Mathematical Physics

Program Description

The main areas of research of the group are:

  • classical and quantum integrable systems;
  • completely solvable statistical methods;
  • direct and inverse spectral transform methods;
  • applications to coherent nonlinear systems in fluids, solids, optics and plasmas;
  • spectral theory of random matrices and random operators;
  • integrable random processes, random partitions, random growth;
  • Laplacian growth and log Coulomb gases;
  • asymptotic methods in spectral analysis;
  • foundational problems in classical and quantum statistical mechanics;
  • solutions of classical nonlinear field equations (gauge theory, gravity);
  • symmetry analysis of P.D.E.'s;
  • quasi-crystals;
  • conformal field theory;
  • representation theory of Lie groups and quantum groups;
  • percolation phenomena;
  • foundational problems in quantization (stochastic and geometric quantization; coherent states);
  • mathematical structure of classical and quantum field theories (gauge theory; quantum gravity).

Most members of the program are also members of the CRM Mathematical Physics Laboratory.

Program Members

Academic Program

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.

  • Physics: classical mechanics; statistical mechanics; classical electrodynamics; quantum mechanics; relativity.
  • Mathematics: Real analysis, functional analysis; complex variables; differential equations; introductory group theory and algebra; measure theory.

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:

  • A= course offered at least annually at one of the participating universities.
  • B= course offered every second year at one of the participating universities.
  • C= course offered according to demand at one of the participating universities.
  • b= basic level (Master's level)
  • i= intermediate level (Master's or doctoral)
  • s= specialized course (Master's or doctoral)

(*) 1. Mathematical methods in physics (A, b)

  • McGill: Phys. 198-612 - Advanced Mathematical Physics I
  • McGill: Phys. 198-613 - Advanced Mathematical Physics II
  • McGill: Math 189-585 - Integral Equations and Transforms
  • McGill: Math 189-586 - Applied Partial Differential Equations
  • Univ. de Montréal: Mat 6435 - Equations de la physique

(*m) 2. Mathematical quantum mechanics (A, b)

  • Concordia: Math 684 /854 - Quantum mechanics / Quantization techniques

(*m) 3. Analytical mechanics (B, b)

  • McGill: Math 189-561 - Analytic Mechanics

4. Quantum field theory (A,i)

  • McGill: Phys. 198-673 Theoretical High Energy Physics
  • Univ. de Montréal: Phys. 6812 - Théorie des champs I
  • Univ. de Montréal: Phys. 6822 - Théorie des champs II

5. Statistical mechanics (A, i)

  • Concordia: Phys 661 - Nonequilibrium statistical mechanics
  • McGill: Phys 198-559 - Statistical mechanics

6. General Relativity (B, b)

7. Selected Topics in Mathematical Physics (C, s)

  • Concordia MAST 856A- Selected Topics in Mathematical Physics

8. Lie algebras and groups (A, b)

  • Concordia: Math 694 - Lie groups
  • Univ. de Montréal: Math 6681Q - Algèbre: sujets spéciaux
  • Univ. de Montréal MAT 6633 - Théorie des groupes de Lie
  • UQAM: Mat 7410 - Groupes et algèbres de Lie

9. Differentiable manifolds (A, b)

  • Concordia: Math 656 - Differential geometry
  • McGill: Math 189-576 - Geometry and topology I
  • McGill: Math 189-577 - Geometry and topology II
  • Univ. de Montréal: Math 6323 - Variétés différentiables
  • UQAM: Mat 8131 - Géométrie différentielle

10. Functional analysis (A, b)

  • Concordia: Math 662 - Functional analysis I
  • McGill: Math 189-635 - Functional analysis
  • Univ. de Montréal: Math 6112- Analyse fonctionelle I

11. Differential equations (A, i)

  • Concordia: Math 666 - Differential equations
  • McGill: Math 189-575 - Partial differential equations
  • Univ. de Montréal: Math 6180 - Equations différentielles

2018-19 Course Listings


Généralisations de l’analyse complexe et leurs applications

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.

Prof. Sébastien Tremblay

UQTR MAP6021-00

Institution: Université du Québec à Trois-Rivières


Topics in Geometry and Topology : Introduction to mathematical treatment of Einstein's general relativity theory

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.

Prof. Gantumur Tsogtgerel

MATH 599

Institution: McGill University

Surfaces de Riemann

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.

Prof. Vasilisa Shramchenko

MAT 737

Institution: Université de Sherbrooke

Topics in Geometry and Topology: Classical mechanics and symplectic geometry

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.

Prof. Brent Pym

MATH 599

Institution: McGill University

Entropy in Quantum Information Theory

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:

–Non-commutative probability
–Quantum entropies
–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.

Prof. Vojkan Jaksic

MATH 595

Institution: McGill University