**MATH 110**

**Linear Algebra**

**Units: 6.00**

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).

**Requirements:**Prerequisite None. Exclusion MATH 111; MATH 112; MATH 212. Recommended At least one 4U Mathematics course.

**Offering Faculty:**Faculty of Arts and Science

**MATH 111**

**Linear Algebra**

**Units: 6.00**

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).

**Requirements:**Prerequisite None. Exclusion MATH 110; MATH 112. Recommended At least one 4U Mathematics course.

**Offering Faculty:**Faculty of Arts and Science

**MATH 112**

**Introduction to Linear Algebra**

**Units: 3.00**

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).

**Requirements:**Prerequisite None. Exclusion MATH 110; MATH 111. Recommended At least one 4U Mathematics course.

**Course Equivalencies:**MATH 110B/112 / APSC 174

**Offering Faculty:**Faculty of Arts and Science

**MATH 120**

**Differential and Integral Calculus**

**Units: 6.00**

A thorough discussion of calculus, including limits, continuity, differentiation, integration, multivariable differential calculus, and sequences and series.

LEARNING HOURS 288 (72L;24T;192P).

**Requirements:**Prerequisite None. Exclusion MATH 121; MATH 123; MATH 124; MATH 126. Recommended MHF4U and MCV4U or 4U AFIC or permission of the Department. Note This course is intended for students who plan to pursue a Major or Medial Plan in Mathematics, Statistics or Physics.

**Offering Faculty:**Faculty of Arts and Science

**MATH 121**

**Differential and Integral Calculus**

**Units: 6.00**

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).

**Requirements:**Prerequisite None. Exclusion MATH 120; MATH 123; MATH 124; MATH 126. Recommended MHF4U and MCV4U or equivalent, or 4U AFIC, or MATH P06, or permission of the Department. Note This course is intended for students who wish to pursue a Major or Medial Plan in a subject other than Mathematics or Statistics.

**Course Equivalencies:**MATH121; MATH121B;MATH122B

**Offering Faculty:**Faculty of Arts and Science

**MATH 123**

**Differential and Integral Calculus I**

**Units: 3.00**

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.

**Requirements:**Prerequisite Permission of the Department. Exclusion MATH 120; MATH 121; MATH 126. Note This course is not intended for students pursuing a MATH or STAT Plan.

**Offering Faculty:**Faculty of Arts and Science

**MATH 124**

**Differential and Integral Calculus II**

**Units: 3.00**

Topics include techniques of integration; differential equations, and multivariable differential calculus.

LEARNING HOURS 126 (36L;12T;78P).

**Requirements:**Prerequisite MATH 123 or permission of the Department. Exclusion MATH 120; MATH 121; MATH 126. Note For students who have credit for a one-term course in calculus. Topics covered are the same as those in the Winter term of MATH 121.

**Course Equivalencies:**APSC172, MATH124

**Offering Faculty:**Faculty of Arts and Science

**MATH 126**

**Differential and Integral Calculus**

**Units: 6.00**

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).

**Requirements:**Prerequisite None. Exclusion MATH 120; MATH 121; MATH 123; MATH 124; MATH 126. Note This course is primarily intended for students in the BAH program. Students in the BSCH, BCMPH or BCOM program should not enrol in this course.

**Offering Faculty:**Faculty of Arts and Science

**MATH 210**

**Rings and Fields**

**Units: 3.00**

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)

**Requirements:**MATH110

**Offering Faculty:**Faculty of Arts and Science

**MATH 211**

**Algebraic Methods**

**Units: 6.00**

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)

**Requirements:**Pre MATH120;121;122;124;126

**Offering Faculty:**Faculty of Arts and Science

**MATH 212**

**Linear Algebra ll**

**Units: 3.00**

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).

**Requirements:**Prerequisite MATH 111 or MATH 112 or MTHE 217. Exclusion MATH 110. Equivalency MATH 312.

**Offering Faculty:**Faculty of Arts and Science

**MATH 217**

**Algebraic Structures**

**Units: 3.50**

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)

**Requirements:**MATH118 OR APSC174

**Offering Faculty:**Faculty of Arts and Science

**MATH 221**

**Vector Calculus**

**Units: 3.00**

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.

