Candidates will be selected on the basis of their past academic record, letters of recommendation and the relevance of the proposed area of research to the areas of specialization of the Department. The normal requirement for admission to the program is a Master of Science degree, with high standing in Mathematics or an allied discipline from a recognized university. Exceptional candidates who have successfully completed one-year's study at the Master's level may, upon approval by the Graduate Studies Committee, be exempted from the required completion of the Master's degree and admitted directly into the Ph.D. program.
The principal aim of the Ph.D. program is to enable students to attain levels of mastery in one of the Department's areas of specializaiton, commensurate with carrying on independent mathematical research at a high level. These areas are listed below. Our Department has over thirty faculty members who specialize in these areas, and every year visiting and research professors, post-doctoral fellows, and adjunct faculty join the Department to collaborate on research and sometimes teach specialized courses.
In addition to the courses offered at Concordia University, students can take advantage of the Institut des sciences mathématiques (ISM), an organization which facilitates students taking courses at any of its other member universities in Montreal and across Quebec.
To see a list of Faculty members in the following research areas, view Full-Time Faculty (Index by Research Area)
Students must complete a program of 90 credits, consisting of the following components:
b. Six Courses or seminars (18 credits);
2. Academic Standing
The 18 credits associated with seminar and coursework must be completed with a grade point average of at least 3.00. The specific program of courses and seminars, chosen from the list, will be determined by the Graduate Studies Committee in consultation with the student's Advisory Committee.
The minimum period of residence is two years of full-time graduate study, beyond the master's, or the equivalent in part-time study. (A minimum of one year of full-time study is normally expected).
4. Comprehensive Examination
The comprehensive examination is composed of the following two parts:
The comprehensive B examination is an oral presentation of the candidate's plan of his or her doctoral thesis in front of the student's Advisory Committee. It is normally taken within two years (6 terms) of the candidate's entry in to the program or the equivalent of part-time study.
Concurrently with the preparation for the Part B exam, the students will be engaging in their research work towards the dissertation. After submitting the doctoral thesis, the candidate is required to pass an oral defence of the thesis. The doctoral thesis must make an original contribution to mathematical knowledge, at a level suitable for publication in a reputable professional journal in the relevant area.
6. Average Time to Completion
Normally, a student completes all requirements for the degree, except for the thesis, within two years of entering the program. The normal period for completion of the program, for a student already holding the equivalent of a master's degree, is three to four years.
The doctoral level courses described below are offered when appropriate. All courses are worth 3 credits unless otherwise indicated.
Number Theory and Computational Algebra
MAST 830 - Cyclotomic Fields
L-series, Dirichlet theorem, Gauss sums, Stickelberger theorem, class groups and class number, circular units, analytic formulae.
MAST 831 - Class Field Theory
Local and global class field theory, ideles and adeles, reciprocity laws, existence theorem.
MAST 832 - Elliptic curves
Introduction to elliptic curves over finite fields, local and global fields, rational points, Mordell-Weil theorem, formal group.
MAST 833 - Selected Topics in Number Theory
MAST 834 - Selected Topics in Computational Algebra
MAST 837 - Selected Topics in Analysis
MAST 838 - Selected Topics in Pure Mathematics
Mathematical Physics and Differential Geometry
MAST 840 - Lie Groups
The mathematical theory of Lie groups and introduction to their representation theory with applications to mathematical physics. Topics will include classical Lie groups, one-parameter subgroups, Lie algebras and the exponential mapping, adjoint and coadjoint representations, roots and weights, the Killing form, semi-direct products, Haar measure and decompositions such as those of Cartan and Iwasawa. The theory of unitary representations on Hilbert spaces. Physical applications of compact Lie groups (such as SU(2) and SU(3)) and non-compact groups (such as the Lorentz and Poincaré groups).
MAST 841 - Partial Differential Equations (P.D.E.'s)
Introduction to the mathematical theory of PDE's, including applications to mathematical physics. Topics will include Sturm-Liouville systems, boundary value and eigenvalue problems, Green's functions for time-independent and time-dependent equations, Laplace and Fournier transform methods. Additional topics will be selected from the theory of elliptic equations (e.g. Laplace and Poisson equations), hyperbolic equations (e.g., the Cauchy problem for the wave equation) and parabolic equations (e.g., the Cauchy problem for the heat equation). Links will be made with the theory of differential operators and with analysis on manifolds.
