**Admission Requirements. **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 MSc 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 PhD program.

**Requirements for the Degree**

**Credits.**Students must complete a program of 90 credits, consisting of the following components:- Comprehensive examinations (12 credits);
- Six courses or seminars (18 credits);
- Thesis (60 credits).

**Academic Standing.**The 18 credits associated with seminar and course work 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.

**Residence.**The minimum period of residence is two years of full-time graduate study, beyond the MA/MSc, or the equivalent in part-time study. (A minimum of one year of full-time study is normally expected).

**Comprehensive Examination.**The comprehensive examination is composed of the following two parts:**Part A**(6 credits)

This is a written examination, consisting of two parts. The first part of the Comprehensive A examination is to test the candidate’s general knowledge of fundamental mathematical concepts. 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. 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. Candidates are allowed at most one failure in the Part A examination.**Part B**(6 credits)

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-three years of the candidate’s entry into the program (or the equivalent of part-time study) and at least one year before the expected completion of the thesis.

**Thesis.**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.

**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 an MA/MSc degree, is three to four years.

**Academic Regulations**

**GPA Requirement.**The 18 course and seminar credits 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.

**C Rule.**Students who receive more than one*C*grade during the course of their PhD studies will be required to withdraw from the program. Students may apply for re-admission. Students who receive another*C*after re-admission will be required to withdraw from the program and will not be considered for re-admission.

**F Rule.**Students who receive a failing grade in the course of their PhD studies will be withdrawn from the program. Students may apply for re-admission. Students who receive another failing grade after re-admission will be withdrawn from the program and will not be considered for re-admission.

**Time Limit.**All work must be completed within 18 terms (6 years) of full-time study or 24 terms (8 years) of part-time study, from the time of initial registration in the program.

**Graduation Requirement.**In order to graduate, students must have a cumulative GPA of at least 3.00.

**Number Theory and Computational Algebra**

**MAST 830 Cyclotomic Fields** (3 credits)

L-series, Dirichlet theorem, Gauss sums, Stickelberger theorem, class groups and class number, circular units, analytic formulae.

**MAST 831 Class Field Theory **(3 credits)

Local and global class field theory, ideles and adeles, reciprocity laws, existence theorem.

**MAST 832 Elliptic Curves **(3 credits)

Introduction to elliptic curves over finite fields, local and global fields, rational points, Mordell-Weil theorem, formal groups.

**MAST 833 Selected Topics in Number Theory **(3 credits)

**MAST 834 Selected Topics in Computational Algebra **(3 credits)

**Analysis**

**MAST 837 Selected Topics in Analysis **(3 credits)

**MAST 838 Selected Topics in Pure Mathematics** (3 credits)

**Mathematical Physics and Differential Geometry**

**MAST 840 Lie Groups **(3 credits)

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)** (3 credits)

Introduction to the mathematical theory of P.D.E.’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 Fourier 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 **(3 credits)

Manifolds, differential systems, Riemannian, Kahlerian and symplectic geometry, bundles, supermanifolds with applications to relativity, quantization, gauge field theory and Hamiltonian systems.

**MAST 852 Algebro-Geometric Methods in Physics **(3 credits)

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 **(3 credits)

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 **(3 credits)

Geometric quantization, Borel quantization, Mackey quantization, stochastic and phase space quantization, the problems of prequantization and polarization, deformation theory, dequantization.

**MAST 855 Spectral Geometry **(3 credits)

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** (3 credits)

**MAST 857 Selected Topics in Differential Geometry **(3 credits)

**Dynamical Systems**

**MAST 860 Differentiable Dynamical Systems **(3 credits)

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** (3 credits)

Review of functional analysis, Frobenius-Perron operator 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 **(3 credits)

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 **(3 credits)

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 **(3 credits)

**Statistics and Actuarial Mathematics**

**MAST 871 Advanced Probability Theory **(3 credits)

Definition of probability spaces, review of convergence concepts, conditioning and the Markov property, introduction to stochastic processes and martingales.

**MAST 872 Stochastic Processes **(3 credits)

Stochastic sequences, martingales and semi-martingales, Gaussian processes, processes with independent increments, Markov processes, limit theorems for stochastic processes.

**MAST 873 Advanced Statistical Inference **(3 credits)

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, uniformly most powerful tests, unbiased tests, locally best tests, general linear hypothesis, multiple decision problems.