**Requirements:**MATH120 OR MATH121 OR MATH126 OR APSC172 OR MATH122 OR MATH124

**Offering Faculty:**Faculty of Arts and Science

**MATH 224**

**Applied Math For Civil Eng.**

**Units: 4.50**

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)

**Requirements:**(APSC171 AND APSC172) OR (APSC142 AND APSC172) OR (APSC142 AND APSC174) OR (APSC171 AND APSC174)

**Offering Faculty:**Faculty of Arts and Science

**MATH 225**

**Ordinary Differential Equations**

**Units: 3.00**

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)

**Requirements:**(MATH128 AND MATH118) OR (MATH128 AND MATH129) OR (APSC171 AND APSC174) OR (APSC171 AND APSC172) OR (MATH120 AND MATH110) OR (MATH120 AND MATH111) OR (MATH120 AND MATH112)

**Offering Faculty:**Faculty of Arts and Science

**MATH 227**

**Vector Analysis**

**Units: 3.00**

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)

**Requirements:**(MATH128 AND MATH118) OR (MATH128 AND MATH129) OR (MATH12## AND MATH11##) OR (APSC171 AND APSC172) OR (APSC171 AND APSC174)

**Offering Faculty:**Faculty of Arts and Science

**MATH 228**

**Complex Analysis**

**Units: 3.00**

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).

**Requirements:**Prerequisite (MATH 110 or MATH 111 or MATH 112) and (MATH 120 or MATH 121 or MATH 124). Exclusion MATH 326; PHYS 312; PHYS 317.

**Offering Faculty:**Faculty of Arts and Science

**MATH 231**

**Differential Equations**

**Units: 3.00**

An introduction to ordinary differential equations and their applications. Intended for students concentrating in Mathematics or Statistics.

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

**Requirements:**Prerequisite (MATH 110 or MATH 111 or MATH 212) and (MATH 120 or MATH 121 or MATH 124). Exclusion MATH 225; MATH 226; MATH 232.

**Offering Faculty:**Faculty of Arts and Science

**MATH 235**

**Diff Equations For Elec & Comp**

**Units: 3.00**

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)

**Requirements:**(MATH128 AND MATH118) OR (MATH128 AND MATH129) OR (APSC171 AND APSC172) OR (APSC171 AND APSC174) OR (MATH120 AND MATH11##)

**Offering Faculty:**Faculty of Arts and Science

**MATH 237**

**Diff Equations & Computer Meth**

**Units: 3.50**

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)

**Requirements:**MATH110 or MATH120 or (APSC171

**Offering Faculty:**Faculty of Arts and Science

**MATH 239**

**Applied Math Modeling**

**Units: 3.00**

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 ~

**Requirements:**APSC172 OR MATH120 OR MATH121 OR MATH120 OR MATH126 OR MATH121 OR APSC172 OR MATH126

**Course Equivalencies:**MATH239, MATH339

**Offering Faculty:**Faculty of Arts and Science

**MATH 272**

**Applications of Numerical Methods**

**Units: 3.00**

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).

**Requirements:**Prerequisite (MATH 110 or MATH 111 or MATH 212) and (CISC 101 or CISC 121) Corequisite (MATH 225 or MATH 231 or MATH 232). Exclusion CISC 271; PHYS 213; PHYS 313.

**Offering Faculty:**Faculty of Arts and Science

**MATH 280**

**Advanced Calculus**

**Units: 3.00**

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)

**Requirements:**(MATH120 AND MATH110) OR (MATH120 AND MATH111) OR (MATH120 AND APSC174) OR (APSC172 AND MATH110) OR (APSC172 AND MATH111) OR (APSC172 AND APSC174) OR (MATH120 AND MATH112) OR (APSC172 AND MATH112)

**Offering Faculty:**Faculty of Arts and Science

**MATH 281**

**Introduction to Real Analysis**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 310**

**Group Theory**

**Units: 3.00**

Permutation groups, matrix groups, abstract groups, subgroups, homomorphisms, cosets, quotient groups, group actions, Sylow theorems.

