Note that not all courses are offered in every session. Refer to the applicable timetable for details.
Students must ensure that prerequisites are met. Students may be deregistered, at the request of the instructor, from any course for which prerequisites and/or restrictions have not been met.
MATHEMATICS COURSES
*MATH 5F90
MSc Thesis
(also offered as STAT 5F90)
Research project involving the preparation of a thesis which will demonstrate a capacity for independent work. The research shall be carried out under the supervision of a faculty member.
*MATH 5P09
Solitons and Nonlinear Wave Equations
(also offered as PHYS 5P09)
Introduction to solitons: Linear and nonlinear travelling waves. Nonlinear evolution equations (Korteweg de Vries, nonlinear Schrodinger, sineGordon). Soliton solutions and their interaction properties. Lax pairs, inverse scattering, zerocurvature equations and Backlund transformations, Hamiltonian structures, conservation laws.
Note: taught in conjunction with MATH 4P09.
MATH 5P10
Groups, Rings, and Groups Rings
Advanced group theory and ring theory, such as group actions, pgroups and Sylow subgroups, solvable and nilpotent groups, finiteness conditions in rings, semisimplicity, the WedderburnArtin theorem. Introduction to group rings, such as unit groups, augmentation ideals, several important types of units, and the isomorphism problem.
MATH 5P11
Advanced Algebraic Structures
Topics may include: Algebraic coding theory; Combinatorial group theory; Advanced structures in ring theory.
Note: Taught in conjunction with either 4P11 or 4P13
*MATH 5P20
Computational Methods for Algebraic and Differential Systems
(also offered as PHYS 5P20)
Computer algebra applications of solving polynomial systems of algebraic and differential systems of equations are covered, including the necessary algebraic background. Polynomials and ideals, Groebner bases, affine varieties, solving by elimination, Groebner basis conversion, solving equations by resultants, differential algebra, differential Groebner bases.
MATH 5P21
High Performance Computing
Parallel computing architectures, new programming models, pilot parallel framework, parallel programming with MPI, threadbased parallelism, and a final project regarding the application of parallel computing to a mathematical problem.
Note: Students entering this course are expected to have a good grasp of basic procedural programming in a language such as C or FORTRAN.
*MATH 5P30
Dynamical Systems
(also offered as PHYS 5P68)
Introduction to dynamical systems and their applications in mathematical modeling. Linear flows, local theory of nonlinear flows, linearization theorems, stable manifold theorem. Global theory: limit sets and attractors, Poincare´Bendixson theorem. Structural stability and bifurcations of vector fields. Low dimensional phenomena in discrete dynamics. Chaotic dynamics: routes to chaos, characterization of chaos and strange attractors.
MATH 5P31
Ergodicity, Entropy and Chaos
Introduction to ergodic theory, invariant measures, Birkhoff ergodic theorem. The first return formula, Kac's lemma, recurrence theorems. Entropy, coding maps, ShannonMcMillian Breiman theorem. Chaos, predictability, Lyapunov exponent, speed of divergence, Pesin theorem. Ergodicity, entropy and chaos in shift dynamical systems and cellular automata.
MATH 5P35
Graph Theory
Basic definitions, paths and cycles, connectivity, trees and forests, bipartite graphs, Eulerian graphs; Matchings in bipartite graphs and in general graphs; Planar graphs, Euler's formula and Kuratowski's theorem. Graph colourings, Brooks' and Vizing's theorem and colouring of planar graphs; Network flows, MinMax Theorem.
MATH 5P36
Algorithmic Game Theory
Basic definitions, games, strategies, costs and payoffs, equilibria, cooperative games; Complexity of finding Nash equilibria; Mechanism design; Combinatorial auctions; Profit maximization in mechanism design; Cost sharing; Online mechanisms; Inefficiency of equilibria; Selfish routing; Network formation games; Potential function method; The price of anarchy; Sponsored search auctions.
MATH 5P40
Functional Analysis
Basic theory of Hilbert spaces, including the Projection Theorem, the Riesz Representation Theorem and the weak topology; weak derivatives, Sobolev spaces and the Sobolev Imbedding Theorem; the variational formulation of boundary value problems for ordinary and partial differential equations, the LaxMilgram Lemma and its applications; the finite element method.