MAST 851 - Differential Geometric Methods in Physics
Manifolds, differential systems, Riemannian, Kahlerian and symplectic geometry, bundles, super manifolds with applications to relativity, quantization, gauge field theory and Hamiltonian systems.
MAST 852 - Algebro-Geometric Methods in Physics
Algebraic curves, Jacobi varieties, theta functions, moduli spaces of holomorphic bundles and algebraic curves, rational maps, sheaves and cohomology with applications to gauge theory, relativity and integrable systems.
MAST 853 - Gauge Theory and Relativity
Yang-Mills theory, connections of fibre bundles, spinors, twistors, classical solutions, invariance groups, instantons, monopoles, topological invariants, Einstein equations, equations of motion, Kaluza-Klein, cosmological models, gravitational singularities.
MAST 854 - Quantization Methods
Geometric quantization, Borel quantization, Mackey quantization, stochastic and phase space quantization, the problems of prequantization and polarization, deformation theory, dequantization.
MAST 855 - Spectral Geometry
Schrödinger operators; min-max characterization of eigenvalues, geometry of the spectrum in parameter space, kinetic potentials, spectral approximation theory, linear combinations and smooth transformations of potentials, applications to the N-body problem.
MAST 856 - Selected Topics in Mathematical Physics
MAST 857 - Selected Topics in Differential Geometry
MAST 860 - Differentiable Dynamical Systems
The study of dynamical properties of diffeomorphisms or of one-parameter groups of diffeomorphisms (flows) defined on differentiable manifolds. Periodic points, the non-wandering set, and more general invariant sets, Smale's horseshoe, Anosov, and Morse-Smale systems, general hyperbolic systems, the stable manifold theorem, various forms of stability, Markov partitions and symbolic dynamics.
MAST 861 - Absolutely Continuous Invariant Measures
Review of functional analysis. Frobenius-Perronoperator and its properties, existence of absolutely continuous invariant measures for piecewise expanding transformations, properties of invariant densities, compactness of invariant densities, spectral decomposition of the Frobenius-Perron operator, bounds on the number of absolutely continuous invariant measures, perturbations of absolutely continuous invariant measures.
MAST 862 - Numerical Analysis of Nonlinear Problems
Continuation of solutions, homotopy methods, asymptotic stability, bifurcations, branch switching, limit points and higher order singularities, Hopf bifurcation, control of nonlinear phenomena, ODE with boundary and integral constraints, discretization, numerical stability and multiplicity, periodic solutions, Floquet multipliers, period doubling, tori, control of Hopf bifurcation and periodic solutions, travelling waves, rotations, bifurcation phenomena in partial differential equations, degenerate systems.
MAST 863 - Bifurcation Theory of Vector Fields
Local and global bifurcations. Generalized Hopf bifurcation and generalized homoclinic bifurcation. Hamiltonian systems and systems close to Hamiltonian systems, local codimension two bifurcations of flows.
MAST 865 - Selected Topics in Dynamical Systems
Statistics & Actuarial Mathematics
MAST 871 - Advanced Probability Theory
Definition of probability spaces, review of convergence concepts, conditioning and the Markov property, introduction to stochastic processes and martingales.
MAST 872 - Stochastic Processes
Stochastic sequences, martingales and semi-martingales, Gaussian processes, processes with independent increments, Markov processes, limit theorems for stochastic processes.
MAST 873 - Advanced Statistical Inference
Decision functions, randomization, optimal decision rules, the form of Bayes rule for estimation problems, admissibility and completeness, minimax, rules, invariant statistical decisions, admissible and minimax decision rules, uniformaly most powerful tests, unbiased tests, locally best tests, general linear hypothesis, multiple decision problems.
MAST 874 - Advanced Multivariate Inference
Wishart distribution, analysis of dispersion, tests of linear hypotheses, Rao's test for additional information, test for dimensionality, principal component analysis, discriminant analysis, Mahalanobis distance, cluster analysis, relations with sets of variates.
