XII. Course Descriptions
Mathematics
Department of Mathematics and Statistics
Suggested initial course sequence:

For students with 4U or OAC Calculus and expecting to pursue further studies in mathematics or the physical sciences: MATH*1200, MATH*1210.

For students interested in applications to the biological sciences: MATH*1080, MATH*2080.

For students not expecting to pursue further studies in mathematics: MATH*1030, one STAT*XXXX course.
MATH*1030 Business Mathematics F,W (30) [0.50] 
Primarily intended for business and economics students, this course is designed to introduce and reinforce the essential mathematical
skills needed to understand, analyze, and solve mathematical problems related to business and economics. Topics covered include
basic algebra; functions, including a review of exponential and logarithmic functions; sequences and series with financial
applications; limits; continuity; and differential calculus including derivatives, higher order derivatives, and curve sketching.

Offering(s): 
Also offered through Distance Education format. 
Prerequisite(s): 
4U Advanced Functions 
Restriction(s): 
MATH*1080, MATH*1200 Not available to students registered in the BSC program.

Department(s): 
Department of Mathematics and
Statistics

MATH*1080 Elements of Calculus I F,W (31) [0.50] 
This course provides an introduction to the calculus of one variable with emphasis on mathematical modelling in the biological
sciences. The topics covered include elementary functions, sequences and series, difference equations, differential calculus
and integral calculus.

Offering(s): 
Also offered through Distance Education format. 
Prerequisite(s): 
1 of 4U Advanced Functions, 4U Advanced Functions and Calculus or equivalent 
Restriction(s): 
IPS*1500, MATH*1200 
Department(s): 
Department of Mathematics and
Statistics

MATH*1090 Elements of Calculus II W (31) [0.50] 
This course will expand on integration techniques, and introduce students to difference and differential equations, vectors,
vector functions, and elements of calculus of two or more variables such as partial differentiation and multiple integration.
The course will emphasize content relevant to analyzing biological systems, and methods will be illustrated by application
to biological systems.

Prerequisite(s): 
1 of IPS*1500, MATH*1080, MATH*1200 
Restriction(s): 
IPS*1510, MATH*1210, MATH*2080 
Department(s): 
Department of Mathematics and
Statistics

MATH*1160 Linear Algebra I F,W (30) [0.50] 
This course provides an introduction to linear algebra in Euclidean space. Topics covered include: Ndimensional vectors,
dot product, matrices and matrix operations, systems of linear equations and Gaussian elimination, linear independence, subspaces,
basis and dimension, matrix inverse, matrix rank and determinant, eigenvalues, eigenvectors and diagonalization, orthogonalization
and projections, linear transformations. Some fundamental proofs and applications of these topics will be included.

Prerequisite(s): 
4U Calculus and Vectors or 4U Advanced Functions 
Restriction(s): 
ENGG*1500, MATH*2150, MATH*2160 
Department(s): 
Department of Mathematics and
Statistics

MATH*1200 Calculus I F (31) [0.50] 
This is a theoretical course intended primarily for students who expect to pursue further studies in mathematics and its applications.
Topics include inequalities and absolute value; compound angle formulas for trigonometric functions; limits and continuity
using rigorous definitions; the derivative and derivative formulas (including derivatives of trigonometric, exponential and
logarithmic functions); Fermat's theorem; Rolle's theorem; the meanvalue theorem; applications of the derivative; Riemann
sums; the definite integral; the fundamental theorem of calculus; applications of the definite integral; the mean value theorem
for integrals.

Prerequisite(s): 
1 of 4U Calculus and Vectors, 4U Advanced Functions and Calculus or Grade 12 Calculus 
Restriction(s): 
IPS*1500, MATH*1080 
Department(s): 
Department of Mathematics and
Statistics

MATH*1210 Calculus II W (31) [0.50] 
This course is a continuation of MATH*1200. It is a theoretical course intended primarily for students who need or expect to pursue further studies in mathematics,
physics, chemistry, engineering and computer science. Topics include inverse functions, inverse trigonometric functions, hyperbolic
functions, indeterminate forms and l'Hopital's rule, techniques of integration, parametric equations, polar coordinates, Taylor
and Maclaurin series; functions of two or more variables, partial derivatives, and if time permits, an introduction to multiple
integration.

Prerequisite(s): 
MATH*1080 or MATH*1200 
Restriction(s): 
IPS*1510, MATH*1090, MATH*2080 
Department(s): 
Department of Mathematics and
Statistics

MATH*2000 Proofs, Sets, and Numbers F (31) [0.50] 
This course exposes the student to formal mathematical proof, and introduces the theory of sets and number systems. Topics
include relations and functions, number systems including formal properties of the natural numbers, integers, and the real
and complex numbers. Equivalence relations and partial and total orders are introduced. The geometry and topology of the real
number line and Cartesian plane are introduced. Techniques of formal proof are introduced including wellordering, mathematical
induction, proof by contradiction, and proof by construction. These techniques will be applied to fundamental theorems from
linear algebra.