MATH 5P41
Nonlinear Functional Analysis
Topological fixed point theory with applications to dynamical systems and optimization. Topics include the theorems of Brouwer, Borsuk, SchauderTychnoff, and Kakutani as well as the KnasterKuratowskiMazurkiewicz principle. Applications of these landmark results to the solvability and qualitative analysis of dynamical systems as well as convex and nonconvex optimization are discussed.
MATH 5P50
Algebraic Number Theory
Introduction to algebraic aspects of number theory. Topics include the general theory of factorization of ideals in Dedekind domains and number fields, Kummer's theory on lifting of prime ideals in extension fields, factorization of prime ideals in Galois extensions, local fields, the proof of Hensel's lemma, arithmetic of global fields.
*MATH 5P60
Partial Differential Equations
(also offered as PHYS 5P60)
Review of linear and nonlinear equations in two variables. Existence and uniqueness theory, fundamental solutions, initial/boundaryvalue formulas for the heat equation, wave equation, Laplace equation in multidimensions. Exact solution techniques for 1st and 2nd order linear and nonlinear equations. Analysis of solutions, variational formulations, conservation laws, Noether's theorem.
*MATH 5P64
Differential Geometry and Mathematical Physics
(also offered as PHYS 5P64)
Topics may include: Lagrangian and Hamiltonian mechanics, field theory, differential geometric structures, Lie groups and Lie algebras, Gbundles, manifolds, introduction to algebraic topology. Applications to theoretical physics.
*MATH 5P66
Matrix Groups and Linear Representations
(also offered as PHYS 5P66)
Abelian groups, permutation groups, rotation groups. Representations of discrete and continuous groups by linear transformations (matrices). General properties and constructions of group representations. Representations of specific groups. Lie groups and Lie algebras. Applications in various areas of Mathematics, including invariant theory and group algebras, and Theoretical Physics, including crystallography and symmetries in quantum systems.
#MATH 5P69
Introduction to Scientific Computing
(also offered as PHYS 5P10)
Survey of computational methods and techniques commonly used in condensed matter physics research; use of common subroutine libraries; symbolic computing systems; case studies from various areas of computational science; an independent study term project. Use of graphing and visualization software. Numerical differentiation and integration. Use of special functions. Monte Carlo and molecular dynamics simulation of structure, energetic and thermodynamic properties of metallic, semiconducting and ionic solids and nanoparticles.
MATH 5P70
Topology
Introduction to point set topology concepts and principles. Metric spaces; topological spaces; continuity, compactness; connectedness; countability and separation axioms; metrizability; completeness; Baire spaces.
MATH 5P75
Theory of Financial Mathematics
Mathematical models arising in modern investment practices. Compound interest, annuities, the timevalue of money, Markowitz portfolio theory, efficient frontier, random walks, Brownian processes, future contracts, European and American options, and putcall parity. Introduction to BlackScholes.
Note: Taught in conjunction with MATH 3P75
MATH 5P92
Cryptography and Number Theory
Topics may include RSA cryptosystems, ElGamal cryptosystem, algorithms for discrete logarithmic problem, elliptic curves, computing point multiples on elliptic curves, primality testing and factoring algorithms.
Note: taught in conjunction with MATH 4P92.
MATH 5P96
Technology and Mathematics Education
Critical appraisal of mathematics learning and teaching with digital technology, such as computer algebra systems and dynamic geometry software. Topics may include contemporary research concerning impact on curriculum; use in assessment; implementation in the classroom; mathematics computer games; use in modeling, probability, and statistics; data science and computational thinking; and the design of a rich technologybased mathematical activity.
Note: taught in conjunction with MATH 4P96.
*MATH 5P99
Major Research Paper
(also offered as STAT 5P99)
Survey paper on a topic chosen in consultation with a supervisor from one of the research areas of specialization.
MATH 5V755V79
Selected Topics in Mathematics and Statistics
Investigation of a specific area or group of related topics in mathematics or statistics.
STATISTICS COURSES
#STAT 5F90
MSc Thesis
(also offered as MATH 5F90)
Research project involving the preparation of a thesis which will demonstrate a capacity for independent work. The research shall be carried out under the supervision of a faculty member.
STAT 5P81
Sampling Theory
Theory of finite population sampling; simple random sampling; sampling proportion; estimation of sample size; Stratified sampling; optimal allocation of sample sizes; ratio estimators; regression estimators; systematic and cluster sampling; multistage sampling; error in surveys; computational techniques and computer packages, and related topics. Case studies.
