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)
Les thèmes principaux qui seront étudiés dans ce cours sont les quaternions, les algèbres de Clifford ainsi que la théorie des fonctions analytiques généralisées (fonctions pseudo-analytiques). Ces structures seront également utilisées pour considérer certaines applications, principalement en physique quantique. Pour toutes ces structures, nous allons porter une attention particulière aux généralisations des fonctions analytiques complexes. Dans le cas des quaternions et des algèbres de Clifford, les propriétés algébriques ainsi que géométriques seront considérées. La théorie des fonctions pseudo-analytiques généralise et préserve plusieurs caractéristiques de la théorie des fonctions analytiques complexes. Le système de Cauchy-Riemann est alors substitué par un système plus général, appelé équations de Vekua, qui apparaît dans plusieurs problèmes de la physique mathématique.
If you have taken or are taking the physics GR course, the two courses should complement each other nicely. In particular, there will not be much overlap. While a considerable part of the physics course is (probably) spent on introducing differential geometry, we will assume that the students are comfortable with basic differential geometry. Exact solutions with high degree of symmetry will be studied as prototypical examples of spacetimes, but our focus will be on the properties of realistic spacetimes with no or very little symmetry.
The following topics will be treated.
• Some exact solutions, including black hole and cosmological solutions.
• Lorentzian geometry, geodesic congruences, variational characterization of geodesics.
• Singularity theorems of Penrose and Hawking. These theorems are the highlight of the course, and basically show that spacetimes cannot avoid developing singularities.
• Cauchy problem, if time permits. This result says that the state of the universe "today" completely determines what happens in the future in a certain sense.
The grading will be based on a few homework, and a course project, where the student studies a special topic and gives a presentation.
Ce cours est une introduction à la théorie des surfaces de Riemann. Le préalable exigé est une connaissance de base de l'analyse complexe.
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.
Location: Monday and Wednesday 14h35-15h55 BURN 1214
This course will serve as an introduction to the Hamiltonian formulation of classical mechanics, and the underlying differential geometry of symplectic and Poisson manifolds. We will cover:
- examples of mechanical systems, e.g. oscillators, pendulums and tops - Hamilton's equations of motion
- definitions and basic properties of symplectic and Poisson manifolds
- Liouville's theorem on phase space volumes
- Lagrangian submanifolds and Weinstein's neighbourhood theorem - momentum, symmetries and symplectic reduction
- integrable systems and action-angle variables
Further advanced topics may be selected based on the tastes and background of the audience. Possibilities include local normal forms and stability of equilibiria; classification of toric integrable systems via Delzant polytopes; perturbations of integrable systems and the Kolmogorov-Arnold-Moser (KAM) theorem; Arnold's conjecture on periodic orbits and rudiments of Floer theory; or links with quantum mechanics via geometric/deformation quantization.
This course is a modern introduction to quantum information theory centred around the notions of quantum entropies. The presentation will be essentially self-contained. Pre-requisites are Honours Algebra II and Honours Analysis II, or the permission of the instructor. The topics to be covered are:
–Quantum statistics (hypothesis testing and parameter estimation)
–Mathematical theory of quantum channels, finitely correlated states, and repeated quantum measurements
Additional topics may include quantum Shannon-McMillan-Breiman theorem and theory of quantum spin systems (time permitting).
The course will cover some of the most recent developments in quantum information theory and the lecture notes will be provided.