Prerequisite(s): 
1 of IPS*1500, MATH*1080, MATH*1160, MATH*1200 
Department(s): 
Department of Mathematics and
Statistics

MATH*2130 Numerical Methods W (31) [0.50] 
This course provides a theoretical and practical introduction to numerical methods for approximating the solution(s) of linear
and nonlinear problems in the applied sciences. The topics covered include: solution of a single nonlinear equation; polynomial
interpolation; numerical differentiation and integration; solution of initial value and boundary value problems; and the solution
of systems of linear and nonlinear algebraic equations.

Prerequisite(s): 
(CIS*1300 or CIS*1500), (1 of IPS*1510, MATH*1090, MATH*1210, MATH*2080)

Department(s): 
Department of Mathematics and
Statistics

MATH*2200 Advanced Calculus I F (30) [0.50] 
The topics covered in this course include infinite sequences and series, power series, tests for convergence, Taylor's theorem
and Taylor series for functions of one variable, planes and quadratic surfaces, limits, and continuity, differentiability
of functions of two or more variables, partial differentiation, directional derivatives and gradients, tangent planes, linear
approximation, Taylor's theorem for functions of two variables, critical points, extreme value problems, implicit function
theorem, Jacobians, multiple integrals, and change of variables.

Prerequisite(s): 
1 of IPS*1510, MATH*1090, MATH*1210, MATH*2080 
Department(s): 
Department of Mathematics and
Statistics

MATH*2210 Advanced Calculus II W (30) [0.50] 
This course continues the study of multiple integrals, introducing spherical and cylindrical polar coordinates. The course
also covers vector and scalar fields, including the gradient, divergence, curl and directional derivative, and their physical
interpretation, as well as line integrals and the theorems of Green and Stokes.

Prerequisite(s): 
MATH*2200 
Department(s): 
Department of Mathematics and
Statistics

MATH*2270 Applied Differential Equations F (31) [0.50] 
This course covers the solution of differential equations that arise from problems in engineering and science. Topics include
linear equations of first and higher order, systems of linear equations, Laplace transforms, series solutions of secondorder
equations, and an introduction to partial differential equations.

Prerequisite(s): 
(1 of ENGG*1500, MATH*1160, MATH*2150, MATH*2160), (1 of IPS*1510, MATH*1090, MATH*1210, MATH*2080)

Department(s): 
Department of Mathematics and
Statistics

MATH*3100 Differential Equations II W (31) [0.50] 
This course continues the study of differential equations. Power series solutions around regular singular points including
Bessel equations are presented. First order linear systems and their general solution by matrix methods are thoroughly covered.
Nonlinear systems are introduced along with the concepts of linearization, stability of equilibria, phase plane analysis,
Lyapunov's method, periodic solutions and limit cycles. Twopoint boundary value problems are discussed and an introduction
to linear partial differential equations and their solution by separation of variables and Fourier series is given.

Prerequisite(s): 
MATH*2270, (1 of ENGG*1500, MATH*1160, MATH*2150, MATH*2160)

Department(s): 
Department of Mathematics and
Statistics

MATH*3130 Abstract Algebra F (30) [0.50] 
This course is an introduction to abstract algebra, covering both group theory and ring theory. Specific topics covered include
an introduction to group theory, permutations, symmetric and dihedral groups, subgroups, normal subgroups and factor groups.
Group theory continues through the fundamental homomorphism theorem. Ring theory material covered includes an introduction
to ring theory, subrings, ideals, quotient rings, polynomial rings, and the fundamental ring homomorphism theorem.

Offering(s): 
Offered in evennumbered years. 
Prerequisite(s): 
MATH*2000, (1 of MATH*1160, MATH*2150, MATH*2160)

Department(s): 
Department of Mathematics and
Statistics

MATH*3160 Linear Algebra II W (30) [0.50] 
The topics in this course include complex vector spaces, direct sum decompositions of vector spaces, the CayleyHamilton theorem,
the spectral theorem for normal operators and the Jordan canonical form.

Prerequisite(s): 
(MATH*1160 or MATH*2160), 1.00 credits in MATH or STAT at the 2000 level or above

Department(s): 
Department of Mathematics and
Statistics

MATH*3200 Real Analysis F (30) [0.50] 
This course provides a basic foundation for real analysis. The rigorous treatment of the subject in terms of theory and examples
gives students the flavour of mathematical reasoning and intuition for other advanced topics in mathematics. Topics covered
include the real number line and the supremum property; metric spaces; continuity and uniform continuity; completeness and
compactness; the Banach fixedpoint theorem and its applications to ODEs; uniform convergence and the rigorous treatment of
the Riemann integral.

