# Mathematics and Statistics Courses

## Teaching of Mathematics MTM Courses

Each year the Department of Mathematics and Statistics offers a selection of the following courses.**Notes:**

- 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.

**Description**: 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.

**Component(s)**: Reading

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

**Component(s)**: Research

**Description**: Topics are chosen from the area of Number Theory.

**Notes:**

- 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.

**Description**: 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.).

**Component(s)**: Lecture

**Description**: Topics are chosen from the area of the Application of Mathematics.

**Notes:**

- 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.

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

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Notes:**

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

**Component(s)**: Lecture

**Description**: The course is an extension of undergraduate courses in mathematical analysis in the real domain (Analysis I, II; Real Analysis; Measure Theory).

**Component(s)**: Lecture

**Notes:**

Students may substitute this course with any of the MAST 660-669 courses in the MA/MSc program.

**Description**: The course is an extension of undergraduate courses in mathematical analysis in the complex domain (Complex Analysis I, II).

**Component(s)**: Lecture

**Notes:**

- Students may substitute this course with any of the MAST 660-669 courses in the MA/MSc program.

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

**Notes:**

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

**Component(s)**: Lecture

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

**Component(s)**: Lecture

**Description**: This course discusses theoretical and applied aspects of statistics and probability.

**Component(s)**: Lecture

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

**Notes:**

**Description**: Topics are chosen from the area of Mathematical Logic.

**Notes:**

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

**Component(s)**: Lecture

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

**Component(s)**: Lecture

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

**Notes:**

**Description**: 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.

**Component(s)**: Research

**Description**: 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.

**Component(s)**: Reading

**Description**: Topics are chosen from the area of the History of Mathematics.

**Notes:**

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

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Seminar

**Description**: 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.

**Component(s)**: Thesis Research

## Mathematics MA/MSc Courses

The MA/MSc courses offered by the Department of Mathematics and Statistics fall into the following categories:

Mathematics History and Methods Courses

Topology and Geometry Courses

Analysis Courses

Statistics and Actuarial Mathematics Courses

Applied Mathematics Courses

Algebra and Logic Courses

The course content will be reviewed each year

### Mathematics History and Methods Courses

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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).

**Notes:**

- 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.

**Description**: 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.

**Notes:**

- 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.

**Description**: 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.

**Notes:**

- 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.

### Topology and Geometry Courses

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

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

### Analysis Courses

**Component(s)**: Lecture

**Notes:**

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Component(s)**: Reading

**Notes:**

**Description**: 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.

**Component(s)**: Lecture

### Statistics and Actuarial Mathematics 600-level Courses

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

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

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: A review of statistical techniques and simple random sampling, varying probability sampling, stratified sampling, cluster and systematic sampling-ratio and product estimators.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

### Applied Mathematics Courses

**Component(s)**: Lecture

**Notes:**

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Component(s)**: Reading

**Notes:**

**Description**: Linear algebraic background material, linear differential and control systems, controllability and observability, properties of the attainable set, the maximal principle and time-optimal control.

**Component(s)**: Lecture

### Algebra and Logic Courses

**Description**: Field extensions, normality and separability, normal closures, the Galois correspondence, solution of equations by radicals, application of Galois theory, the fundamental theorem of algebra.

**Component(s)**: Lecture

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

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Component(s)**: Reading

**Notes:**

### Statistics and Actuarial Mathematics 700-level Courses

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

**Component(s)**: Lecture

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

**Component(s)**: Lecture

**Description**: Valuation methods, gains and losses, stochastic returns, dynamic control.

**Component(s)**: Lecture

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

**Component(s)**: Lecture

**Description**: General risk models; renewal processes; Cox processes; surplus control.

**Component(s)**: Lecture

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

**Component(s)**: Lecture

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

**Component(s)**: Lecture

**Description**: Cluster analysis, principal components, discriminant analysis, Mahalanobis distance, special topics.

**Component(s)**: Lecture

**Component(s)**: Reading

**Notes:**

### Mathematics MA/MSc Thesis and Literature Courses.

## Mathematics PhD Courses

The PhD courses offered by the Department of Mathematics and Statistics fall into the following categories:Number Theory and Computational Algebra Courses

Analysis Courses

Physics and Differential Geometry Courses

Dynamical Systems Courses

Statistics and Actuarial Mathematics Courses

### Elective Courses

### Number Theory and Computational Algebra Courses

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

**Component(s)**: Lecture

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

**Component(s)**: Lecture

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

**Component(s)**: Lecture

**Description**: 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.

### Analysis Courses

**Notes:**

### Mathematical Physics and Differential Geometry Courses

**Description**: 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).

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

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

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

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

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Notes:**

### Dynamical Systems Courses

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

### Statistics and Actuarial Mathematics 800-level Courses

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

**Component(s)**: Lecture

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

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

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

**Component(s)**: Lecture

**Description**: 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.

**Component(s)**: Lecture

**Description**: Generalizations of the classical risk model, renewal processes, Cox processes, diffusion models, ruin theory and optimal surplus control.

**Component(s)**: Lecture

### Seminars

### Thesis and Comprehensive Examinations Courses

**Description**: 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.

**Component(s)**: Thesis Research

**Description**: 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.

**Component(s)**: Thesis Research

**Description**: 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.

**Component(s)**: Thesis Research