For students intending a medial or major concentration in Mathematics or Statistics. Provides a thorough introduction to linear algebra up to and including eigenvalues and eigenvectors.
LEARNING HOURS 264 (72L;24T;168P).

An introduction to matrices and linear algebra. Emphasis on applications to biological and economic systems and to computer applications. Topics covered will include systems of equations, eigenvalues, recursions, orthogonality, regression analysis, and geometric transformations.
LEARNING HOURS 240 (72L;168P).

A brief introduction to matrix algebra, linear algebra, and applications. Topics include systems of linear equations, matrix algebra, determinants, the vector spaces Rn and their subspaces, bases, co-ordinates, orthogonalization, linear transformations, eigenvectors, diagonalization of symmetric matrices, quadratic forms.
LEARNING HOURS 120 (36L;84P).

A thorough discussion of calculus, including limits, continuity, differentiation, integration, multivariable differential calculus, and sequences and series.
LEARNING HOURS 288 (72L;24T;192P).

Differentiation and integration with applications to biology, physics, chemistry, economics, and social sciences; differential equations; multivariable differential calculus.
NOTE Also offered online. Consult Arts and Science Online. Learning Hours may vary.
NOTE Also offered at the Bader International Study Centre. Learning Hours may vary.
LEARNING HOURS 262 (48L;11G;72O).

Differentiation and integration of elementary functions, with applications to physical and social sciences.  Topics include limits, related rates, Taylor polynomials, and introductory techniques  and applications of integration. 

Topics include techniques of integration; differential equations, and multivariable differential calculus.
LEARNING HOURS 126 (36L;12T;78P).

Differentiation and integration of the elementary functions with applications to the social sciences and economics; Taylor polynomials; multivariable differential calculus.
LEARNING HOURS 240 (72L;24T;144P).

Integers, polynomials, modular arithmetic, rings, ideals, homomorphisms, quotient rings, division algorithm, greatest common divisors, Euclidean domains, unique factorization, fields, finite fields.
NOTE Students with MATH 112/3.0 may ask for admission with the permissions of the Department.
LEARNING HOURS 132 (36L;12T;84P)

Algebraic techniques used in applied mathematics, statistics, computer science and other areas. Polynomials, complex numbers; least squares approximations; discrete linear systems; eigenvalue estimation; non-negative matrices - Markov chains; permutation groups; linear Diophantine equations; introduction to algebraic structures.
LEARNING HOURS 240 (72L;168P)

Vector spaces, direct sums, linear transformations, eigenvalues, eigenvectors, inner product spaces, self-adjoint operators, positive operators, singular-value decomposition, minimal polynomials, Jordan canonical form, the projection theorem, applications to approximation and optimization problems.
LEARNING HOURS 120 (36L;12T;72P).

The purpose of the course is to provide an introduction to abstract algebraic systems, and to illustrate the concepts with applications to communication engineering. Topics covered are: symbolic logic, switching and logic circuits; set theory and mappings; equivalence relations; the integers; introduction to Boolean algebras; group theory, groups, subgroups, cyclic groups, cosets and Lagrange's theorem, quotient groups, homomorphisms and isomorphisms; applications to error-control codes, binary block codes for noisy communication channels, nearest neighbor decoding, code error detection/correction capabilities, group (linear) codes, coset decoding, generator and parity check matrices, syndrome decoding; basic properties of rings and fields. (30/0/0/12/0)

Double and triple integrals, including polar and spherical coordinates. Parameterized curves and line integrals. Gradient, divergence, and curl. Green's theorem. Parameterized surfaces and surface integrals. Stokes' and Gauss' Theorems.
LEARNING HOURS 120 (36L;84P)
RECOMMENDATION Some linear algebra.

The course will discuss the application of linear differential equations with constant coefficients, and systems of linear equations within the realm of civil engineering. Additionally, the course will explore relevant data analysis techniques including: graphical and statistical analysis and presentation of experimental data, random sampling, estimation using confidence intervals, linear regression, residuals and correlation. (54/0/0/0/0)

An introduction to solving ordinary differential equations. Topics include first order differential equations, linear differential equations with constant coefficients, Laplace transforms, and systems of linear equations.
NOTE Some knowledge of linear algebra is assumed.
LEARNING HOURS 120 (36L;12T;72P)

Review of multiple integrals. Differentiation and integration of vectors; line, surface and volume integrals; gradient, divergence and curl; conservative fields and potential. Spherical and cylindrical coordinates, solid angle. Green's and Stokes' theorems, the divergence theorem. (36/0/0/0/0)

Complex arithmetic, complex plane. Differentiation, analytic functions. Elementary functions. Contour integration, Cauchy's Theorem, and Integral Formula. Taylor and Laurent series, residues with applications to evaluation of integrals.
LEARNING HOURS 120 (36L;12T;72P).