LEARNING HOURS 132 (36L;96P).

**Requirements:**Prerequisite MATH 210.

**Course Equivalencies:**MATH310, MATH313

**Offering Faculty:**Faculty of Arts and Science

**MATH 311**

**Elementary Number Theory**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 314**

**Representations of the Symmetric Group**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 326**

**Functions of a Complex Variable**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 328**

**Real Analysis**

**Units: 3.00**

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).

**Requirements:**Prerequisite MATH 281.

**Offering Faculty:**Faculty of Arts and Science

**MATH 334**

**Mathematical Methods for Engineering and Physics**

**Units: 3.00**

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).

**Requirements:**Prerequisite (MATH 110 or MATH 111 or MATH 212) and MATH 281. Exclusion PHYS 312; PHYS 316.

**Offering Faculty:**Faculty of Arts and Science

**MATH 335**

**Mathematics of Engineering Systems**

**Units: 3.00**

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).

**Requirements:**Prerequisite MATH 334 and (MATH 228 or MATH 326). Exclusion PHYS 312. Equivalency MATH 236.

**Course Equivalencies:**MATH236, MATH335

**Offering Faculty:**Faculty of Arts and Science

**MATH 337**

**Stochastic Models in Operations Research**

**Units: 3.00**

Some probability distributions, simulation, Markov chains, queuing theory, dynamic programming, inventory theory.

LEARNING HOURS 120 (36L;84P).

**Offering Faculty:**Faculty of Arts and Science

**MATH 338**

**Fourier Methods for Boundary Value Problems**

**Units: 3.00**

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).

**Requirements:**Prerequisite (MATH 221 or MATH 280) and (MATH 225 or MATH 231 or MATH 232) and (MATH 110 or MATH 111). Exclusion PHYS 312; PHYS 316.

**Offering Faculty:**Faculty of Arts and Science

**MATH 339**

**Evolutionary Game Theory**

**Units: 3.00**

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).

**Requirements:**Prerequisite (MATH 120 or MATH 121 or MATH 124) and (MATH 110 or MATH 111 or MATH 112). Equivalency MATH 239. Recommended MATH 110 or MATH 111.

**Course Equivalencies:**MATH239, MATH339

**Offering Faculty:**Faculty of Arts and Science

**MATH 341**

**Differential Geometry**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 381**

**Mathematics with a Historical Perspective**

**Units: 3.00**

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).

**Requirements:**Prerequisite (MATH 110 or MATH 111 or MATH 212) and (MATH 120 or MATH 121 or MATH 126).

**Offering Faculty:**Faculty of Arts and Science

**MATH 382**

**Mathematical Explorations**

**Units: 3.00**

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)

**Requirements:**(MATH221 AND MATH210) OR (MATH221 AND MATH211) OR (MATH212 AND MATH280) OR (MATH212 AND MATH281) OR (MATH210 AND MATH280) OR (MATH231 AND MATH210) OR (MATH210 AND MATH281) OR (MATH231 AND MATH211) OR (MATH232 AND MATH210) OR (MATH232 AND MATH211) OR (MATH280 AND MATH210) OR (MATH280 AND MATH211)

**Offering Faculty:**Faculty of Arts and Science

**MATH 384**

**Mathematical Theory of Interest**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 385**

**Life Contingencies**

**Units: 3.00**

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).

**Requirements:**Prerequisite ([MATH 120 or MATH 121 or MATH 124 or MATH 126] and MATH 384 and [STAT 268 or STAT 351]) or permission of the Department.

**Offering Faculty:**Faculty of Arts and Science

**MATH 386**

**Our Number System - an Advanced Perspective**

**Units: 3.00**

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).

**Requirements:**Prerequisite MATH 281.

**Offering Faculty:**Faculty of Arts and Science

**MATH 387**

**Elementary Geometry - an Advanced Perspective**

**Units: 3.00**

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).

**Requirements:**Prerequisite (Level 3 or above and [MATH 221 or MATH 280 or MATH 281]) or permission of the Department.

**Offering Faculty:**Faculty of Arts and Science

**MATH 401**

**Graph Theory**

**Units: 3.00**

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).

