# Mathematics MA/MSc

## 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.

**Proficiency in English. **Applicants whose primary language is not English must demonstrate that their knowledge of English is sufficient to pursue graduate studies in their chosen field. Please refer to the Graduate Admission page for further information on the Language Proficiency requirements and exemptions.

## Requirements for the Degree

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

**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

**Academic Standing.**Please refer to the Academic Standing section of the Calendar for a detailed review of the Academic Regulations.

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

**Time Limit.**Please refer to the Academic Regulation page for further details regarding the Time Limit requirements.

**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.

## Courses

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.