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.
This course is a topic course on elliptic curves, which can be taken as a first or second course on the subject. We will first study elliptic curves (and general curves) over finite fields, proving the Weil conjectures for curves over finite fields, and making the link with random matrix theory. Other related possible topics: the Sato-Tate conjecture and the modularity of complex multiplication elliptic curves.
The course will, for the first part, follow Hartshorne's book Algebraic Geometry, chapter III. We will develop the general theory of derived functors and hyper derived functors, in particular right derived functors of a left exact functor, respectively, left derived functors of a right exact functor. This will provide the conceptual framework for defining sheaf cohomology and de Rham cohomology. We will also study Chech cohomology of a coherent sheaf on scheme and its relationship to sheaf cohomology and de Rham cohomology. Finally, duality theorems for both sheaf cohomology (of quasi coherent sheaves on a projective scheme) and de Rham cohomology will be investigated.
Evaluation: there will be homework assignments during the semester and a final, written exam at the end of it.
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).
Ce cours se veut une introduction à la géométrie algébrique et sera divisé en deux parties. Dans la première partie on étudiera les courbes algébriques sur les complexes avec une emphase sur les courbes projectives et lisses (i.e. les surfaces de Riemann compactes). La théorie sera développée en faisant appel simultanément à des notions d'algèbre, d'analyse complexe et de topologie. Voici les thèmes qui seront traités dans cette première partie: courbes affines, courbes projectives, applications holomorphes et méromorphes, fonctions thêtas, courbes hyperelliptiques, revêtements et formule d'Hurwitz, théorème de Riemann-Roch, singularités (noeud, cusp). Dans la deuxième partie du cours nous aborderons la théorie générale des variétés algébriques sur un corps quelconque avec une emphase sur celles qui sont définies sur le corps des complexes. Voici les thèmes qui seront couverts dans cette deuxième partie: anneaux noethériens et ensembles algébriques affines, variétés affines et théorème du Nullstellensaz de Hilbert, variétés projectives, anneaux des coordonnées affines et homogènes, morphismes et applications birationnelles, topologie de Zariski et équivalence de catégorie pour le cas affine.
• 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.
Nombres et entiers algébriques. Unités. Norme, trace, discriminant et ramification. Base intégrale. Corps quadratiques, cyclotomiques. Groupes de classes. Décomposition en idéaux premiers. Équations diophantiennes.
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, 17<+>e<+> problème de Hilbert.
We will develop local and global class field theory through the lenses of Galois cohomology. We will spend a substantial amount of time developing from scratch the cohomology of discrete modules for a profinite group and only then move towards the intended arithmetic target. As a background, some earlier exposure to algebraic number theory and Galois theory would be helpful.
We will begin considering several notions of height (naive height, Weil height, Neron-Tate height, dynamical height), which are quantitative measurements for the "complexity" of an algebraic number. With this notion we will explore several techniques to prove finiteness results in the theory of Diophantine equations. We will follow mostly (but not only) the book of Bombieri—Gübler "Heights in Diophantine Geometry".
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
This course is cross-listed. Undergraduate students should register for MATH 596 and graduate students for MATH 726.
The course will be devoted to the emerging subject of unlikely intersections. This is a topic in arithmetic geometry — the discipline that merges number theory and algebraic geometry — and deals with phenomena where one does not expect an intersection between certain subvarieties of a given variety and, in light of this, when such an intersection exists seeks to provide a classification of the conditions when such unlikely phenomena occur. It is quite similar in spirit to the purely algebraic geometrical term of excess intersection. Some of the outstanding conjectures and theorems in this area are the Andre-Oort conjecture (now a theorem) and the Zilber-Pink conjecture.
The course will provide a gentle introduction to these topics. The exact syllabus will be determined so as to accommodate the audience. The course will also be accessible to motivated undergraduate students and so a lot of background material will be provided during the course. Strong background in algebra at the level of the undergraduate courses will be assumed and background in algebraic number theory and algebraic geometry is desired but not required. Undergraduates wishing to take the course are advised to contact me during the Fall semester.
The course will be based on the book Zannier: Some problems of unlikely intersections in arithmetic and geometry, Annals of Mathematical Studies, vol. 181. Although this should be thought of more as a road map than a text book.
The method of evaluation in this course will be based on a final paper, a presentation, and exercises.
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.
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.
Représentations de carquois, représentations projectives et injectives, algèbres et modules, algèbres de carquois liés, théorie d'Auslander-Reiten.
La théorie de la représentation des groupes et une théorie algébrique dont les ramifications s’étendent à de très nombreux domaines des mathématiques ainsi qu’à la Physique te à la Chimie. L’apprentissage de cette théorie permettra entre autre à l’étudiant d’appréhender d’autres théories algébriques de la représentation.
Descripteur : Représentations linéraires des groupes finis. Sous-représentations, théorème de Mashke; représentations irréductibles. Théorie des caractères. Décomposition en composantes isotypiques. Produits tensoriels; représentation induites. Représentations linéaires des groupes compacts. Exemples: groupes cycliques, diédraux, symétriques: tores, groupes de rotations.