**Requirements:**Prerequisite MATH 210 or MATH 211. Recommended Experience with abstract mathematics and mathematical proof, and a good foundation in linear algebra.

**Offering Faculty:**Faculty of Arts and Science

**MATH 402**

**Enumerative Combinatorics**

**Units: 3.00**

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).

**Requirements:**Prerequisite MATH 210 or MATH 211. Recommended Experience with abstract mathematics and mathematical proof, and a good foundation in linear algebra.

**Offering Faculty:**Faculty of Arts and Science

**MATH 406**

**Introduction to Coding Theory**

**Units: 3.00**

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).

**Requirements:**Prerequisite MATH 210.

**Offering Faculty:**Faculty of Arts and Science

**MATH 413**

**Introduction to Algebraic Geometry**

**Units: 3.00**

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).

**Requirements:**Prerequisite MATH 210.

**Offering Faculty:**Faculty of Arts and Science

**MATH 414**

**Introduction to Galois Theory**

**Units: 3.00**

An introduction to Galois Theory and some of its applications.

LEARNING HOURS 132 (36L;96P).

**Requirements:**Prerequisite MATH 310.

**Offering Faculty:**Faculty of Arts and Science

**MATH 418**

**Number Theory and Cryptography**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 421**

**Fourier Analysis**

**Units: 3.00**

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).

**Requirements:**Prerequisite MATH 281 or permission of the Department.

**Offering Faculty:**Faculty of Arts and Science

**MATH 427**

**Introduction to Deterministic Dynamical Systems**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 429**

**Functional Analysis and Quantum Mechanics**

**Units: 3.00**

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).

**Requirements:**Prerequisite ([MATH 110 or MATH 111 or MATH 112] and MATH 281) or permission of the Department.

**Offering Faculty:**Faculty of Arts and Science

**MATH 433**

**Continuum Mechanics with Applications**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 434**

**Optimization Theory with Applications to Machine Learning**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 436**

**Partial Differential Equations**

**Units: 3.00**

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).

**Requirements:**Prerequisite MATH 231 and MATH 280. Recommended MATH 328 or MATH 334 or MATH 338 or PHYS 312.

**Offering Faculty:**Faculty of Arts and Science

**MATH 437**

**Topics in Applied Mathematics**

**Units: 3.00**

Subject matter to vary from year to year. Given jointly with MATH 837.

LEARNING HOURS 132 (36L;96P).

**Requirements:**Prerequisite Permission of the Department.

**Offering Faculty:**Faculty of Arts and Science

**MATH 439**

**Lagrangian Mechanics, Dynamics, and Control**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 474**

**Information Theory**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 477**

**Data Compression and Source Coding**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 497**

**Topics in Mathematics IV**

**Units: 3.00**

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

**Requirements:**Note The prerequisite can vary depending on specific course content, please consult the Instructor or visit the Department of Mathematics and Statistics webpage for more information.

**Offering Faculty:**Faculty of Arts and Science

**MATH 498**

**Topics in Mathematics V**

**Units: 3.00**

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

**Requirements:**Note The prerequisite can vary depending on specific course content, please consult the Instructor or visit the Department of Mathematics and Statistics webpage for more information.

**Offering Faculty:**Faculty of Arts and Science

**MATH 499**

**Topics In Mathematics**

**Units: 3.00**

Important topics in mathematics not covered in any other courses.

**Requirements:**Prerequisite Permission of the Department.

**Offering Faculty:**Faculty of Arts and Science

**MATH 594**

**Independent Study**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 595**

**Independent Study**

**Units: 6.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 603**

**Mathematics**

**Units: 6.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 620**

**Diff. & Int. Calculus**

**Units: 6.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 800**

**Seminar**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 801**

**Graph Theory**

**Units: 3.00**

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*.)

**Offering Faculty:**Faculty of Arts and Science

**MATH 802**

**Enumerative Combinatorics**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 806**

**Introduction To Coding Theory**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 812**

**Topics In Number Theory**

**Units: 3.00**

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

**Offering Faculty:**Faculty of Arts and Science

**MATH 813**

**Introduction to Algebraic Geometry**

**Units: 3.00**

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).

