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)
This topic course concerns the interplay between the Large Deviation Principle (LDP) of probability theory and the mathematical foundations of statistical mechanics (SM). Although this fundamental link goes back to the pioneering work of Boltzmann and has played a central role in the development of both subjects, it is rarely discussed at the introductory level. The goal of the course is to describe the basic theory of LDP and SM with an emphasis on the foundational link between them.
Topics to be covered:
LDP. Cramér’s theorem in the i.i.d. setting. General structure of LDP. Gärtner-Ellis theorem. Boltzmann-Sanov theorems. Method of Ruelle-Lanford’s functions. Varadhan’s Lemma. Applications.
SM of Lattice Gasses. Interactions and pressure. Entropy. Boltzmann and Gibbs equilibrium states. Equivalence of ensembles. Theory of Gibbs states. Hausdorff dimension and Boltzmann entropy. Information theory perspective. Beyond Gibbsianity.
Additional topics will include: LDP and SM in the general dynamical systems setting. Thermodynamic formalism of dynamical systems. Rotators, dynamics, and the 0- Law of Thermodynamics.
Prerequisites. Honours Analysis 3-Math 454, Honours Probability-Math 356, and willingness to pick up on the pre-requisite topics (which are of independent interest) as we proceed. The references, and in some cases pre-recorded videos with pre-requisites, will be provided. In exceptional cases (and this in particular applies to the Joint Honours Math. Phys students), the course can be taken with Math 454 and Math 356 or Phys 362 as co-prerequisites. If you are interested to do so, please contact the instructor.
Représentations et module d'un groupe G, représentations équivalentes, sous-module. Représentations indécomposable, réductible, irréductible. Théorème de Maschke. Morphisme, lemme de Schur.
Algèbre de groupe, fonctions sur cette algèbre, fonction de classe. Caractères, relations d'orthogonalité, tables de caractères. Représentation régulière. Analyse de Fourier sur les groupes finis, identité de Parseval, théorème de Wedderburn.
Nombres algébriques, théorème de la dimension, théorème de Burnside. Action de groupe, lemme de Burnside, paires de Gelfand.
Représentations induites, théorème de réciprocité de Frobenius, critère d'irréductibilité de Mackey.
Marche aléatoire sur les groupes finis. Modèles de Gilbert–Shannon–Reeds, théorèmes de Diaconis.