**MAST 874 Advanced Multivariate Inference **(3 credits)

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 **(3 credits)

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 **(3 credits)

Failure time models, inference in parametric models, proportional hazards, non-parametric inference, multivariate failure time data, competing risks.

**MAST 877 Reliability Theory **(3 credits)

Reliability performance measures, unrepairable systems, repairable systems, load-strength reliability models, distributions with monotone failure rates, analysis of performance effectiveness, optimal redundancy, heuristic methods in reliability.

**MAST 878 Advanced Risk Theory **(3 credits)

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 **(3 credits)

**Seminars**

**MAST 858 Seminar in Mathematical Physics **(3 credits)

**MAST 859 Seminar in Differential Geometry **(3 credits)

**MAST 868 Seminar in Dynamical Systems **(3 credits)

**MAST 889 Seminar in Probability, Statistics and Actuarial Mathematics **(3 credits)

**MAST 898 Seminar in Number Theory **(3 credits)

**MAST 899 Seminar in Computational Algebra **(3 credits)

**Thesis and Comprehensive Examinations**

**MAST 890 Comprehensive Examination A **(6 credits)

**MAST 891 Comprehensive Examination B** (6 credits)

**MAST 892 Doctoral Thesis **(60 credits)

**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.

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

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

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

**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.

**Academic Regulations**

**GPA Requirement.**The academic progress of students is monitored on a periodic basis. To be permitted to continue in the program, students must obtain a cumulative grade point average (GPA) of 3.00 based on a minimum of 12 credits. Students whose GPA falls below 3.00 are considered to be on academic probation during the following review period. Students whose GPA falls below 3.00 for two consecutive review periods are withdrawn from the program.

**C Rule.**Normally a student receiving a grade of*C*in two courses will be required to withdraw from the program. Students withdrawing for this reason may petition the Graduate Studies Committee for special consideration. In cases of extenuating circumstances probationary continuation in the program will be considered.

**F Rule.**Students who receive a failing grade in the course of their studies will be withdrawn from the program. Students may apply for re-admission. Students who receive another failing grade after re-admission will be withdrawn from the program and will not be considered for re-admission.

**Time Limit.**All work for a MA/MSc degree for full-time students must be completed within 12 terms (4 years) from the time of initial registration in the program at Concordia University; for part-time students the time limit is 15 terms (5 years).

**Graduation Requirement.**In order to graduate, students must have a cumulative GPA of at least 3.00.

**Master of/Magisteriate in Science/Arts with Thesis (Option A)**

Candidates are required to take six 3-credit courses, or equivalent, and MAST 700.

**Master of/Magisteriate in Science/Arts without Thesis (Option B)**

Candidates are required to take ten 3-credit courses, or equivalent, and MAST 701.

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

**MAST 650-654 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 and 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.

**History and Methods**

**MAST 651 The Contributions of Mathematics to Intellectual Life** (3 credits)

This course examines several major mathematical advances over the centuries in the historical and intellectual contexts of the day and also focuses on the developments of a particular branch of mathematics over the more recent past. Examples may include recent advances in number theory and geometry leading to a proof of Fermat’s Last Theorem and applications of number theory to cryptography.

**MAST 652 Topics in Research in Mathematics Education** (3 credits)

The general aim of this course is to acquaint students with research problems in mathematics education and ways of approaching them (theoretical frameworks and research methodologies).

**Note:** The content varies from term to term and from year to year. Students may re-register for this course provided the course content has changed. Changes in content are indicated by the title of the course.

**MAST 653 Topics in the Foundations of Mathematics** (3 credits)

This course focuses on foundational issues and developments in mathematics, with topics chosen from particular branches of mathematics, e.g., geometry (Euclidean and non-Euclidean geometries; comparison of Euclid’s “Elements” with Hilbert’s “Grundlagen der Geometrie”, etc.), or logic (evolution of logic from Aristotle to Boole; Hilbert’s program; Gödel’s Incompleteness theorems, etc.). It may also look at foundational problems in mathematics suggested by physics and other sciences. More general, philosophical, epistemological and methodological questions about the nature of mathematics may also be chosen as topics for the course.

**Note:** The content varies from term to term and from year to year. Students may re-register for this course provided the course content has changed. Changes in content are indicated by the title of the course.

