# Mathematics and Statistics

The department offers a selection of courses from the following list each academic year. Course offerings for the current academic year can be found on the Department of Mathematics and Statistics website.

**Courses in Mathematics**

**MATH-800* Seminar **

Students are expected to participate in a weekly seminar in which they are required to present material on a topic that relates to their research.

**MATH-801* Graph Theory **

An introduction to graph theory, one of the central disciplines of discrete mathematics. Topics include: graphs,subgraphs, trees, connectivity, Euler tours, Hamiltonian cycles, matchings, independent sets,cliques, colourings, and planarity.(Offered jointly with MATH-401*.) Three term-hours; lectures.

**MATH-802* Enumerative Combinatorics **

Enumerative combinatorics is concerned with counting the number of elements of finite sets with prescribed conditions. The techniques covered include inclusion-exclusion, bijective proofs, double-counting arguments, recurrence relations, and generating functions. (Offered jointly with MATH-402.) Three term hours; lectures.

**MATH-806* Introduction to Coding Theory **

Construction and properties of finite fields. Polynomials, vector spaces, block codes over finite fields. Hamming distance and other code parameters. Bounds relating code parameters. Cyclic codes and their structure as ideals. Weight distribution. Special codes and their relation to designs and projective planes. Decoding algorithms. (Offered jointly with MATH/MTHE 406*.) Three term-hours; lectures.

**MATH-812* Topics in Number Theory **

Subject matter may vary from year to year. Three term-hours; lectures.

**MATH-813* Introduction to Algebraic Geometry **

An introduction to the study of systems of polynomial equations in one or many variables. Topics covered include the Hilbert basis theorem, the Nullstellenstaz, the dictionary between ideals and affine varieties, and projective geometry (Offered jointly with MATH 413*). Three term-hours; lectures.

**MATH-818* Number Theory and Cryptography **

Time estimates for arithmetic and elementary number theory algorithms (division algorithm, Euclidean algorithm, congruences), modular arithmetic, finite fields, quadratic residues. Design of simple cryptographic systems; public key, RSA systems. Primality and factoring: pseudoprimes, Pollard's rho-method, index calculus. Elliptic curve cryptography. (Offered jointly with MATH/MTHE-418*.) Three term hours; lectures.

**MATH-827* Introduction to Deterministic Dynamical Systems **

Topics include: global properties of flows and diffeomorphisms; invariant sets and dynamics; bifurcations of fixed and periodic points; stability and chaos. (Offered jointly with MATH-427*.) Three term-hours; lectures.

**MATH-830* Modern Control Theory **

This course covers core topics in modern control theory: Linearization, existence and uniqueness of trajectories for nonlinear and linear systems, the transition matrix,controllability, observability, minimal realizations, feedback stabilization, linear state observers, optimal control theory, the linear quadratic regulator, dynamic programming. (Offered jointly with MTHE-430*.) Three term-hours; lectures.

**MATH-834* Optimization Theory and Applications **

Theory of convex sets and functions; separation theorems; primal-dual properties; geometric treatment of optimization problems; algorithmic procedures for solving constrained optimization programs;engineering and economics applications. (Offered jointly with MATH-434). Three term-hours; lectures.

**MATH-836* Lagrangian Mechanics, Dynamics, and Control **

Geometric modelling, including configuration space, tangent bundle, kinetic energy, inertia, and force. Euler-Lagrange equations using affine connections. The last part of the course develops one of the following three applications: mechanical systems with nonholonomic constraints; control theory for mechanical systems; equilibria and stability. (Offered jointly with MATH/MTHE-439*) Three term-hours; lectures.

**MATH-837* Topics in Applied Mathematics **

Subject matter may vary from year to year. Three term-hours; lectures.

**MATH-838* Topics in Mathematical Biology **

Subject matter may vary from year to year. Three term-hours; lectures.

**MATH-844* Differentiable Manifolds **

Differentiable structures, smooth manifolds and submanifolds, immersions and submersions, vector fields and differential forms, orientation and integration, de Rham cohomology. Three term-hours; lectures.

**MATH-872* Control of Stochastic Systems **

Stabilization and optimization of controlled dynamical systems under probabilistic uncertainty. Topics include: review of probability, controlled Markov chains, martingale and Lyapunov methods for stochastic stability, dynamic programming, partially observed models and non-linear filtering, the Kalman Filter, average cost problems, learning and computational methods, decentralized stochastic control, and stochastic control in continuous-time. (Offered jointly with MTHE- 472*.) Three term hours, fall or winter; lectures.

