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

2021-22 Course Listings

Fall

Topics in Mathematics and Statistics: Large Deviation Principle and Statistical Mechanics of Lattice Gases

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.

Prof. Vojkan Jaksic

MATH 594

Institution: McGill University

Théorie de la représentation des groupes finis

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.

Prof. Yvan Saint-Aubin

MAT 6621

Institution: Université de Montréal