Note: taught in conjunction with STAT 4P81
STAT 5P82
Nonparametric Statistics
Order statistics; rank tests and statistics; methods based on the binomial distribution; contingency tables; KolmogorovSmirnov statistics; nonparametric analysis of variance; nonparametric regression; comparisons with parametric methods. Computational techniques and use of SAS, Maple or other statistical packages, Case Studies.
Note: taught in conjunction with STAT 4P82.
STAT 5P83
Linear Models
Classical linear model, generalized inverse matrix, distribution and quadratic forms, regression model, nested classification and classification with interaction, covariance analysis, variance components, binary data, polynomial data, log linear model, linear logit models, generalized linear model, conditional likelihoods, quasilikelihoods, estimating equations, computational techniques and related topics.
Prerequisite(s): STAT 3P86 (or equivalent) or permission of the instructor.
STAT 5P84
Time Series Analysis and Stochastic Processes
Multivariate, marginal and conditional Normal distributions; partial correlation coefficient. Probability density function of sample correlation coefficient. Autoregressive and Moving Average models; stability analysis; serial correlation; forecasting; maximum likelihood estimation. Spectral theory; spectrum of ARMA models; aliasing. Spectral analysis of timeseries data; periodogram; data filtering; spectrum smooting by spectral or lag window; using a kernel. Estimating serial correlation; largesample theory.
STAT 5P85
Mathematical Statistical Inference
Revision of probability theory, convergence of random variables, statistical models, sufficiency and ancillarity, point estimation, likelihood theory, optimal estimation, Bayesian methods, computational methods, minimum variance estimation, interval estimation and hypothesis testing, linear and generalized linear models, goodnessoffit for discrete and continuous data, robustness, large sample theory, Bayesian inference.
STAT 5P86
Multivariate Statistics
Theory of multivariate statistics, matrix algebra and random vector, sample geometry and random sampling, multivariate normal distribution, inference about means, covariance matrix, generalized Hotelling's Tsquare distribution, sample covariance and sample generalized variance, Wishart distribution, general hypothesis testing, analysis of variance and linear regression model, principle components, factor analysis, covariance analysis, canonical correlation analysis, discrimination and classification, cluster analysis and related topics.
Prerequisite(s): STAT 3P86 (or equivalent) or permission of the instructor.
STAT 5P87
Computational Statistics
Classification: logistic regression, linear and quadratic discriminant analysis. Resampling methods: crossvalidation and bootstrap. Linear model selection and regularization: subset selection, shrinkage methods, dimension reduction methods, considerations in high dimensions. Nonlinear regression: polynomial regression, regression splines, smoothing splines, local regression and generalized additive models. Treebased methods: decision trees, bagging, random forests, and boosting. Support vector machines: maximal margin classifier, support vector classifier, support vector machines (SVMs), SVMs with more than two classes. Unsupervised learning: principal component analysis, clustering methods.
Prerequisite(s): STAT 3P82 and STAT 3P86 (or their equivalences), or permission of the instructor.
STAT 5P88
Advanced Statistics
Topics may vary year to year. Advanced methods and theory in statistical inference, survival analysis, risk analysis, sampling techniques, bootstrapping, Jackknife, generalized linear models, mixed models, modern computational statistics, quality control, life data modeling, biostatistics, multivariate analysis, time series analysis and related topics.
#STAT 5P89
Bayesian and Causal Bayesian Networks
(also offered as STAT 4P89)
Representation (factorizing joint probabilistic distributions, exploiting probabilistic independence properties, dseparation, IMap, naive Bayes); parameter learning (maximum likelihood estimation, Bayesian parameter estimation); structure learning (constrainedbased approaches, scorebased approaches, Bayesian model averaging); learning with incomplete data (parameter and structure learning, learning with hidden variables); causal Bayesian networks (intervention, SCMs, inference, learning). Use of R or Python.
Prerequisite(s): STAT 2P81, STAT 3P85, or the permission of the instructor.
STAT 5P95
Statistics Seminar
Independent study and presentation of major research papers in areas of specialization.
Note: this course will be evaluated as Credit/NoCredit.
#STAT 5P99
Major Research Paper
(also offered as MATH 5P99)
Survey paper on a topic chosen in consultation with a supervisor from one of the research areas of specialization.