**MATH-874* Information Theory **

An introduction to the fundamental principles of the theory of communication. Topics include:information measures, entropy, mutual information, divergence; modeling of information sources, discrete memoryless sources, Markov sources, entropy rate, source redundancy, fundamentals of lossless data compression, block encoding, variable-length encoding, Kraft inequality, design of Shannon-Fano and Huffman codes; fundamentals of channel coding, channel capacity, noisy channel coding theorem, channels with memory, lossless information transmission theorem; continuous-alphabet sources and channels, differential entropy, capacity of discrete-time and band-limited continuous-time Gaussian channels; rate-distortion theory, lossy data compression, rate-distortion theorem, lossy information transmission theorem. (Offered jointly with MATH/MTHE-474*). Three term hours; lectures.

**MATH-877* Data Compression and Source Coding **

Fundamentals of the theoretical and practical (algorithmic) aspects of lossless and lossy data compression. Topics include: adaptive Huffman coding, arithmetic coding, the fundamental performance limits of universal lossless coding, Lempel-Ziv and related dictionary based methods, the Burrows-Wheeler transform, elements of Kolmogorov complexity theory, rate-distortion theory, scalar and vector quantization, applications to speech and image coding. (Offered jointly with MATH/MTHE-477*.) PREREQUISITE: MATH-874*

**MATH-884* Data Networks **

This course covers performance models for data networking, delay models and loss models; analysis of multiple access systems, routing, and flow control; multiplexing; priority systems; satellite multiple access, wireless networking, wireless sensor networks. Knowledge of networking protocols is not required. (Offered jointly with MATH/MTHE-484*.) Three term hours; lectures. PREREQUISITE: STAT-855*

**MATH-891* Core Course in Analysis I **

This course provides basic knowledge in real and complex analysis at the graduate level on the following topics: Lebesgue measure and integration theory; elementary Hilbert space theory; examples of Banach space techniques. Three term-hours, fall; lectures.

**MATH-892* Core Course in Analysis II **

This course provides basic knowledge in real and complex analysis at the graduate level on the following topics: basic theory of Fourier transforms; basic elements of spectral theory and Banach algebras; complex analysis. Three term-hours, winter; lectures.

**MATH-893* Core Course in Algebra I **

This course provides basic knowledge in algebra at the graduate level on the following topics: elementary theory of groups; elementary theory of rings and modules; Galois theory. Three term-hours, fall; lectures.

**MATH-894* Core Course in Algebra II **

This course provides basic knowledge in algebra at the graduate level on the following topics: representation theory of finite groups through characters; advanced theory of modules; advanced theory of rings. Three term-hours, winter; lectures.

**MATH-895* Core Course in Probability Theory **

This course provides basic knowledge in probability at the graduate level. Topics will include: basic notions and concepts of Probability Theory; characteristic functions; law of large numbers and central limit theorem; martingales; stochastic processes. Three term-hours, winter; lectures.

**MATH-896* Core Course in Mathematical Statistics **

This course provides basic knowledge in mathematical statistics at the graduate level. Topics will include: Classical and Bayesian inference, Multivariate Gaussian distribution and its applications in Statistics; decision theory; basic techniques of non-parametric estimation. Three term-hours, fall; lectures.

**MATH-898 Master's Project **

**MATH-899 Master's Thesis Research **

**MATH-901* Research Institute Course **

Advanced topics course, normally offered in the summer term, by a research institute in Canada or abroad can be taken for credit with the permission of the Supervisor and Coordinator of Graduate Studies and in cooperation with Institute organizers. Grades are assigned on a PASS - FAIL basis.

**MATH-902* Topics in Algebra **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-903* Topics in Algebra **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-905* Topics in Algebra **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-912* Topics in Number Theory **

Subject matter will vary from year to year. Three term-hours; seminar or reading course.

**MATH-913* Topics in Number Theory **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-915* Topics in Number Theory **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-922* Topics in Analysis **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-923* Topics in Analysis **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-925* Topics in Analysis **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-932* Topics in Applied Mathematics **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-933* Topics in Applied Mathematics **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-935* Topics in Applied Mathematics **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-936* Topics in Control Theory **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-937* Topics in Control Theory **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-939* Topics in Control Theory **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-942* Topics in Topology and Geometry **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-943* Topics in Topology and Geometry **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-945* Topics in Topology and Geometry **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-972* Topics in Communication Theory **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-973* Topics in Communication Theory **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-975* Topics in Communication Theory **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**MATH-999 Ph.D. Thesis Research **

**COURSES IN PROBABILITY AND STATISTICS**

**STAT-853* Statistical Inference **

Decision theory and Bayesian inference; principles of optimal statistical procedures; maximum likelihood principle; large sample theory for maximum likelihood estimates; principles of hypotheses testing and the Neyman-Pearson theory; generalized likelihood ratio tests; the chi-square, t, F and other distributions. (Offered jointly with STAT-463*.) Three term hours; lectures.