**MAST 654 Topics in the History of Mathematics** (3 credits)

This course may focus on a particular epoch and place in the history of mathematics (e.g., Ancient Greek, Indian and Chinese mathematics; the development of mathematics in Europe in the 17th to 19th centuries, etc.), or on the history of a particular area of mathematics (history of geometry, algebra, analysis, number theory, etc.). Aspects related to the history of approaches to teaching mathematics may also be addressed.

**Note:** The content varies from term to term and from year to year. Students may re-register for this course provided the course content has changed. Changes in content are indicated bythe title of the course.

**Topology and Geometry**

**MAST 655 Topology **(3 credits)

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

**MAST 656 Differential Geometry **(3 credits)

Mappings, functions and vectors fields on R^{n}, 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 **(3 credits)

**MAST 658 Lie Groups **(3 credits)

**Analysis**

**MAST 661 Topics in Analysis **(3 credits)

**MAST 662 Functional Analysis I **(3 credits)

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 **(3 credits)

This course covers the following topics: measurable transformations, functional analysis review, the Birkhoff Ergodic Theorem, the Mean Ergodic Theorem, recurrence, ergodicity, mixing, examples, entrophy, invariant measures and existence of invariant measures.

**MAST 664 Dynamical Systems **(3 credits)

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 *F*r (*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 **(3 credits)

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 **(3 credits)

**MAST 667 Reading Course in Analysis **(3 credits)

**MAST 668 Transform Calculus **(3 credits)

**MAST 669 Measure Theory **(3 credits)

Measure and integration, measure spaces, convergence theorems, Radon-Nikodem theorem, measure and outer measure, extension theorem, product measures, Hausdorf measure, L^{p}-spaces, Riesz theorem, bounded linear functionals on C(X), conditional expectations and martingales.

**Statistics and Actuarial Mathematics**

**MAST 670 Mathematical Methods in Statistics **(3 credits)

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 **(3 credits)

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 **(3 credits)

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 **(3 credits)

Testing of hypotheses; Neyman-Pearson theory; optimal tests; linear hypotheses; invariance; sequential analysis.

**MAST 674 Multivariate Analysis **(3 credits)

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 T^{2} and Wishart distribution. Other multivariate techniques including MANOVA; canonical correlations and principal components may also be introduced.

**MAST 675 Sample Surveys **(3 credits)

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 **(3 credits)

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 **(3 credits)

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 **(3 credits)

**MAST 679 Topics in Statistics and Probability **(3 credits)

**MAST 720 Survival Analysis **(3 credits)

Parametric and non-parametric failure time models; proportional hazards; competing risks.

**MAST 721 Advanced Actuarial Mathematics **(3 credits)

General risk contingencies; advanced multiple life theory; population theory; funding methods and dynamic control.

**MAST 722 Advanced Pension Mathematics **(3 credits)

Valuation methods, gains and losses, stochastic returns, dynamic control.

**MAST 723 Portfolio Theory **(3 credits)

Asset and liability management models, optimal portfolio selection, stochastic returns, special topics.

**MAST 724 Risk Theory **(3 credits)

General risk models; renewal processes; Cox processes; surplus control.

**MAST 725 Credibility Theory **(3 credits)

Classical, regression and hierarchical Bayes models, empirical credibility, robust credibility, special topics.

**MAST 726 Loss Distributions **(3 credits)

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

**MAST 727 Risk Classification **(3 credits)

Cluster analysis, principal components, discriminant analysis, Mahalanobis distance, special topics.

**MAST 728 Reading Course in Actuarial Mathematics **(3 credits)

**MAST 729 Selected Topics in Actuarial Mathematics **(3 credits)

**Applied Mathematics**

**MAST 680 Topics in Applied Mathematics **(3 credits)

**MAST 681 Optimization **(3 credits)

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 **(3 credits)

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 **(3 credits)

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 **(3 credits)

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 **(3 credits)

**MAST 686 Reading Course in Applied Mathematics **(3 credits)

**MAST 687 Control Theory **(3 credits)

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 **(3 credits)

**MAST 689 Variational Methods **(3 credits)

**Algebra and Logic**

**MAST 691 Mathematical Logic **(3 credits)

**MAST 692 Advanced Algebra I **(3 credits)

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 **(3 credits)

Dedekind domains; ideal class groups; ramification; discriminant and different; Dirichlet unit theorem; decomposition of primes; local fields; cyclotomic fields.

