## 2023-2024 : Algèbre avancée (M1)

Le cours a lieu le lundi à 10h15 en amphi A.

Programme :

1. Modules
• Definition of a module. Homomorphisms, submodules, quotient modules...
• Free modules, finitely generated modules
• Matrix of a linear map, Cayley–Hamilton theorem
2. Finitely generated modules over principal ideal domains
• Structure theorems
• Applications: finitely generated abelian groups, reduction of endomorphisms
3. Rings
• Local rings, Nakayama's lemma
• Localization of a ring, of a module
• Ring extensions, integrality, finiteness, integrally closed rings
4. Tensor product
• Tensor product of modules
• Tensor product and exact sequences, notion of flatness
• Tensor product of algebras
• Extension of scalars
If time allows, additional topics may be addressed: discrete valuation rings, Dedekind rings, the Nullstellensatz, criteria for flatness...

## 2023-2024 : Riemann surfaces (M1)

This course is an introduction to Riemann surfaces. Starting with examples like the Riemann sphere and the complex tori, we set up the basics of theory (ramification, divisors, differential forms, genus...) in order to tackle the important theorems (Riemann-Hurwitz, Riemann-Roch...). We end up with an opening to important concepts for number theory: modular curves and monodromy theory.

We will cover the following topics:
• Definition of Riemann surfaces.
• The Riemann sphere, elliptic functions, complex tori.
• Holomorphic maps between Riemann surfaces.
• Ramification theory, Riemann-Hurwitz formula.
• Meromorphic functions, Riemann-Roch theorem.
• Differential forms, genus of a Riemann surface.
• Quotients of Riemann surfaces. Statement of the uniformization theorem.
• Modular curves and monodromy representations.

## 2022-2023 : Algèbre (Agrégation)

Feuilles d'exercices :

## 2022-2023 : Option C (Agrégation)

Feuilles d'exercices :