Master of Science in Mathematics

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  • 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).
  • Academic title
    Master of Science in Mathematics
  • Course description

    Actuarial Mathematics & Statistics

    Areas of concentration are statistical inference in linear models, the estimation of various components, computationally intensive methods for inference, sampling and distribution theory. In applied probability, the areas of efficient techniques in network reliability and models in credibility, risk theory and actuarial statistics are currently under study. Faculty members currently doing research in this area: Y.P. Chaubey, P. Gaillardetz, C. Hyndman, J. Garrido, A. Sen, W. Sun, Q. Tang, and X. Zhou. (CRM Lab associated with the Statistics area: Statistics Research Group)

    Research topics include asymptotic analysis, harmonic analysis, inverse spectral methods, inverse monodromy problems, partial differential equations, and several complex variables. Faculty members currently doing research in this area: G. Dafn, A. Shnirelman and A. Stancu. (CRM Lab associated with this area: Mathematical Analysis Research Group)

    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, scientific computing and linear algebraic methods, nonsmooth analysis and control theory. Faculty members currently doing research in this area: A. Boyarsky, W. Byers, P. Gora, H. Proppe, and R. Stern.

    Mathematical Physics & Differential Geometry

    Present research deals with classical and quantum gauge theory, integrable systems (both classical and quantum) and coherent non-linear wave phenomena, isomonodromic deformations, quantization techniques, square integrable group representations, spectral analysis in quantum physics and spectral theory of random matrices. Faculty members currently doing research in this area: S.T. Ali, M. Bertola, C. Cummins, R. Hall, J. Harnad, and D. Korotkin. (CRM Labs associated with this area: Mathematical Physics Research Group and CIRGET)

    Number Theory & Computational Algebra
    Current work is centered around the Iwasawa theory, algebraic number fields, p-adic L-functions, the Artin L-series and arithmetic associated with algebro-geometric objects and the connection between modular functions and the finite simple groups. Faculty members currently doing research in this area: E. Cohen, C. David, A. Iovita, H. Kisilevsky, J. McKay, M. Szabo, and F. Thaine. (CRM Lab associated with the Number Theory area: CICMA).


    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


    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 guidelines of a faculty member.

Other programs related to mathematics, applied mathematics

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