An introduction to ordinary differential equations and their applications. Intended for students concentrating in Mathematics or Statistics.
LEARNING HOURS 132 (36L;12T;84P).

First order differential equations, linear differential equations with constant coefficients. Laplace transforms. Systems of linear differential equations. Examples involving the use of differential equations in solving circuits will be presented. (30/0/0/6/0)

Topics include Newton's Law, conservative and central force fields; linear systems with constant coefficients; matrix exponentials and the variation of constants formula; Euler's method; separable and exact differential equations; Laplace Transforms, transfer functions and convolution; introduction to Fourier Series and boundary value problems: the heat equation, the wave equation. Computer methods (symbolic and numerical) are introduced in the laboratory component of the course. (20/7/0/15/0)

A survey of important mathematical techniques used to model processes in a variety of fields. Topics include multivariable calculus and optimization, game theory, discrete-time dynamical systems, and dynamic optimization. Examples will be drawn from several areas including biology, economics, and medicine.  (18/9/5/4/0) ~ COURSE DELETED IN 2008/09 ~

An introductory course on the use of computers in science. Topics include: solving linear and nonlinear equations, interpolation, integration, and numerical solutions of ordinary differential equations. Extensive use is made of MATLAB, a high level interactive numerical package.
LEARNING HOURS 120 (36L;12Lb;12T;60P).

Limits, continuity, C1 and linear approximations of functions of several variables. Multiple integrals and Jacobians. Line and surface integrals. The theorems of Green, Stokes, and Gauss.
LEARNING HOURS 132 (36L;12T;84P)

Taylor's theorem, optimization, implicit and inverse function theorems. Elementary topology of Euclidean spaces. Sequences and series of numbers and functions. Pointwise and uniform convergence. Power series.
LEARNING HOURS 132 (36L;12T;84P).

Permutation groups, matrix groups, abstract groups, subgroups, homomorphisms, cosets, quotient groups, group actions, Sylow theorems.
LEARNING HOURS 132 (36L;96P).

Congruences; Euler's theorem; continued fractions; prime numbers and their distribution; quadratic forms; Pell's equation; quadratic reciprocity; introduction to elliptic curves.
LEARNING HOURS 120 (36L;84P).

The symmetric group consists of all permutations of a finite set or equivalently all the bijections from the set to itself. This course explores how to map the symmetric group into a collection of invertible matrices. To handle, count, and manipulate these objects, appropriate combinatorial tools are introduced.
LEARNING HOURS 132 (36L;96L).

Complex numbers, analytic functions, harmonic functions, Cauchy's Theorem, Taylor and Laurent series, calculus of residues, Rouche's Theorem.
LEARNING HOURS 120 (36L;12T;72P).

Metric spaces, topological spaces, compactness, completeness, contraction mappings, sequences and series of functions, uniform convergence, normed linear spaces, Hibert space.
LEARNING HOURS 132 (36L;96P).

Banach and Hilbert spaces of continuous- and discrete-time signals; spaces of continuous and not necessarily continuous signals; continuous-discrete Fourier transform; continuous-continuous Fourier transform; discrete-continuous Fourier transform; discrete-discrete Fourier transform; transform inversion using Fourier series and Fourier integrals.
LEARNING HOURS 132 (36L;12T;84P).

Linear input/output systems and their stability. Frequency-domain and time-domain analysis. Continuous and discrete time-modeling. Fourier, Laplace, and Z-transforms. Sampling and the discrete-time Fourier transform. Application to modulation of communications signals, filter design, and digital sampling.
LEARNING HOURS 132 (36L;12T;84P).

Some probability distributions, simulation, Markov chains, queuing theory, dynamic programming, inventory theory.
LEARNING HOURS 120 (36L;84P).

Methods and theory for ordinary and partial differential equations; separation of variables in rectangular and cylindrical coordinate systems; sinusoidal and Bessel orthogonal functions; the wave, diffusion, and Laplace's equation; Sturm-Liouville theory; Fourier transform.
LEARNING HOURS 118 (36L;12T;70P).

