The main areas of research of the group are:
Most members of the program are also members of the CRM Mathematical Physics Laboratory.
It is expected that students entering the program have an adequate preparation in both physics and mathematics. The normal requirement is either a master's degree or the equivalent of a Canadian honours degree in one of the two disciplines, with strong preparation in the other. The following is a list of subjects with which the incoming student is assumed to have familiarity at this level.
Besides the specific courses offered this year, the following general framework of courses is recommended to students doing their degrees within this program. The needs and background of each individual student will determine which of these courses is required; the choice and timing should be determined in consultation with the student's thesis advisor. In any particular year, these courses might be offered at only one of the participating universities, but the titles and course numbers are listed in order to facilitate cross registration. In the following, an asterisk (*) signifies a (Master's level) course that is obligatory for all students in the programme, and (*m) signifies a course that is obligatory for students who have not already completed the equivalent at a level equivalent to an honours level undergraduate degree. The following notation is used to distinguish the level and frequency of courses offered:
(*) 1. Mathematical methods in physics (A, b)
(*m) 2. Mathematical quantum mechanics (A, b)
(*m) 3. Analytical mechanics (B, b)
4. Quantum field theory (A,i)
5. Statistical mechanics (A, i)
6. General Relativity (B, b)
7. Selected Topics in Mathematical Physics (C, s)
8. Lie algebras and groups (A, b)
9. Differentiable manifolds (A, b)
10. Functional analysis (A, b)
11. Differential equations (A, i)
The course aims at introducing the notion of Lie group and Lie algebra, with focus on concrete matrix representations. We will start with a minimalistic review of necessary notions of differential geometry and move progressively into the theory of the classification of semisimple Lie algebras (and groups). We will also discuss some representation theory.
After the introduction, the main topics we aim at covering are: Connection between Lie groups and Lie Algebras (Baker-Cambpell-Hausdorff); general theory of Lie algebras; nilpotent, solvable and semisimple algebras; Classical simple Lie groups and algebras; classification (Dynkin diagrams); representation theory (Weyl theorem, weight spaces, irreducible reps, characters); integration over groups and Haar measure.
We will keep the prerequisites to a minimum.
The course concerns modern entropic information theory centred around coding algorithms. It will follow the book “The Ergodic Theory of Discrete Sample Paths” by Paul C. Shields AMS Graduate Studies in Mathematics, Vol 13, 1996, with digressions concerning more recent developments. The course will continue with a research seminar in the Winter 2021. Various research opportunities (post-doctoral, PhD, masters, and undergraduate level) related to the material of the course and the seminar will be available starting Summer 2021. Besides the students in mathematics and statistics, this course might be of interest to mathematically inclined students in electrical engineering and computer science. Staring with the seminal work of Shannon, most of the central topics of the course originated in the field of electrical engineering/information transmission.
Ce cours est une introduction à la théorie des surfaces de Riemann. Le préalable exigé est une connaissance de base de l'analyse complexe.
Contenu: Surfaces de Riemann compactes. Structures complexes engendrées par une métrique. Applications holomorphes. Revêtements ramifiés de la sphère de Riemann, formule de Riemann-Hurwitz. Topologie et formes différentielles sur les surfaces de Riemann. Différentielles abéliennes, Jacobien. Fonctions méromorphes sur les surfaces de Riemann compactes. Théorème d'Abel. Théorème de Riemann-Roch. Fonctions théta, fonctions de Weierstrass. Aperçu des courbes algébriques.
L'objectif du concours est de présenter les notions principales derésolution des équations aux dérivées partielles (EDP). Dans ce cours,nous présentons les sujets suivants :
EDP non linéaires du premier ordre. Solutions à l'aide de la méthode deMonge, Formulation du problème de Cauchy, Intégrations complètes et lescrochets de Jacobi et de Piosson, Les équations de Hamilton-Jacobi.
EDP du deuxième ordre hyperbolique, elliptique et parabolique.Classification des EDP du second ordre par la méthode de Beltrami,Théorème d'existence des solutions et théorème de Cauchy-Kowalewska,Intégrale intermédiaire pour les équations linéaires de type hyperbolique,Problème de Sturm-Liouville et polynômes orthogonaux, Résolution par laméthode de cascade de Laplace, Transformation d'homographe et de Legendre,Méthode d'intégration de Riemann, Méthode de la moyenne sphérique, Méthoded'Hadamard et le principe de Duhamel, Fonction de Green et solutionfondamentale. Unification de presque toutes les méthodes connues pourtrouver les solutions exactes et analytiques par la théorie des groupes.