Admission Requirements

Applicants must have a Bachelor's degree with Honours in Mathematics, or equivalent. Qualified applicants requiring prerequisite courses may be required to take up to 12 undergraduate credits in addition to and as a part of the regular graduate program. Promising candidates who lack the equivalent of an Honours degree in Mathematics may be admitted after having completed a qualifying program.

 

Program Objectives

The goal of the M.A./M.Sc. program is to provide students with basic knowledge sufficient for undertaking doctoral study, as well as applying mathemtical expertise in industry should doctoral studies not be pursued.  Students generally follow a program tailored to their individual needs and interests, with either a thesis or a project option, and many receive financial support through fellowships or teaching and research assistantships.  Master's students may also avail themselves of the Institut des sciences mathématiques (ISM).

  

Areas of Study

To see a list of Faculty members in the following research areas, view Full-Time Faculty (Index by Research Area)

• Actuarial Mathematics & Mathematical Finance
  This group is active in commodity market models, credibility theory, forward-backward stochastic differential equations, insurance statistics, risk theory, stochastic analysis and valuation of financial guarantees embedded in life insurance contracts. 

• Analysis, Partial Differential Equations & Applied Mathematics
  Research topics include harmonic analysis on Euclidean space, in particular the theory of singular integral operators, Hardy spaces, and other function spaces, applications of harmonic analysis to partial differential equations, applications of PDE's and curvature flows to convex geometry, and the study of geometric properties of solutions of PDE's, including evolution equations of curves and surfaces by curvature driven speed, fluid dynamics and turbulence, mathematical neuroscience. (CRM Lab associated with this area: Mathematical Analysis Laboratory)

• Dynamical Systems
  Current research topics include ergodic theory and absolutely continuous invariant measures, the interplay between ergodic theory and topological analysis, computer modeling, small stochastic perturbations, random maps theory with applications to modelling financial markets, scientific computing and linear algebraic methods, nonsmooth analysis and control theory, bifurcation theory and chaos.

• Mathematical Physics & Differential Geometry
  Present research deals with quantum and classical integrable systems, inverse spectral and inverse monodromy methods, moduli spaces of Riemann surfaces, relativistic field theory, spectral theory of random matrices and random processes, quajtization techniques, square integrable group representations, wavelets and signal analysis, symmetry reduction techniques and spectral analysis in quantum physics.   (CRM Labs associated with this area: Mathematical Physics Laboratory and CIRGET)

• Number Theory & Computational Algebra
  Current work is centered around the algebraic, analytic and p-adic theory arising from the arithmetic of cyclotomic fields, elliptic and modular curves and higher dimensional algebro-geometric objects, their associated L-functions, Galois representations, Iwasawa theory, modular functions and forms and their group theoretic connections (for example, "Moonshine").   (CRM Lab associated with the Number Theory area: CICMA).

• Statistics & Probability
  Areas of concentration are curve estimation, distribution theory, estimation of variance components, inference, linear models, modeling data, non-parametric methods, survey sampling, survival analysis, stochastic processes and applicaitons, stochastic analysis and non-linear filtering.   (CRM Lab associated with the Statistics area: Statistics Laboratory)

 

Degree Requirements

1. Credits
A candidate is required to complete a minimum of 45 credits.

2. Residence
The minimum residence requirement is one year (3 terms) of full-time study, or the equivalent in part-time study.

3. Courses
Students may enter one of the two options below. The choice of the option, the selection of courses and the topic of the thesis, must be approved by the Graduate Program Director.

• Master of Science/Arts with Thesis (Option A)
  Candidates are required to take six 3-credit courses, or equivalent, and MAST 700- Thesis.

• Master of Science/Arts without Thesis (Option B)
  Candidates are required to take ten 3-credit courses, or equivalent, and MAST 701- Project.

4. Course Load
A full-time student will take at least two courses during the first term. A part-time student will normally take one course during the first term. The course load during subsequent terms will be determined by the Graduate Program Director, in consultation with the student.

 

Courses

The Master of Science/Arts courses offered by the Department of Mathematics and Statististics fall into the following categories.

MAST 650          History and Methods
MAST 655-659    Topology and Geometry
MAST 660-669    Analysis
MAST 670-679    Statistics and Actuarial Mathematics
MAST 680-689    Applied Mathematics
MAST 690-699    Algebra and Logic
MAST 720-729    Statistics & Actuarial Mathematics

The course content will be reviewed each year in light of the interests of the students and faculty. In any session only those courses will be given for which there is sufficient demand.  Courses are worth 3 credits unless otherwise indicated.

