Teaching
Courses¶
Course at USTC: Rings and Modules (2023)¶
In 2023, I give an algebra course at USTC in Hefei (China). Here are the exercise sheets:
Exercise sheet 1: Rings and ideals.
Exercise sheet 2: Euclidean, principal and unique factorization domains.
Exercise sheet 3: Noetherian rings and polynomial rings.
Exercise sheet 4: Modules, exact sequences, generating families, free families and bases.
Exercise sheet 5: Finite type modules, Noetherian modules, tensor product and localization.
Exercise sheet 6: Flat and projective modules, indecomposable and simple modules.
Exercise sheet 7: Finite type modules over principal ideal domains.
Exercise sheet 8: Brief introduction to algebraic geometry.
And here is the general layout of the course:
Chap. 1: Rings and arithmetic.
1) Rings and algebras.
2) Ring and algebra morphisms.
3) Subrings and ideals.
4) Quotient of a ring by a two-sided ideal.
5) Structure of (Z/nZ)^*.
6) Divisibility in a ring.
7) Prime and maximal ideals.
8) Euclideans domains.
9) Unique factorization domains.
10) Noetherian rings.
11) Polynomial rings.
Chap. 2: Modules.
1) Definition and examples.
2) Submodules and quotient modules.
3) Products, direct sums and exact sequences.
4) Annihilators and torsion.
5) Generating families, free families, bases.
6) Finite type modules.
7) Nakayama's lemma.
8) Noetherian modules.
9) Tensor product.
10) Localization.
11) Projective and flat modules.
12) Indecomposable modules.
13) Simple modules.
Chap. 3: Finite type modules over principal ideal domains.
1) Torsion-free modules.
2) Torsion modules.
3) Matrices with coefficients in a principal ideal domain.
4) The aligned basis theorem.
5) Applications to finitely generated abelian groups and to endomorphisms of finite-dimensional vector spaces.
Chap. 4: Brief introduction to algebraic geometry.
1) Algebraic sets.
2) Zariski topology.
3) Hilbert's Nullstellensatz.
4) The spectrum of a ring.
Chapter 4 is bonus material that will not enter in the final exam.
Course MAT562 Introduction to Algebraic Geometry and Elliptic Curves (2021-2023)¶
The course notes can be found on Moodle. Here are the exercise sheets:
Exercise sheet 1 : Affine varieties 1.
Exercise sheet 2 : Affine varieties 2.
Exercise sheet 3 : Projective varieties 1.
Exercise sheet 4 : Projective varieties 2.
Exercise sheet 5 : Elliptic curves.
Exercise sheet 6 : Elliptic curves over the complex numbers.
Exercise sheet 7 : Endomorphisms of elliptic curves and elliptic curves over finite fields.
Exercise sheet 8 : Elliptic curve cryptography and torsion of elliptic curves over the rational numbers.
Exercise sheet 9 : Mordell-Weil theorem for elliptic curves.
The solutions of some of the exercise sheets can be requested by e-mail. Here are the final exams:
Answers to the exam 2021-2022.
Mini-course for the Journées de rentrée des Masters de la FMJH (2021)¶
Here you can find the current version of the course notes.
Course MAA303 Algebra and Arithmetic at École Polytechnique (2019-2023)¶
Here you can have access to the course notes. As for the exercise sheets:
Exercise sheet 1 : Groups and morphisms.
Exercise sheet 2 : Subgroups.
Exercise sheet 3 : Quotients.
Exercise sheet 4 : Cutting groups into smaller pieces.
Exercise sheet 5 : Symmetric group.
Exercise sheet 6 : Group actions.
Exercise sheet 6bis : Bonus exercises on group actions.
Exercise sheet 7 : Sylow theorems.
Exercise sheet 8 : Rings.
Exercise sheet 9 : Field extensions.
Exercise sheet 10 : Finite fields and cyclotomic extensions.
The solutions can be requested by e-mail. Here are the final exams:
Answers to the exam 2022-2023.
Answers to the exam 2021-2022.
Answers to the exam 2020-2021.
Answers to the exam 2019-2020.
All other resources can be found on the Moodle page of the course.
Course on Rational Points at the University of Bonn (2018-2019)¶
Here you can have access to the notes of the course by Yifan Wu. As for the exercise sheets:
Exercise sheet 1 : Ci-fields.
Exercise sheet 2 : Galois cohomology.
Exercise sheet 3 : Brauer group (part 1).
Exercise sheet 4 : Brauer group (part 2).
Exercise sheet 5 : The Hasse principle.
