M2 Analysis, Number Theory and Geometry

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Our master's program M2 offers three crash courses (in differential geometry, algebra and complex analysis) in September, several fundamental advanced courses in the first term (such as algebraic geometry, number theory, ergodic theory and dynamical systems, groups and geometry - the subjects may vary slightly from year to year) and a number of more specialized courses in the second term (the subjects change regularly and completely, we try to be sufficiently diverse in our offer). Of course you are also expected to write and defend a master thesis under the guidance of a senior researcher. This can be done at Orsay mathematical department but also at another research unit if necessary.

Course Prerequisites

M1 in fundamental mathematics or an equivalent level of qualification. Admission upon competitive exam.

  • Understand and proficiently use high-level mathematical tools and methods.

  • Conceive and write a rigorous mathematical proof.

  • Analyse a research paper with a view to summarising it and using it.

  • Clearly explain a theory and mathematical results.

  • Understand and mathematically model a problem in order to resolve it.

Post-graduate profile

Students will have acquired basic knowledge of modern mathematics and studied a particular area in enough depth to begin research (usually in the context of thesis preparation).

Career prospects

The main outcome of this course is to prepare students for doctoral studies in fundamental mathematics.


Laboratoire de mathématiques d'Orsay.

CMLS (Ecole Polytechnique).


Seul un groupe des UEs, "Cours Fondamentaux, compte au 1er semestre. On peut les choisir comme on veut, à condition de valider 30 ECTS. De plus on peut compléter par certains cours de AMS ("cours communs AAG-AMS, à préciser dans la maquette AMS)AMS).

Matières ECTS Cours TD TP Cours-TD Cours-TP TD-TP A distance Projet Tutorat
Surfaces de Riemann et variétés abéliennes 15 50 25
Représentations des algèbres de Lie 7.5 24 12
Algebre Homologique 7.5 24
AAG - Théorie ergodique 7.5 25 12.5
AAG - Théorie des schémas 15 48 24
AAG - Théorie des Nombres 15 48 24
AAG - Techniques d'analyse harmonique 15 50 25
AAG - Systèmes Dynamiques topologiques et différentiables 7.5 25 12.5
AAG - Groupes et Géométries 15 48 24

Au second semestre, comptent deux groupes d'UE: Cours Spécialisés (6 ECTS) et Cours complémentaires (3 ECTS). Le reste, soit 21 ECTS, est acquis en effectuant un stage de recherche et soutenant un mémoire de M2. En outre, les étudiants peuvent suivre des cours de langues/FLE, histoire de maths ou séminaire des étudiants (ce dernier pouvant exceptionnellement remplacer un cours accéléré).

Matières ECTS Cours TD TP Cours-TD Cours-TP TD-TP A distance Projet Tutorat
Histoire des Mathématiques 3 12 12
Anglais/FLE 3
AAG - Cours accélérés Analyse réelle et complexe 3 20
AAG - Cours accéléré Géométrie Différentielle 3 20
AAG - Cours accéléré Algèbre 3 20
A2 - Séminaire Etudiant 3 20
Matières ECTS Cours TD TP Cours-TD Cours-TP TD-TP A distance Projet Tutorat
AAG - Mémoire 21 576
Matières ECTS Cours TD TP Cours-TD Cours-TP TD-TP A distance Projet Tutorat
Théorie métrique des nombres 6 20
Introduction aux groupes quantiques compacts 6 20
Introduction aux algèbres vertex 6 40
Théorie ergodique des groupes 6 36
Géométrie différentielle algébrique 6 20
Dynamique des difféomorphismes de surface en entropie strictement positive 6 20
Dynamique arithmétique 6 20
Modalités de candidatures
Application period
From 01/02/2021 to 09/07/2021
Compulsory supporting documents
  • Motivation letter.

  • All transcripts of the years / semesters validated since the high school diploma at the date of application.

  • Curriculum Vitae.

Additional supporting documents
  • Curriculum EU (description of the units of education followed) of the last two years.

  • Choice sheet of M2 (obligatory for the candidates registered in M1 at the University Paris-Saclay) to download on https://www.universite-paris-saclay.fr/en/admission/apply-master-programmes.

  • VAP file (obligatory for all persons requesting a valuation of the assets to enter the diploma).

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