History and Methods

MAST 650 - Development of Mathematical Ideas (6 credits)

Topology and Geometry

MAST 655 - Topology
Topological spaces. Order, product, subspace, quotient topologies. Continuous functions. Compactness and connectedness. The fundamental group and covering spaces.

MAST 656 - Differential Geometry
Mappings, functions and vectors fields on Rn, inverse and implicit function theorem, differentiable manifolds, immersions, submanifolds, Lie groups, transformation groups, tangent and cotangent bundles, vector fields, flows, Lie derivatives, Frobenius' theorem, tensors, tensor fields, differential forms, exterior differential calculus, partitions of unity, integration on manifolds, Stokes' theorem, Poincaré lemma, introduction to symplectic geometry and Hamiltonian systems.

MAST 657 - Manifolds

MAST 658 - Lie Groups

Analysis

MAST 661 - Topics in Analysis

MAST 662 - Functional Analysis I
This course will be an introduction to the theory of Hilbert spaces and the spectral analysis of self-adjoint and normal operators on Hilbert spaces.  Applications could include Stone's theorem on one parameter groups and/or reproducing kernel Hilbert spaces.

MAST 663 - Introduction to Ergodic Theory
This course covers the following topics: measurable transformations, functional analysis review, the Birkhoff Ergodic Theorem, the Mean Ergodic Theorem, recurrence, ergodicity, mixing, examples, entropy, invariant measures and existence of invariant measures.

MAST 664 - Dynamical Systems
An introduction to the range of dynamical behaviour exhibited by one-dimensional dynamical systems. Recurrence, hyperbolicity, chaotic behaviour, topological conjugacy, structural stability, and bifurcation theory for one-parameter families of transformation. The study of unimodal functions on the interval such as the family Fr(X) = rx (1-x), where 0 < r < 4. For general continuous maps of the interval, the structure of the set of periodic orbits, for example, is found in the theorem of Sarkovskii.

MAST 665 - Complex Analysis
Review of Cauchy-Riemann equations, holomorphic and meromorphic functions, Cauchy integral theorem, calculus of residues, Laurent series, elementary multiple-valued functions, periodic meromorphic functions, elliptic functions of Jacobi and Wierstrass, elliptic integrals, theta functions.  Riemann surfaces, uniformization, algebraic curves, Abelian integrals, the Abel map, Riemann theta functions, Abel's theorem, Jacobi varieties, Jacobi inversion problem.  Applications to differential equations.

MAST 666 - Differential Equations

MAST 667 - Reading Course in Analysis

MAST 668 - Transform Calculus

MAST 669 - Measure Theory
Measure and integration, measure spaces, convergence theorems, Radon-Nikodem theorem, measure and outer measure, extension theorem, product measures, Hausdorf measure, LP-spaces, Riesz theorem, bounded linear functionals on C(X), conditional expectations and martingales.

Statistics & Actuarial Mathematics

MAST 670 - Mathematical Methods in Statistics
This course will discuss mathematical topics which may be used concurrently or subsequently in other statistics stream courses. The topics will come mainly from the following broad categories: 1) geometry of Euclidean space; 2) matrix theory and distribution of quadratic forms; 3) measure theory applications (Reimann-Stieltjes integrals); 4) complex variables (characteristic functions and inversion); 5) inequalities (Cauchy-Schwarz, Holder, Minkowski, etc.) and numerical techniques (Newton-Raphson algorithm, scoring method, statistical differentials); 6) some topics from probability theory.

MAST 671 - Probability Theory
Axiomatic construction of probability; characteristic and generating functions; probabilistic models in reliability theory; laws of large numbers; infinitely divisible distributions; the asymptotic theory of extreme order statistics.

MAST 672 - Statistical Inference I
Order statistics; estimation theory; properties of estimators; maximum likelihood method; Bayes estimation; sufficiency and completeness; interval estimation; shortest length confidence interval; Bayesian intervals; sequential estimation.

MAST 673 - Statistical Inference II
Testing of hypotheses; Neyman-Pearson theory; optimal tests; linear hypotheses; invariance; sequential analysis.

MAST 674 - Multivariate Analysis
An introduction to multivariate distributions will be provided; multivariate normal distribution and its properties will be investigated. Estimation and testing problems related with multivariate normal populations will be discussed with emphasis on Hotelling's generalized T2 and Wishart distribution. Other multivariate techniques including MANOVA; canonical correlations and principal components may also be introduced.

MAST 675 - Sample Surveys
A review of statistical techniques and simple random sampling. varying probability sampling, stratified sampling, cluster and systematic sampling-ratio and product estimators.