Study groups¶
2017-2018: Algebraic theory of quadratic forms (ENS)¶
In the first semester 2017/2018, I organized a study on the algebraic theory of quadratic forms for Master students at the École normale supérieure. The main reference was the book Introduction to quadratic forms over fields by Tsit Yuen Lam. Here is the programme.
2016-2017: Class field theory (ENS)¶
In 2016, I organized with Arthur-César Le Bras an informal study group on class field theory for Master students at the École normale supérieure. We covered both local and global class field theory. We followed the approach of the book Algebraic Number Theory by Jürgen Neukirch.
Problem sessions¶
2022: Around the inverse Galois problem (PCMI Graduate Summer School)¶
In August 2022, I gave problem sessions for the course on the inverse Galois problem by Olivier Wittenberg at the PCMI Graduate Summer School Number Theory informed by Computation. Here is the exercise sheet.
2017-2018: Complex analysis (ENS)¶
In the second semester 2017/2018, I gave problem sessions for the Complex Analysis course of Julien Barral at the École normale supérieure. Here are the exercise sheets.
| Sujet | Thème |
| TD1 | Cauchy-Riemann equations, power series |
| TD2 | Logarithm, Cauchy's formula and first consequences |
| TD3 | Maximum principle; logarithm of a function |
| TD4 | Uniform limits of holomorphic functions |
| TD5 | Harmonic functions |
| TD6 | Meromorphic functions, residue theorem |
| TD7 | Elliptic functions |
| TD8 | Entire functions |
| TD9 | Special functions 1 |
| TD10 | Special functions 2 |
| TD11 | Conformal mapping theorem |
| Bonus | On Schwarz-Christoffel's theorem |
2016-2017: Algebra 2 (ENS)¶
In the first semester 2016/2017, I gave problem sessions for the Algebra 2 course of Jean-François Dat at the École normale supérieure. Here are the exercise sheets.
| Sujet | Corrigé | Thème |
| TD1 | Corrigé du TD1 | Generalities about rings |
| TD2 | Corrigé du TD2 | Generalities about rings and modules |
| TD3 | Corrigé du TD3 | Algebraic sets, Zariski topology, noetherian rings and modules |
| TD4 | Corrigé du TD4 | Noetherian rings and modules, completions |
| TD5 | Corrigé du TD5 | Limits, polynomials |
| TD6 | Corrigé du TD6 | Euclidean, principal anf unique factorization domains |
| TD7 | Corrigé du TD7 | Localization |
| TD8 | Corrigé du TD8 | Tensor product (and complements of homological algebra) |
| TD9 | Corrigé du TD9 | Modules of finite length |
| Midterm | Solutions of the midterm | Partiel |
| TD10 | Corrigé du TD10 | Finite type modules over a principal domain |
| TD11 | Corrigé du TD11 | Field extensions, Nullstellensatz |
| TD12 | Corrigé du TD12 | Finite fields, normal extensions |
| TD13 | Corrigé du TD13 | Separable extensions |
| TD14 | Corrigé du TD14 | Galois theory I |
| TD15 | Corrigé du TD15 | Galois theory I I |
| TD16 | Corrigé du TD16 | Integrality |
2015-2016: Algebra 2 (ENS)¶
In the first semester 2015-2016, I gave problem sessions for the Algebra 2 course of Jan Nekovář at the École normale supérieure. Here are the exercise sheets.
| Sujet | Corrigé | Thème |
| TD1 | Corrigé du TD1 | Tools from group theory |
| TD2 | Corrigé du TD2 | Rings, ideals, principal and unique factorization domains |
| TD3 | Corrigé du TD3 | Prime and maximal ideals; polynomials; approximation |
| TD4 | Corrigé du TD4 | Noetherian rings and modules |
| TD5 | Corrigé du TD5 | Modules over a principal domain; discriminant |
| TD6 | Corrigé du TD6 | Field extensions |
| TD7 | Corrigé du TD7 | Normal, separable and Galois extensions |
| TD8 | Corrigé du TD8 | Galois theory I |
| TD9 | Corrigé du TD9 | Galois theory II |
| TD10 | Corrigé du TD10 | Finiteness in commutative algebra |
| TD11 | Corrigé du TD11 | Noether normalization, localization |
| TD12 | Corrigé du TD12 | Algebra and geometry, Nullstellensatz, Nakayama's lemma |
| TD13 | Corrigé du TD13 | Zariski topology, irreducibility, dimension |
| TD14 | Corrigé du TD14 | Smoothness, discrete valuation rings, Dedekind domains |
| TD15 | Corrigé du TD15 | Review |
2014-2015: Algebra 2 (ENS)¶
In the second semester 2014-2015, I gave problem sessions for the Algebra 2 course of Jan Nekovář at the École normale supérieure. Here are the exercise sheets.
