Algebra and Number Theory

Program Description

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.

Program Members

Academic Program

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.

2018-19 Course Listings

Fall

Topics in Algebra: Galois theory for schemes

The course will develop the theory of the etale fundamental group of a connected scheme in parallel to the Galois theory of a field and the theory of the fundamental group of a topological space. Some knowledge of Algebraic Geometry would be helpful but not necessary as the course will be self  contained.

Prof. Adrian Iovita

MAST 699/2 ou/or MAST 833

Institution: Concordia University

Elliptic Curves

This course will cover the basic theory of elliptic curves and algebraic curves. The prerequisites for this course is the standard background in abstract algebra (groups, rings, field extensions, Galois theory etc). A first course in algebraic number theory is recommended, but not mandatory. The course is open to all graduate students, either at the master or the Ph.D. level.

Recommended Textbook: J. Silverman, The Arithmetic of Elliptic Curves.

Prof. Chantal David

MAST 699/2 E / 833/2

Institution: Concordia University

Introduction to Class Field Theory

The course will describe the statement and proofs of the main results of class field theory, both local and global, following the treatment given in the textbook of Cassels-Frolich, which shall be followed fairly closely.

Prof. Henri Darmon

MATH 596 - 001

Institution: McGill University

Théorie algébrique des nombres

Les sujets traités comprennent:

  • Nombres et entiers algébriques
  • Unités
  • Norme, trace, discriminant et ramification
  • Base intégrale
  • Corps quadratiques, cyclotomiques
  • Groupes de classes
  • Décomposition en idéaux premiers
  • Équations diophantiennes.

Prof. Matilde Lalin

MAT 6617

Institution: Université de Montréal

Sieve methods and applications

This is an introductory course to sieve methods and their applications. After reviewing some background material in probabilistic number theory, we will introduce and study the main objects of sieve theory. In particular, we will develop the combinatorial sieve and Selberg's sieve and use them to prove various estimates about prime numbers. We will then use sieve theory to study L-functions and establish Linnik's theorem. The proof of this result provides a natural entry point to the theory of bilinear sum estimates and the development of the Large Sieve. As an application of this circle of ideas, we will prove the Bombieri-Vinogradov theorem. The course will conclude with a discussion of the recent spectacular developments about bounded and large gaps between primes.

Prof. Dimitris Koukoulopoulos

MAT6684V

Institution: Université de Montréal

Compact Lie Groups and their Representations

Prerequisites. Strong background in linear algebra. Familiarity with differential geometry (e.g. tangent bundles, Frobenius’s theorem on integrable subbundles of the tangent bundle), real analysis (e.g. Stone-Weierstrass theorem, Hilbert spaces, compact operators), and representation theory of finite groups would be very useful too.

Course textbooks. All optional.
Representations of Compact Lie Groups by Brocker and Tom Dieck.
Lie Groups by Bump.

Syllabus. Topics will include: examples of classical compact Lie groups and Lie algebras, Haar measure, the Peter-Weyl Theorem, the exponential map, maximal tori and their conjugacy, the Weyl group, Weyl integration formula, roots and root systems, Dynkin diagrams, highest weight theory, Weyl character formula.

Time permitting, we may also discuss applications of the above to random matrix theory.

Evaluation. To be determined after our organizational meeting.

Prof. Michael Lipnowski

MATH 595

Institution: McGill University

Winter

Topics in Algebra: p-adic modular forms

This course will present the theory of p-adic and overconvergent modular forms as they appear in the work of J.-P. Serre, N. Katz and R. Coleman. Basic knowledge of classical modular forms (for example as they appear in "A course in Arithmetic" by J.-P. Serre) and of algebraic geometry is required.     

Prof. Adrian Iovita

MAST 699/4 C ou/or MAST 833

Institution: Concordia University

Topics in Algebra & Number Theory: Elliptic Curves II

This course will cover advanced topics in the theory of elliptic curves. It is intended as a continuation of the course “Elliptic Curves” to be taught by Prof. David in the Fall at Concordia University. Although it is not required to take David’s course, I will assume that students know the material covered in David’s course. The exact selection of topics will be determined once a more precise syllabus for David’s course becomes available, but they will mostly be chosen from Silverman’s books on elliptic curves.

Prof. Eyal Goren

MATH 596

Institution: McGill University

Analytic Number Theory

The goal of this course is to give a complete proof of the prime number theorem, a proof that will help the student appreciate many of the important theorems in the subject. We will review in detail the motivation for the prime number theorem (and other conjectures and theorems about prime numbers), and focus on the background needed in both number theory and analysis (so that students feel comfortable with the techniques used). Then we will prove the prime number theorem and begin to appreciate the importance of the Riemann Hypothesis.  Having gone slowly over this we will be ready to use these ideas in many different directions. Our only scheduled goal will be to prove Dirichlet's theorem, that there are infinitely many primes in each reduced residue class a mod q, though we will at least sketch how to estimate how many primes like in each such class.  Moreover, if things go well, we will apply these ideas to primes in short intervals, in short arithmetic progressions, study least quadratic non-residues, ....

Tues and Thurs 10:30-12:30 in room 5183

Prof. Andrew Granville

MAT6627

Institution: Université de Montréal

Analyse p-adique et groupe de Lie p-adique

Les sujets traités comprennent: Nombres p-adiques, corps p-adiques, topologie p-adique, théorie d'évaluation, théorie de ramification, groupe pro-fini, groupe pro-p, p-groupe puissant, groupe uniformément puissant, structure de Lie sur des groupes pro-p, algèbre d'Iwasawa, théorème de structure sur les modules d'Iwasawa.

Prof. Antonio Lei

MAT 7390

Institution: Université Laval

Algèbre: sujets spéciaux (Algèbres et modules)

Algèbres et morphismes, modules sur une algèbre, morphismes et suites exactes, modules de morphismes. Catégorie de modules, produits et sommes directs, équivalence de catégories. Foncteurs Hom, exactitude de foncteurs, modules projectifs et injectifs. Produits tensoriels de modules, théorèmes de Watts, algèbre tensorielle et extérieure. Algèbres des chemins d'un carquois. Modules artiniens et noethériens, suite de composition, théorème de Jordan-Hölder. Radicaux de modules, socle et coiffe.

Sources principales

• I Assem, Algèbres et modules, Masson et Les presses de l'Université d'Ottawa (1997).
• I Assem, D Simson, A Skowronski, Elements of the representation theory of associative algebras, Cambridge University Press (2006).

Prof. Yvan Saint-Aubin

MAT6681Q

Institution: Université de Montréal