**STAT-854* Statistical Spectrum Estimation **

Many systems evolve with an inherent amount of randomness in time and/or space. The focus of this course is on developing and analyzing methods for analyzing time series. Because most of the common time--domain methods are unreliable, the emphasis is on frequency--domain methods, i.e. methods that work and expose the bias that plagues most time--domain techniques. Slepian sequences (discrete prolate spheroidal sequences) and multi--taper methods of spectrum estimation are covered in detail. (Offered jointly with MTHE-454*.) Three term-hours; lectures.

**STAT-855* Stochastic Processes and Applications **

Markov chains, birth and death processes, random walk problems, elementary renewal theory, Markov processes, Brownian motion and Poisson processes, queuing theory, branching processes. (Offered jointly with MTHE/STAT-455*.) Three term hours; lectures.

**STAT-856* Bayesian Analysis**

This course is an introduction to Bayesian analysis and decision theory. Topics covered will include: elements of decision theory; Bayesian point estimation, set estimation, and hypothesis testing; special priors; computations for Bayesian analysis. (Offered jointly with STAT-456.)

**STAT-857* Statistical Computing**

Introduction to the theory and application of statistical algorithms. Topics may include classification, smoothing, model selection, optimization, sampling, supervised and unsupervised learning. (Offered jointly with STAT-457).

**STAT-862* Computational Data Analysis **

Introduction to the statistical packages SAS and R; classification; spline and smoothing spline; regularization, ridge regression and Lasso; model selection; resampling methods; importance sampling; Markov chain Monte Carlo; Metropolis-Hasting algorithm; Gibbs sampling; optimization. (Offered jointly with STAT-462.) Three term hours; lectures.

**STAT-864* Discrete Time Series Analysis **

Autocorrelation and autocovariance, stationarity; ARIMA models; model identification and forecasting; spectral analysis. Applications to biological, physical and economic data. (Offered jointly with STAT-464*.) Three term-hours; lectures.

**STAT-865* Quality Management **

An overview of the statistical and lean manufacturing tools and techniques used in the measurement and improvement of quality in business, government and industry today. Topics include management and planning tools, Six Sigma approach, statistical process charting, process capability analysis, measurement system analysis. (Offered jointly with STAT-465*.) Three term-hours; lectures.

**STAT-871* Sampling and Experimental Design **

Simple random sampling; Unequal probability sampling; Stratified sampling; Cluster sampling; Multi‐stage sampling; Analysis of variance and covariance; Block designs; Fractional factorial designs; Split‐plot designs; Response surface methodology; Robust parameter designs for products and process improvement. (Offered jointly with STAT-471*.) Three term hours; lectures.

**STAT-873* Generalized Linear Models **

An introduction to advanced regression methods for binary, categorical, and count data. Major topics include maximum-likelihood method, binomial and Poisson regression, contingency tables, log linear models, and random effect models. The generalized linear models will be discussed both in theory and in applications to real data from a variety of sources.(Offered jointly with STAT-473*.)

**STAT-886* Survival Analysis **

Introduces the theory and application of survival analysis: survival distributions and their applications, parametric and nonparametric methods, proportional hazards models, counting process and proportional hazards regression, planning and designing clinical trials. (Offered jointly with STAT-486*.) Three term-hours; lectures.

**STAT-888 Master’s Practicum **

Under the guidance of the supervisor, students will carry out a practicum project in a health research group/site and practise biostatistical methods and data analysis, or conduct methodology research in a biostatistical project. Students will summarize the results of the project in a written report that will be reviewed and orally defended.

**STAT-898 Master's Project **

**STAT-899 Master's Thesis Research **

**STAT-952* Topics in Probability **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**STAT-953* Topics in Probability **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**STAT-955* Topics in Probability **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**STAT-962* Topics in Statistics **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**STAT-963* Topics in Statistics **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**STAT-965* Topics in Statistics **

Subject matter will vary from year to year. Three term-hours; Seminar or reading course.

**STAT-999 Ph.D. Thesis Research**