This course highlights the usefulness of game theoretical approaches in solving problems in the natural sciences and economics. Basic ideas of game theory, including Nash equilibrium and mixed strategies; stability using approaches developed for the study of dynamical systems, including evolutionary stability and replicator dynamics; the emergence of co-operative behaviour; limitations of applying the theory to human behaviour.
LEARNING HOURS 120 (36L;84P).

Introductory geometry of curves/surfaces: directional/covariant derivative; differential forms; Frenet formulas; congruent curves; surfaces in R3: mappings, topology, intrinsic geometry; manifolds; Gaussian/mean curvature; geodesics, exponential map; Gauss-Bonnet Theorem; conjugate points; constant curvature surfaces.
LEARNING HOURS 132 (36L;96P).

A historical perspective on mathematical ideas focusing on a selection of important and accessible theorems. A project is required.
LEARNING HOURS 120 (36L;12G;72P).

Elementary mathematical material will be used to explore different ways of discovering results and mastering concepts. Topics will come from number theory, geometry, analysis, probability theory, and linear algebra. Much class time will be used for problem solving and presentations by students.
LEARNING HOURS 120 (36L;84P)

Interest accumulation factors, annuities, amortization, sinking funds, bonds, yield rates, capital budgeting, contingent payments. Students will work mostly on their own; there will be a total of six survey lectures and six tests throughout the term, plus opportunity for individual help.
LEARNING HOURS 102 (12L;84P).

Measurement of mortality, life annuities, life insurance, premiums, reserves, cash values, population theory, multi-life functions, multiple-decrement functions. The classroom meetings will be primarily problem-solving sessions, based on assigned readings and problems.
LEARNING HOURS 108 (36L;72P).

Integers and rationals from the natural numbers; completing the rationals to the reals; consequences of completeness for sequences and calculus; extensions beyond rational numbers, real numbers, and complex numbers.
LEARNING HOURS 120 (36L;84P).

In-depth follow-up to high school geometry: striking new results/connections; analysis/proof of new/familiar results from various perspectives; extensions (projective geometry, e.g.); relation of classical unsolvable constructions to modern algebra; models/technology for geometric exploration.
LEARNING HOURS 120 (36L;84P).

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. Given jointly with MATH 801.
LEARNING HOURS 120 (36L;84P).

Enumerative combinatorics is concerned with counting the number of elements of a finite set. The techniques covered include inclusion-exclusion, bijective proofs, double-counting arguments, recurrence relations, and generating functions. Given jointly with MATH 802.
LEARNING HOURS 120 (36L;84P).

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.
LEARNING HOURS 120 (36L;84P).

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.
LEARNING HOURS 132 (36L;96P).

An introduction to Galois Theory and some of its applications.
LEARNING HOURS 132 (36L;96P).

Time estimates for arithmetic and elementary number theory algorithms (division algorithm, Euclidean algorithm, congruences), modular arithmetic, finite fields, quadratic residues. Simple cryptographic systems; public key, RSA. Primality and factoring: pseudoprimes, Pollard's rho-method, index calculus. Elliptic curve cryptography.
LEARNING HOURS 120 (36L;84P).

An exploration of the modern theory of Fourier series: Abel and Cesàro summability; Dirichlet's and Fejér's kernels; term by term differentiation and integration; infinite products; Bernoulli numbers; the Fourier transform; the inversion theorem; convolution of functions; the Plancherel theorem; and the Poisson summation theorem.
LEARNING HOURS 132 (36L;96P).

Topics include: global properties of flows and diffeomorphisms, Invariant sets and dynamics, Bifurcations of fixed and periodic points; stability and chaos. Examples will be selected by the instructor. Given jointly with MATH 827.
LEARNING HOURS 120 (36L;84P).

A generalization of linear algebra and calculus to infinite dimensional spaces. Now questions about continuity and completeness become crucial, and algebraic, topological, and analytical arguments need to be combined. We focus mainly on Hilbert spaces and the need for Functional Analysis will be motivated by its application to Quantum Mechanics.
LEARNING HOURS 132 (36L;12G;84P).

Continuum mechanics lays the foundations for the study of the mechanical behavior of materials. After a review of vector and tensor analysis, the kinematics of continua are introduced. Conservation of mass, balance of momenta and energy are presented with the constitutive models. Applications are given in elasticity theory and fluid dynamics.
NOTE This is the MATH version of MTHE 433 in FEAS.
LEARNING HOURS 120 (36L;84P).

