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.
• Categories and functors, adjoint and equivalence, tensor products, localization of rings and module, limits.
• Affine schemes. Integral extensions.
• Noetherian and artinian rings and modules. Hilbert’s basis theorem, Noether’s normalization lemma and Hilbert’s Nullstellensatz. The affine space.
• Representations of finite groups.
Représentations des groupes, algèbre d’un groupe fini, table de caractères, représentations des groupes symétriques, groupes de Lie, algèbre de Lie, représentations des groupes classiques.
Nombres et entiers algébriques. Unités. Norme, trace, discriminant et ramification. Base intégrale. Corps quadratiques, cyclotomiques. Groupes de classes. Décomposition en idéaux premiers. Équations diophantiennes.
Modèles probabilistes en théorie des nombres (Kubilius, fonctions aléatoires multiplicatives, matrices aléatoires); théorèmes centraux limites en théorie des nombres (Erdös-Kac, Selberg); répartition des diviseurs d'entiers; champs log-corrélés et maxima de zeta.
Polynômes symétriques. L'anneau des fonctions symétriques et sa structure : générateurs, produit scalaire de Hall, bases orthogonales, et identités combinatoires. Liens avec la théorie des représentations des groupes symétriques. Connexions avec la combinatoire, l’algèbre et la théorie des représentations.
Lemme de Zorn. Catégories et foncteurs: notions et exemples de base: catégories de structures mathématiques, monoïde, catégorie des ensembles; section, rétraction, exemples géométriques et algébriques. Foncteurs et transformations naturelles: exemples de base, catégories de foncteurs. Équivalence de catégories: exemples de base. Modules. Théorèmes d'homomorphisme et d'isomorphisme. Sommes et produits directs, modules libres. Modules de type fini sur un anneau principal et applications aux formes canoniques des matrices. Modules noethériens et artiniens: exemples et propriétés de base. Modules indécomposables, théorème de Krull-Schmidt. Anneaux et polynômes: nilradical et localisation; élimination classique, ensembles algébriques, théorème des zéros de Hilbert. Théorie des corps: groupe de Galois, résolution par radicaux; indépendance algébrique, degré de transcendance, dimension des ensembles algébriques irréductibles; corps ordonnables, 17e problème de Hilbert.
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.
We will first cover the Riemann zeta function and prove its analytic properties. We will relate its values at negative integers to Bernoulli numbers and show that these satisfies p-adic congruences that allow one to define a p-adic meromorphic zeta function.
We will then study L-functions associated with modular forms, in particular showing that critical values are (essentially) algebraic and how these, for many modular forms, vary p-adically too.
We will then analyze other automorphic L-functions, such as but not limited to Rankin–Selberg and triple product and conclude with an overview on several conjectures on L-functions, such as Birch–Swinnerton-Dyer and Bloch–Kato, and their p-adic avatar, the Iwasawa Main conjecture.
Completion of the topics of MATH 570. Rudiments of algebraic number theory. A deeper study of field extensions; Galois theory, separable and regular extensions. Semi-simple rings and modules. Representations of finite groups
Anneaux commutatifs, idéaux premiers, rudiments de géométrie algébrique, Nullstellensatz de Hilbert, localisation, complétion, théorie de la dimension.
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.