**MAST 694 Group Theory **(3 credits)

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 **(3 credits)

**MAST 697 Reading Course in Algebra **(3 credits)

**MAST 698 Category Theory **(3 credits)

**MAST 699 Topics in Algebra **(3 credits)

**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.

#### Note: Admissions have been suspended.

**Admission Requirements. **A Bachelor’s degree with a minimum GPA of 3.00, an interest in the teaching of pre-university mathematics, as well as an adequate mathematical background including courses equivalent to: a) 6 credits in statistics-probability; b) 6 credits in advanced calculus; c) 6 credits in linear algebra and d) 3 credits in differential equations or algebraic systems. Candidates must be able to demonstrate their capacity for graduate level work in some academic field, not necessarily mathematics. Candidates will normally be interviewed to ensure their suitability for the program. Applicants with a deficiency in their academic background may be required to take up to 12 undergraduate credits in addition to or as a part of the regular graduate program. Promising candidates who lack the requirements for admission may be considered after having completed a qualifying program. Applicants without teaching experience may be admitted to the program provided they satisfy the Graduate Studies Committee of their potential for teaching or for educational research.

**Requirements for the Degree**

**Credits.**A fully-qualified candidate is required to complete a minimum of 45 credits.

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

**Courses.**

Students may enter one of the three options below. The choice of the option, the selection of the courses and the thesis or project topic must be approved by the Graduate Program Director. Besides the courses listed in the present section, Master/Magisteriate in the Teaching of Mathematics (MTM) students may take any MAST 600 or higher level course offered in the MSc program, subject to the Graduate Program Director’s approval. Students aspiring to become College mathematics teachers upon graduation will be encouraged to take at least three MSc mathematics courses.*Thesis Option:*MATH 602, 647, 654 and eight additional 3-credit courses.*Project Option:*MATH 602, 603 and eleven additional 3-credit courses.*Course Option:*Fifteen 3-credit courses.

**Academic Regulations**

**GPA Requirement.**The academic progress of students is monitored on a yearly basis. To be permitted to continue in the program, students must obtain a cumulative grade point average (GPA) of 3.00 based on a minimum of 12 credits. Students whose GPA falls below 3.00 are considered to be on academic probation during the following review period. Students whose GPA falls below 3.00 for two consecutive review periods are withdrawn from the program.

**C Rule.**Normally a student receiving a grade of*C*in two courses will be required to withdraw from the program. Students withdrawing for this reason may petition the Graduate Studies Committee for special consideration. In cases of extenuating circumstances probationary continuation in the program will be considered.

**F Rule.**Students who receive a failing grade in the course of their studies will be withdrawn from the program. Students may apply for re-admission. Students who receive another failing grade after re-admission will be withdrawn from the program and will not be considered for re-admission.

**Time Limit.**All work for a master’s/magisteriate degree for full-time students must be completed within 12 terms (4 years) from the time of initial registration in the program at Concordia University; for part-time students the time limit is 15 terms (5 years).

**Graduation Requirement.**In order to graduate, students must have a cumulative GPA of at least 3.00.

MTM courses fall into six categories:

- Psychology of Mathematics Education (PME): MATH 630, 649.
- Didactics of Mathematics (DM): MATH 624.
- Information and Communication Technology (ICT): MATH 633, 634, and 639.
- Research in Mathematics Education (RME): MATH 641, 642, 645, and 646.
- Mathematics content courses (MC): MATH 601, 613, 616, 618, 621, 622, 625, 626, 627, 637, 640, and 648.
- Thesis or Extended Project (T/P): Seminar MATH 652; Reading courses MATH 602 and 647; Extended Project MATH 603, and Thesis MATH 654.

Each year the Department of Mathematics and Statistics offers a selection of the following courses. Courses are worth 3 credits unless otherwise indicated.

**MATH 601 Topics in Mathematics**

**Note:** The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated bythe title of the course.

**MATH 602 Readings in Mathematics Education I**

This reading course is closely related to the project or thesis. The outcome is a section of the literature review chapter, related to the domain of research that is the focus of the project or thesis.

**MATH 603 Extended Project** (9 credits)

A student investigates a mathematics education topic, prepares a report, and gives a seminar presentation under the guidance of a faculty member.