Theory of convex sets and functions; separation theorems; primal-duel properties; geometric treatment of optimization problems; algorithmic procedures for solving constrained optimization programs; engineering and economic applications.

Quasilinear equations: Cauchy problems, method of characteristics; Cauchy-Kovalevski theorem; generalized solutions; wave equation, Huygens' principle, conservation of energy, domain of dependence; Laplace equation, boundary value problems, potential theory, Green's functions; heat equation, maximum principle.
LEARNING HOURS 132 (36L;96P).

Subject matter to vary from year to year. Given jointly with MATH 837.
LEARNING HOURS 132 (36L;96P).

Geometric modeling, 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.
LEARNING HOURS 132 (36L;12T;84P).

Topics include: information measures, entropy, mutual information, modeling of information sources, lossless data compression, block encoding, variable-length encoding, Kraft inequality, fundamentals of channel coding, channel capacity, rate-distortion theory, lossy data compression, rate-distortion theorem. Given jointly with MATH 874.
LEARNING HOURS 140 (36L;104P).

Topics include: arithmetic coding, universal lossless coding, Lempel-Ziv and related dictionary based methods, rate distortion theory, scalar and vector quantization, predictive and transform coding, applications to speech and image coding.
LEARNING HOURS 120 (36L;12O;72P).

An important topic in mathematics not covered in any other courses.

An important topic in mathematics not covered in any other courses.

Important topics in mathematics not covered in any other courses.

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.

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*.)

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.

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 406*.) Three term-hours, fall or winter; lectures.

Subject matter may vary from year to year. Three term-hours, fall or winter; lectures.

An introduction to the study of systems of polynomial equations in one or more 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).

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-418. Three term hours, fall or winter; lectures.

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, fall or winter; lectures.

A generalization of linear algebra and calculus to infinite dimensional spaces. Now questions about continuity and completeness become crucial, and algebraic, topological, and analytical arguments need to be combined. We focus mainly on Hilbert spaces and the need for Functional Analysis will be motivated by its application to Quantum Mechanics. (Offered jointly with MATH 429.) Three term hours; lectures.
EXCLUSION: MATH -429

This course covers core topics in discrete and continuous time modern control theory: controllability, observability and minimal realizations; Lyapunov stability; the linear quadradic regulator and design of robust controllers; state estimation via Luenberger and deterministic Kalman-Bucy filters. Laboratory experiments illustrate design considerations in implementing the lecture material. Students are required to identify a high order under-actuated torsion disc system; perform model verification experiments; design and implement robust linear feedback controllers; design and implement nonlinear sliding mode controllers; study implementation issues for observers and Kalman-Bucy filters; depending on instructor, some nonlinear control strategies may be implemented.  (Offered jointly with MATH-430*.) Three term hours, fall; lectures.

Theory of convex sets and functions; separation theorems; primal-dual properties; geometric treatment of optimization problems; algorithmic procedures for solving constrained optimization programs; applications of optimization theory to machine learning. (Offered jointly with MATH/MTHE 434.)
EXCLUSIONS: MTHE/MATH 434

This is a course in advanced mathematical methods used to construct models of biological phenomena in ecology, epidemiology, and evolutionary biology. The course will focus on population models, starting with individual-based models based on assumptions on the distribution of individual traits, then scaling up to stochastic models for small populations and deterministic models for large populations.

Configuration space, generalized coordinates, Euler-Lagrange equations. Forces: dissipative, potential. Simple mechanical control systems: modeling, linearization about equilibrium points, linear controllability tests; equivalence with kinematic systems and trajectory generation. (Offered jointly with MATH 439). Three term hours, fall or winter; lectures.

Subject matter may vary from year to year. Three term-hours, fall or winter; lectures.

Subject matter may vary from year to year. Three term-hours, fall or winter; lectures.

Calculus on manifolds; transversality; Sard's Theorem; immersions and submersions; intersection theory; Jordan curve theorem; Lefschetz fixed point theorem; Poincaré-Hopf Theorem. (Offered jointly with MATH-444*.) Three term-hours, winter; lectures.

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.

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-474).  Three term hours, fall; lectures.

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-477*).

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 484.) Three term hours, winter; lectures.

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.

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.

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.

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.

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, fall; lectures.

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, winter; lectures.

This course provides basic knowledge in mathematical statistics at the graduate level. Topics will include: Weak convergence in metric spaces; Delta method; Method of moments; M-estimation; Asymptotic normality and efficiency; Likelihood ratio test; U statistics; Bootstrap; Applications in statistics.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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