An area of major interest in this program is the study of the Galois group of the algebraic closure of the rationals. The methods of study of this group involves its representations in other algebraic, geometric or analytic objects. This leads to connections with algebraic groups, analytic varieties (real, complex and p-adic), and Lie theory. The interconnections are deep, and progress in number theory requires a more profound understanding of each of these connections. For example, the conjecture of Shimura-Tanyama-Weil that all elliptic curves over the rationals are modular implies Fermat's Last Theorem.
In recent years, due to the availability of powerful computers and software such as MAPLE, CAYLEY and PARI, large scale computations have proven themselves extremely important in making and verifying conjectures. Computational algebra is rapidly evolving with the production of better and faster algorithms for making computations.
The participating universities contributing to this Institute have many people working in currently active areas of research in number theory, elliptic curves, arithmetic geometry, algebraic groups, group theory and Lie theory, commutative algebra, representation theory of groups and Lie algebras, Galois theory, profinite groups and computational algebra, representation theory of associative algebras, homologic and categoric algebra, ring theory and modules.
Many members of the group are also members of CICMA (Centre interuniversitaire en calcul mathématique algébrique), an inter-university research center that organizes many scientific activities.
This program is designed for strong graduate students with an interest in algebra, group theory, number theory (algebraic and/or analytic) and algebraic geometry. Both theoretical and computational aspects of these topics are covered.
There are no formal prerequisites beyond those required by the departments. However, the following guidelines should be followed and courses selected in consultation with an advisor in the program.
All students are expected to acquire the fundamentals of algebra, such as are contained in beginning level graduate courses at any of the participating universities (e.g., group theory, commutative algebra, Galois theory, number theory).
Students are then expected to take a number of more specialized courses (in their areas of interest and adjacent or complementary areas).
Students are encouraged to participate in advanced seminars and courses in their areas of specialty.
The course will focus on the study of elliptic curves over the complex and p-adic numbers. It will cover topics such as: complex uniformisation, Weistrass P-functions, the periods of an elliptic curve, the formal group of an elliptic curve, ordinary and supersingular elliptic curves, integral model of elliptic curves, the local Galois representation.
The main objective of the course is to study geometrically algebraic objects, for example commutative rings with identity. To such a ring we will attach a topological space and a sheaf of rings on it, making it into a geometric object called "affine scheme". We will see that affine schemes can be glued together to give other (non-affine) schemes.
The time of the course: Tuesdays, Fridays 9:00-11:00AM.
Groupes de Lie, espaces tangents et champs de vecteurs lisses, algèbres de Lie, application exponentielle, représentations adjointes et coadjointes, algèbres résolubles et nilpotentes, décomposition en espaces de racines, groupes de Weyl, matrices de Cartan, esquisse de la classification des algèbres de Lie semisimples complexes, présentation de Serre, théorème de Weyl, décomposition en espaces de poids, algèbres enveloppantes, modules de Verma, et un choix selon les intérêts et la formation des étudiants: Catégorie O, algèbres de Lie de dimension infinie, théorie géométrique des représentations, formules de caractère Weyl-Kac et propriétés modulaires.
Les mardis: 13h30-16h20
Lie groups, examples and general theory. Structure theory of Lie algebras. Solvable and nilpotent algebras. Engel's and Lie's theorems. Classification of semisimple Lie algebras. Representation theory of semisimple Lie algebras and compact Lie groups.
Nombres et entiers algébriques. Unités. Norme, trace, discriminant et ramification. Base intégrale. Corps quadratiques, cyclotomiques. Groupes de classes. Décomposition en idéaux premiers. Équations diophantiennes.
Introduction to Ring Theory: definitions and examples, ideals, quotients and isomorphisms. Euclidean domains, principal ideal domains and unique factorization domains. Polynomial rings and introduction to modules.
This course will revolve around qualitative and quantitative aspects of the Galois inverse problem (GIP), asking which finite groups are realizable over the rationals. Conjecturally all of them do and, furthermore, Malle has proposed a conjectural asymptotic formula for the number of finite extensions of the rationals with a given Galois group and bounded discriminant.
After a warm-up, establishing the GIP in several examples, we will rapidly prove it for the class of (odd) nilpotent groups and discuss the status of Malle's conjecture for nilpotent groups. We next focus on proving the considerably stronger conclusion where GIP is established for all solvable groups (Shafarevich).
After that we will focus on Galois groups with prescribed ramification and establish Shafarevich's theorem on their number of generators and relations (after pro-l completion), prove the Golod--Shafarevich inequality and give examples of infinite class field towers. Finally, we will focus on the theory of random profinite groups developed by Liu and Wood, conjecturally giving a statistical description of such Galois groups in natural families of number fields.
The course will be an introduction to Shimura varieties. It will cover foundational topics such as the notions of Hermitian symmetric domains, variations of Hodge structures, Shimura data, canonical models of Shimura varieties, the Eichler-Shimura isomorphism, Matsushima’s formula, the L2- cohomology of Siegel Shimura varieties.
Corps (extensions, théorie de Galois, corps finis), Anneaux (noethériens et artiniens, radicaux, idéaux premiers et maximaux, localisation, théorème de Wedderburn, Nullstellensatz), Modules (lemme de Schur, modules projectifs et injectifs, suites exactes, produit tensoriel, catégories).
Algebraic groups. Flag varieties and the Borel-Weil theorem. Quantum groups and crystals.
Class Field Theory describes the abelian Galois extensions of local or global fields (for example, finite extensions of the p-adic numbers or the rational numbers) in terms of the internal arithmetic of the field. We aim to cover aspects of both the local and global theory, and along the way learn a little bit of Lubin-Tate theory and Galois cohomology.
Distribution des nombres premiers. Fonction zêta de Riemann et fonctions-L de Dirichlet. Le théorème des nombres premiers, et de Bombieri-Vinogradov. La répartition des nombres premiers consécutifs.
Carquois d'une algèbre, représentations d'algèbres héréditaires, théorie d'Auslander -Reiten, ensembles partiellement ordonnés et catégories d'espaces vectoriels, revêtements d'une algèbre, algèbres auto-injectives, théorie de l'inclinaison.