MAST 875 - Advanced Sampling
Unequal probability sampling, multistage sampling, super population models, Bayes and empirical Bayes estimation, estimation of variance from complex surveys, non-response errors and multivariate auxiliary information.
MAST 876 - Survival Analysis
Failure time models, inference in parametric models, proportional hazards, non-parametric inference, multivariate failure time data, competing risks.
MAST 877 - Reliability Theory
Reliability performance measures, unrepairable systems, repairable systems; load-strength reliability models, distributions with monotone failure rates, analysis of performance effectiveness, optimal redundance, heuristic methods in reliability.
MAST 878 - Advanced Risk Theory
Generalizations of the classical risk model, renewal processes, Cox processes, diffusion models, ruin theory and optimal surplus control.
MAST 881 - Selected Topics in Probability, Statistics and Actuarial Mathematics
MAST 858 - Seminar in Mathematical Physics
MAST 859 - Seminar in Differential Geometry
MAST 868 - Seminar in Dynamical Systems
MAST 889 - Seminar in Probability, Statistics and Actuarial Mathematics
MAST 898 - Seminar in Number Theory
MAST 899 - Seminar in Computational Algebra
Thesis and Comprehensive Examinations
MAST 890 - Comprehensive Examination A (6credits)
MAST 891 - Comprehensive Examination B(6 credits)
MAST 892 - Doctoral Thesis (60 credits)
1. From the Graduate Calendar
The comprehensive examination is composed of the following two parts:
This is a written examination, consisting of two parts. It will normally be completed within one year (3 terms) of the candidate's entry into the program or the equivalent of part-time study. Candidates are allowed at most one failure in the Part A examination. The first part of the Comprehensive A examination is to test the candidate’s general knowledge of fundamental mathematical concepts. The second part of the Comprehensive A examination tests the candidate’s knowledge of topics in his or her area of specialization. The material will be chosen from the list of course descriptions given by the Graduate Studies Committee in consultation with the candidate’s research supervisor and the student’s Advisory Committee.
The Comprehensive B examination is an oral presentation of the candidate’s plan of his or her doctoral thesis in front of the student’s Advisory Committee. It is normally taken within two years (6 terms) of the candidate's entry into the program or the equivalent of part-time study.
2.1 Common Part A1
The common part of the examination consists of 5 topics, standard in any undergraduate degree in Mathematics:
- Single Variable Real Analysis
Properties of the real numbers, infimum and supremum of sets. Numerical sequences and series. Limits of functions, continuous functions, intermediate value theorem, uniform continuity. Differentiation, mean value theorem, L’Hospital’s rule. Riemann integral, fundamental theorem of calculus. Sequences and series of functions, power series, uniform convergence, Taylor expansion.
References: M. Spivak, Calculus, 3rd edition; W. Rudin, Principles of Mathematical Analysis, McGraw-Hill.
- Linear Algebra
Matrices and systems of linear equations, Gauss reduction, elementary matrix operations. Vector spaces, subspaces, basis, dimension. Determinants. Inner product, orthogonality, orthonormal bases. Linear mappings, change of basis, kernel and image. Eigenvalues and eigenvectors, diagonalization, Jordan normal forms. Bilinear, quadratic and hermitian forms. Spectral theorem, Jordan canonical form, minimal polynomial.
References: S. Lipschutz, Linear Algebra, Schaum’s Outline Series, McGraw-Hill; S.H. Friedberg, A.J. Insel, L.E. Spence, Linear Algebra, 2nd edition, Prentice-Hall.
- Complex Analysis
Analytic functions, Cauchy-Riemann equations. Power series representation. Line integrals, Cauchy’s theorem and Cauchy’s integral formulas. Residue theorem and its application, Rouche’s theorem. Open mapping theorem. Morera theorem. Liouville’s theorem, fundamental theorem of algebra. Meromorphic functions and Laurent expansions. Fractional-linear (Mobius) transformations.
References: J.B. Convay, Functions of One Complex Variable, Springer Verlag; R.A. Silverman, Introductory Complex Analysis, Dover.
- Metric Spaces
Metric spaces, function spaces. Compactness, completeness, fixed-point theorems. Continuous functions and their properties. Ascoli-Arzela theorem. Weierstrass approximation theorem.