MAST 676 - Linear Models
Matrix approach to development and prediction in linear models will be used. Statistical inferences on the parameters will be discussed after development of proper distribution theory. The concept of generalized inverse will be fully developed and analysis of variance models with fixed and mixed effects will be analyzed.

MAST 677 - Time Series
Statistical analysis of time series in the time domain.  Moving average and exponential smoothing methods to forecast seasonal and non-seasonal time series, construction of prediction intervals for future observations, Box-Jenkins ARIMA models and their applications to forecasting seasonal and non-seasonal time series.  A substantial portion of the course will involve computer analysis of time series using computer packages (mainly MINITAB). No prior computer knowledge is required.

MAST 678 - Statistical Consulting and Data Analysis

MAST 679 - Topics in Statistics and Probabality

MAST 720 - Survival Analysis
Parametric and non-parametric failure time models; proportional hazards; competing risks.

MAST 721 - Advanced Actuarial Mathematics
General risk contingencies; advanced mutliple life theory; population theory; funding methods and dynamic control.

MAST 722 - Advanced Pension Mathematics
Valuation methods, gains and losses, stochastic returns, dynamic control.

MAST 723 - Portfolio Theory
Asset and liability management models, optimal portfolio selection, stochastic returns, special topics.

MAST 724 - Risk Theory
General risk models; renewal processes; Cox processes; surplus control.

MAST 725 - Credibility Theory
Classical, regression and hierarchical Bayes models, empirical credibility, robust credibility, special topics.

MAST 726 - Loss Distributions
Heavy tailed distributions, grouped/censured data, point and interval estimation, goodness-of-fit, model selection.

MAST 727 - Risk Classification
Cluster analysis, principal components, discriminant analysis, Mahalanobis distance, special topics.

MAST 728 - Reading Course in Actuarial Mathematics

MAST 729 - Selected Topics in Actuarial Mathematics

Applied Mathematics

MAST 680 - Topics in Applied Mathematics

MAST 681 - Optimization
Introduction to nonsmooth analysis: generalized directional derivative, generalized gradient, nonsmooth calculus; connections with convex analysis. Mathematical programming: optimality conditions; generalized multiplier approach to constraint qualifications and sensitivity analysis. Application of the theory: functions defined as pointwise maxima of a family of functions; minimizing the maximal eigenvalue of a matrix-valued function; variational analysis of an extended eigenvalue problem.

MAST 682 - Matrix Analysis
Jordan canonical form and applications, Perron-Frobenius theory of nonnegative matrices with applications to economics and biology, generalizations to matrices which leave a cone invariant.

MAST 683 - Numerical Analysis
This course consists of fundamental topics in numerical analysis with a bias towards analytical problems involving optimization, integration, differential equations and Fourier transforms.  The computer language C++ will be introduced and studied as part of this course; the use of "functional programming" and graphical techniques will be strongly encouraged.  By the end of the course, students should have made a good start on the construction of a personal library of tools for exploring and solving  mathematical problems numerically.

MAST 684 - Quantum Mechanics
The aim of this course is two-fold: (i) to provide an elementary account of the theory of non- relativistic bound systems, and (ii) to give an introduction to some current research in this area, including spectral geometry.

MAST 685 - Approximation Theory

MAST 686 - Reading Course in Applied Mathematics

MAST 687 - Control Theory
Linear algebraic background material, linear differential and control systems, controllability and observability, properties of the attainable set, the maximal principle and time-optimal control.

MAST 688 - Stability Theory

MAST 689 - Variational Methods

Algebra and Logic

MAST 691 - Mathematical Logic

MAST 692 - Advanced Algebra I
Field extensions, normality and separability, normal closures, the Galois correspondence, solution of equations by radicals, application of Galois theory, the fundamental theorem of algebra.

MAST 693 - Algebraic Number Theory
Dedekind domains; ideal class groups; ramification; discriminant and different; Dirichlet unit theorem; decomposition of primes; local fields; cyclotomic fields.

MAST 694 - Group Theory
Introduction to group theory, including the following topics:  continuous and locally compact groups, subgroups and associated homogeneous spaces.  Haar measures, quasi-invariant measures, group extensions and universal covering groups, unitary representations, Euclidean and Poincaré groups, square integrability of group representations with applications to image processing.

MAST 696 - Advanced Algebra II

MAST 697 - Reading Course in Algebra

MAST 698 - Category Theory

MAST 699 - Topics in Algebra

Thesis and Mathematical Literature

MAST 700 - Thesis (27 credits)

MAST 701 - Project (15 credits)
A student investigates a mathematical topic, prepares a report and gives a seminar presentation under the guidance of a faculty member.

 

The ISM/CRM

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.

 

Other Information

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.