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

2019-20 Course Listings


Symétries et équations différentielles

1) L'application de la théorie de groupes de Lie à la résolution des équations différentielles ordinaires et aux dérivés partiels. La séparation de variables dans les équations linéaires aux dérivés partielles. Les équations non linéaires décrivant des phénomènes non linéaires en physique.

(2) Concept d'intégrabilité pour les systèmes de dimension fini et infini.

(3) Singularités des solutions des équations différentielles ordinaires et aux dérivés partielles. La propriété de Painlevé et les transcendantes de Painlevé.

Les sujets qui seront traités:

1) Résumé des résultats pertinents de la théorie des groupes et des algèbres de Lie. Groupes de Lie locaux. Groupes de transformations locales.

(2)  Champs de vecteurs et leurs prolongations.

(3)  Comment trouver le groupe de symétries d'un système d'équations différentielles. Les équations déterminants.

(4)  Exemples de calcul et d'applications de groupes de symétries

(5)  La méthode de réduction par symétrie pour les équations aux dérivés partielles.

(6)  Classification de sous algèbres d'une algèbre de Lie.

(7)  Solutions des équations différentielles ordinaires.

(8)  Séparation de variables dans les équations aux dérivées partielles linéaires.

(9) Singularités de solutions des équations différentielles. La propriété de Painlevé. Classification des équations avec la propriété de Painlevé. Les fonctions elliptiques

Prof. Pavel Winternitz

MAT 6436

Institution: Université de Montréal