**MATH 613 Topics in Number Theory**

Topics are chosen from the area of Number Theory.

**Note:** The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated bythe title of the course.

**MATH 616 Linear Algebra**

This course is an extension of undergraduate courses in linear algebra, covering a selection of topics in advanced linear algebra (e.g. from the theory of general vector spaces, linear and multilinear algebras, matrix theory, etc.)

**MATH 618 Topics in the Application of Mathematics**

Topics are chosen from the area of the Application of Mathematics.

**Note:** The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated bythe title of the course.

**MATH 621 Geometry**

The course offers an insight into Euclidean and Non-Euclidean geometries.

**MATH 622 Abstract Algebra**

The course looks at objects such as numbers, polynomials, matrices or transformations from an algebraic-structural point of view. The course may aim at proving such “famous impossibilities” as squaring the circle, duplicating the cube, trisecting an angle or solving a polynomial equation of degree 5 or more by radicals.

**MATH 624 Topics in Mathematics Education**

This course is an overview and critical analysis of theories and technologies of mathematics teaching. Applications of the theories to studying and/or developing teaching situations or tools for specific mathematical topics are examined.

**Note:** The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated bythe title of the course.

**MATH 625 Topology**

The course develops elements of the theory of topological spaces and their transformations.

**MATH 626 Analysis I**

The course is an extension of undergraduate courses in mathematical analysis in the real domain (Analysis I, II; Real Analysis; Measure Theory). Students may substitute this course with any of the MAST 660-669 courses in the MA/MSc program.

**MATH 627 Analysis II**

The course is an extension of undergraduate courses in mathematical analysis in the complex domain (Complex Analysis I, II). Students may substitute this course with any of the MAST 660-669 courses in the MA/MSc program.

**MATH 630 Topics in the Psychology of Mathematics Education**

This course studies epistemological, cognitive, affective, social and cultural issues involved in mathematics.

**Note:** The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated bythe title of the course.

**MATH 633 Applications of Technology in Mathematics Curriculum Development**

This course is an overview of the impact of information and communication technology on curricula, textbooks and teaching approaches.

**MATH 634 Computer Software and Mathematics Instruction**

This course is an overview and critical evaluation of computer software designed for use in mathematics instruction.

**MATH 637 Statistics and Probability**

This course discusses theoretical and applied aspects of statistics and probability. Students may substitute this course with any of the MAST 670-677 courses in the MA/MSc program.

**MATH 639 Topics in Technology in Mathematics Education**

This course involves the elaboration, experimentation and critical analysis of individual projects of integration of ICT in mathematics education.

**Note:** The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated bythe title of the course.

**MATH 640 Topics in Logic**

Topics are chosen from the area of Mathematical Logic.

**Note:** The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated bythe title of the course.

**MATH 641 Survey of Research in Mathematics Education**

This course is an overview of recent results in mathematics education research.

**MATH 642 Research Methods for Mathematics Education**

This course is an overview of qualitative and quantitative methods in mathematics education research.

**MATH 645 Topics in Mathematics Education Research**

This course is an overview of research literature on a chosen topic or issue in mathematics education.

**Note:** The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated bythe title of the course.

**MATH 646 Research Internship**

Students conduct a pilot study or participate in a research project as a research assistant under the supervision of a senior researcher. The outcome is a written report of the study.

**MATH 647 Readings in Mathematics Education II**

The course is closely related to project or thesis writing. Its outcome is a section of the literature review chapter, focused on the student’s particular research question.

**MATH 648 Topics in the History of Mathematics**

Topics are chosen from the area of the History of Mathematics.

**Note:** The content varies from term to term and from year to year. Students may re-register for this course, provided the course content has changed. Changes in content are indicated bythe title of the course.

**MATH 649 Heuristics and Problem Solving**

This course examines cognitive processes, tools and strategies involved in solving mathematical problems.

**MATH 652 Seminar in Mathematics Education**

This course is primarily a thesis or project preparation seminar but it is open to students in the Course Option as well. The research related to students’ research projects is presented and critically evaluated.

**MATH 654 Thesis** (15 credits)

Students are required to demonstrate their ability to carry out original, independent research. The thesis is researched and written under the direction of a supervisor and thesis committee. Upon completion of the thesis, the student is required to defend his/her thesis before the thesis committee.