Groups acting on Riemann surface | Lecture 9 | |
| (W) Mar 31 | Monodromy representations | Lecture 10 |
HW 5: [TeX], [PDF] |
| (M) Apr 5 | Function fields | Lecture 11 | |
| (W) Apr 7 | Belyi maps in the LMFDB: search by base field | ||
| (M) Apr 12 | Uniformization of Riemann surfaces; Fuchsian groups | Lecture 12 | |
| (W) Apr 14 | Belyi maps in the LMFDB: search by passport size | LMFDB Github | |
| (M) Apr 19 | Student holiday---no class | ||
| (W) Apr 21 | Belyi maps in the LMFDB: finishing search features | LMFDB Github | |
| (M) Apr 26 | Fuchsian triangle groups; hyperbolic geometry | Lecture 13 | |
| (W) Apr 28 | Primitive permutation groups | Lecture 13.5 | |
| (M) May 3 | The modular group as a triangle group | Lecture 14 |
HW 6: [TeX], [PDF] |
| (W) May 5 | Computing primitivizations | Github repo | |
| (M) May 10 | Computing primitivizations | Github repo | |
| (W) May 12 | Computing primitivizations | Github repo | |
| (M) May 17 | Belyi's Theorem | Lecture 15 | |