**Offering Faculty:**Faculty of Arts and Science

**MATH 818**

**Number Theory & Cryptography**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 823**

**Ordinary Diff. Equations**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 827**

**Deterministic Dynamical System**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 829**

**Functional Analysis**

**Units: 3.00**

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

**Offering Faculty:**Faculty of Arts and Science

**MATH 830**

**Modern Control Theory**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 833**

**Continuum Mechanics**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 834**

**Optimization Theory with Applications to Machine Learning**

**Units: 3.00**

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

**Offering Faculty:**Faculty of Arts and Science

**MATH 835**

**Mathematical Biology**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 836**

**Lagrangian Mechanics. Dynamics & Control**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 837**

**Topics In Applied Mathematics**

**Units: 3.00**

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

**Offering Faculty:**Faculty of Arts and Science

**MATH 838**

**Topics In Mathematical Biology**

**Units: 3.00**

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

**Offering Faculty:**Faculty of Arts and Science

**MATH 844**

**Differentiable Manifolds**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 845**

**Differential Geometry**

**Units: 6.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 848**

**Topics In Graph Theory**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 860**

**Probability Theory**

**Units: 6.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 862**

**Stochastic Processes**

**Units: 6.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 865**

**Stochastic Processes**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 872**

**Control Of Stochastic Systems**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 873**

**Num. Lin. & Nonlin. Algebra II**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 874**

**Information Theory**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 877**

**Data Compression And Source Co**

**Units: 3.00**

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*).

**Offering Faculty:**Faculty of Arts and Science

**MATH 880**

**Foundations Of Mathematics I**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 883**

**Mathematical Logic**

**Units: 6.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 884**

**Data Networks**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 891**

**Core Course In Analysis I**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 892**

**Core Course In Analysis II**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 893**

**Core Course In Algebra I**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 894**

**Core Course In Algebra II**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 895**

**Core Course In Prob. Theory**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 896**

**Core Mathematical Statistics I**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 897**

**Core Mathematical Statistics II**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 898**

**Master's Project**

**Units: 6.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 899**

**Master's Thesis Research**

**Units: 6.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 901**

**Research Institute Course**

**Units: 3.00**

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.

**Offering Faculty:**Faculty of Arts and Science

**MATH 902**

**Topics In Algebra**

**Units: 3.00**

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

**Offering Faculty:**Faculty of Arts and Science

**MATH 903**

**Topics in Algebra**

**Units: 3.00**

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

**Offering Faculty:**Faculty of Arts and Science

**MATH 905**

**Topics In Algebra**

**Units: 3.00**

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

**Offering Faculty:**Faculty of Arts and Science

**MATH 912**

**Topics In Number Theory**

**Units: 3.00**

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

**Offering Faculty:**Faculty of Arts and Science

**MATH 913**

**Topics In Number Theory**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 915**

**Topics in Number Theory**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 916**

**Group Representations IIb**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 920**

**Topological Vector Spaces**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 921**

**Vector Lattices & Integration**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 922**

**Topics In Analysis**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 923**

**Topics In Analysis**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 925**

**Topics In Analysis**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 926**

**Measure & Integration**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 928**

**Abstract Harmonic Analysis**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 930**

**Quantum Mechanics**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 932**

**Topics In Applied Mathematics**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 933**

**Topics In Applied Mathematics**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 935**

**Topics In Applied Mathematics**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 936**

**Topics In Control Theory**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 937**

**Topics In Control Theory**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 939**

**Topics In Control Theory**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 942**

**Topics In Topology & Geometry**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 943**

**Topics In Topology & Geometry**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 945**

**Topics In Topology & Geometry**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 950**

**Pontragin's Maximum Principle**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 972**

**Topics In Communication Theory**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 973**

**Topics In Communication Theory**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 975**

**Topics In Communication Theory**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 986**

**Philosophy Of Mathematics**

**Units: 3.00**

**Offering Faculty:**Faculty of Arts and Science

**MATH 999**

**Ph. D. Thesis Research**

**Units: 6.00**

**Offering Faculty:**Faculty of Arts and Science