| Sujet | Corrigé | Thème |
| TD1 | Corrigé du TD1 | Tools from group theory |
| TD2 | Corrigé du TD2 | Rings, ideals, principal and unique factorization domains |
| TD3 | Corrigé du TD3 | Prime and maximal ideals; polynomials; approximation |
| TD4 | Corrigé du TD4 | Noetherian rings and modules |
| TD5 | Corrigé du TD5 | Modules over a principal domain; discriminant |
| TD6 | Corrigé du TD6 | Field extensions |
| TD7 | Corrigé du TD7 | Normal, separable and Galois extensions |
| TD8 | Corrigé du TD8 | Galois theory I |
| TD9 | Corrigé du TD9 | Galois theory II |
| TD10 | Corrigé du TD10 | Finiteness in commutative algebra |
| TD11 | Corrigé du TD11 | Algebra and geometry, Nullstellensatz |
| TD12 | Corrigé du TD12 | Nakayama's lemma |
| TD13 | Corrigé du TD13 | Zariski topology, dimension, smoothness, discrete valuation rings |
| TD14 | Corrigé du TD14 | Review |
Oral examinations¶
2013-2014: Mathematics, Lycée Henri IV¶
| Sujet 1 | Rappels sur les fonctions de la variable réelle |
| Sujet 2 | Topologie: Vrai ou Faux pour faire réfléchir sur le cours |
| Sujet 3 | Topologie |
| Sujet 4 | Espaces vectoriels normés, séries positives |
| Sujet 5 | Séries numériques |
| Sujet 6 | Séries numériques, suites et séries de fonctions |
| Sujet 7 | Suites et séries de fonctions, séries entières |
| Sujet 8 | Séries entières polynômes |
| Sujet 9 | Algèbre linéaire |
| Sujet 10 | Réduction |
| Sujet 11 | Colle de révision |
| Sujet 12 | Equations différentielles linéaires, révisions d'intégration |
| Sujet 13 | Arithmétique et algèbre non linéaire |
| Sujet 14 | Algèbre bilinéaire |
| Sujet 15 | Algèbre bilinéaire, espaces euclidiens, espaces hermitiens |
| Sujet 16 | Espaces hermitiens, séries de Fourier |
| Sujet 17 | Séries de Fourier, équations différentielles non linéaires |
| Sujet 18 | Equations différentielles non linéaires, géométrie |
| Sujet 19 | Fonctions de plusieurs variables |
2011-2012: Mathematics, Lycée Henri IV¶
| Semaine 1 | Polynômes et fractions rationnelles |
| Semaine 2 | Suites numériques |
| Semaine 3 | Développements asymtotiques |
| Semaine 4 | Séries positives |
| Semaine 5 | Séries positives, séries réelles ou complexes |
| Semaine 6 | Séries réelles ou complexes, suites et séries de fonctions |
| Semaine 7 | Suites et séries de fonctions |
| Semaine 8 | Séries entières |
| Semaine 9 | Séries entières bis |
| Semaine 10 | Fonctions réglées |
| Semaine 11 | Algèbre linéaire |
| Semaine 12 | Réduction des endomorphismes |
| Semaine 13 | Intégration |
| Semaine 14 | Topologie |
| Semaine 15 | Formes quadratiques et espaces euclidiens |
| Semaine 16 | Formes quadratiques et hermitiennes, espaces euclidiens et hermitiens |
| Semaine 17 | Séries de Fourier |
| Semaine 18 | Équations différentielles linéaires |
| Semaine 19 | Groupes, anneaux, corps |
| Semaine 20 | Fonctions de plusieurs variables, calcul différentiel |
| Semaine 21 | Révisions d'algèbre |
Material for highschool students¶
| Constructions à la règle et au compas | Vidéo Twitch sur les constructions à la règle et au compas | Avril 2020 |
| Réseaux et aires | Séance au club Parimaths avec le groupe débutant | Décembre 2014 |
| Quelques problèmes classiques d'arithmétique II | Séance au club Parimaths avec le groupe avancé | Mars 2014 |
| Méli-mélo de preuves graphiques | Séance au club Parimaths avec le groupe débutants | Février 2014 |
| Quelques problèmes classiques d'arithmétique | Séance au club Parimaths avec le groupe avancé | Novembre 2013 |
| Exercices d'arithmétique supplémentaires | Fiche avec trois exercices d'arithmétique type olympiade (avec corrigés) | Février 2012 |
| Exercices d'arithmétique | Séance d'exercices d'arithmétique type olympiade (avec corrigés) | Février 2012 |