References: H. Royden, Real Analysis, 3rd edition, Macmillan; A. Friedman, Foundations of Modern Analysis, Dover.
- Measure Theory
Lebesgue measure and integration on real line, convergence theorems, absolute continuity, Lp spaces. General theory of measure and integration, Radon-Nikodym theorem.
References: H. Royden, Real Analysis, 3rd edition, Macmillan.
2.2 Specialized Part A2
The specialized part is taken in the area of study chosen by the student and the supervisor for the student's thesis work. The following is a partial list of some possible topics.
- Algebra and Number Theory
Group theory: group actions, Sylow’s theorems, finitely generated abelian groups. Ring theory: polynomial rings, Euclidean domains, unique factorization domains, principal ideal domains. Commutative algebra: localization, tensor products, rings and modules of fractions, integral extensions, Noetherian and Artinian rings, local rings, valuation rings, Dedekind domains, Hilbert basis theorem, Hilbert Nullstellensatz. Field theory and Galois theory: normal and separable extensions, solvable extensions, solvability by radicals, cyclic and cyclotomic extensions.
References: M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley; D. Dummit and R. Foote, Abstract Algebra, 2nd edition, Prentice-Hall; N. Jacobson, Basic Algebra I, Freeman; N. Jacobson, Basic Algebra II, Freeman;
S. Lang, Algebra, Addison-Wesley.
- Ergodic Theory and Dynamical Systems
One-dimensional dynamics, higher dimensional dynamics, complex dynamics. Invariant measures, basics of ergodic theory. Absolutely continuous invariant measures, Frobenius-Perron operator.
References: R.L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Addison-Wesley.
- Mathematical Physics
- Probability and Statistics
Random variables. Conditional probability and expectation. Markov chains. Poisson process. Continuous time Markov chains. Distributions, random vectors, random samples. Point estimation, testing of hypotheses, interval estimation. Decision theory. Non-parametric methods. Analysis of variance. Linear regression.
References: S.M. Ross, Introduction to Probability Models, Academic Press; G. Grimmett and D. Stirzaker, Probability and Random Processes, Oxford University Press; A.M. Mood, F.A. Graybill and D.C. Boes, Introduction to the Theory of Statistics, McGraw-Hill; R.V. Hogg and A.T. Craig, Introduction to Mathematical Statistics, Macmillan; G. Casella and R.L. Berger, Statistical inference, Duxbury Press.
The Department is part of the Institut des sciences mathématiques (ISM), a unique centre of excellence for graduate training in Mathematics, combining the resources and expertise of the mathematics departments at the four Montreal-area universities, as well as Université de Sherbrooke and Université Laval. The Department is also associated with the Centre de recherches mathématiques (CRM), a NSERC National Research Centre whose mission is to provide leadership in the development of mathematical sciences in Canada. The Department has close collaboration with Computer Science, as well as other departments and universities, which means that students can often engage in interdisciplinary research. A wide range of research seminars and lectures are held regularly, many of them being organized in conjunction with other Montreal universities, the ISM and the CRM.
The Concordia University Library subscribes to more than 80 mathematics and statistics journals and has reciprocal lending arrangements with the other three universities in Montreal. The University's Computer Centre offers excellent computing facilities on both campuses, through its research computers with their extensive software support. Additionally, the Department itself has a variety of personal computers and workstations, connected via ethernet and supporting a wide range of mathematical and text-editing software. Most full-time graduate students are provided with office space inside the Department. The common rooms are shared by faculty and students to facilitate academic and social contact.
Concordia University, one of the two English-language universities in Montreal, is located on two campuses: the Sir George Williams campus in downtown Montreal, and the Loyola campus in the west end of the city. The Department of Mathematics and Statistics is based on the Sir George Williams downton campus, on the 9th floor of the Library building.
Montreal is home to four universities and has a rich and cosmopolitan cultural life; virtually every international community is represented here. Concordia graduate students come from every part of the world, making for an enriching international work environment. Montreal is noted for its dynamic cultural life, its wide selection of sporting and recreational activities, safe environment, attractive residential neighbourhoods and a lively atmosphere. It is in close proximity to many other major centres in Canada and the U.S., as well as attractive vacationing areas.