Prerequisite(s): 
MATH*2000, MATH*2210, (MATH*1160 or MATH*2160)

Department(s): 
Department of Mathematics and
Statistics

MATH*3240 Operations Research F (30) [0.50] 
This is a course in mathematical modelling which has applications to engineering, economics, business and logistics. Topics
covered include linear programming and the simplex method, network models and the shortest path, maximum flow and minimal
spanning tree problems as well as a selection of the following: nonlinear programming, constrained optimization, deterministic
and probabilistic dynamic programming, game theory and simulation.

Offering(s): 
Offered in oddnumbered years. 
Prerequisite(s): 
(1 of MATH*1160, MATH*2150, MATH*2160), 0.50 credits in statistics

Corequisite(s): 
MATH*2200 
Department(s): 
Department of Mathematics and
Statistics

MATH*3260 Complex Analysis W (30) [0.50] 
This course extends calculus to cover functions of a complex variable; it introduces complex variable techniques which are
very useful for mathematics, the physical sciences and engineering. Topics include complex differentiation, planar mappings,
analytic and harmonic functions, contour integration, Taylor and Laurent series, the residue calculus and its application
to the computation of trigonometric and improper integrals, conformal mapping and the Dirichlet problem.

Prerequisite(s): 
MATH*2200 
Department(s): 
Department of Mathematics and
Statistics

MATH*3510 Biomathematics W (30) [0.50] 
This course will convey the fundamentals of applying mathematical modelling techniques to understanding and predicting the
dynamics of biological systems. Students will learn the development, analysis, and interpretation of biomathematical models
based on discretetime and continuoustime models. Applications may include examples from population biology, ecology, infectious
diseases, microbiology, and genetics.

Prerequisite(s): 
MATH*2270, (1 of ENGG*1500, MATH*1160, MATH*2150, MATH*2160)

Department(s): 
Department of Mathematics and
Statistics

MATH*4050 Topics in Mathematics I W (30) [0.50] 
In this course students will discuss selected topics at an advanced level. It is intended mainly for mathematics students
in the 6th to 8th semester. Content will vary from year to year. Sample topics include: probability theory, Fourier analysis,
mathematical logic, operator algebras, number theory combinatorics, philosophy of mathematics, fractal geometry, chaos, stochastic
differential equations.

Offering(s): 
Offered in oddnumbered years. 
Prerequisite(s): 
MATH*3200 
Department(s): 
Department of Mathematics and
Statistics

MATH*4200 Advanced Analysis W (30) [0.50] 
This senior course in analysis will cover basic operator theory on Hilbert spaces, including selfadjoint operators and the
spectral theorem. Other topics may include weak solutions, Sobolev spaces, inverse problems, measure theoretic probability
or advanced topics from linear or nonlinear functional analysis.

Offering(s): 
Offered in evennumbered years. 
Prerequisite(s): 
MATH*3200 
Department(s): 
Department of Mathematics and
Statistics

MATH*4240 Advanced Topics in Modeling and Optimization F (30) [0.50] 
This course is a study of advanced topics in the areas of optimization and modeling. Topics may include continuous and discrete
models together with techniques for their analysis and design, and optimization topics such as game theory, networks, nonlinear
problems, Markov chains, queuing theory, agentbased models, computational intelligence based techniques and computational
optimization techniques.

Prerequisite(s): 
0.50 credits in Mathematics at the 3000 level. 
Department(s): 
Department of Mathematics and
Statistics

MATH*4270 Partial Differential Equations F (30) [0.50] 
This course focuses on first and secondorder partial differential equations, with examples and applications from selected
fields such as physics, engineering and biology. Topics may include the wave equation, the heat equation, Laplace's equation,
linearity and separation of variables, solution by Fourier series, Bessel, Legendre and Green's functions, an introduction
to the method of characteristics and Fourier transforms. The classification of linear secondorder partial differential equations
is discussed.

Prerequisite(s): 
MATH*3100 
Department(s): 
Department of Mathematics and
Statistics

MATH*4310 Combinatorics and Graph Theory W (30) [0.50] 
This course is an introduction to combinatorics with a focus that includes graph theory. Specific topics covered are enumerative
combinatorics up to inclusionexclusion, the theory of simple graphs, Latin squares and orthogonal Latin squares, and introductory
coding theory. Instructors may continue to one or more of the Polya theory of counting, graph coloring and embedding, combinatorial
design theory, Ramsey theory, or advanced topics in enumerative combinatorics.

Offering(s): 
Offered in oddnumbered years. 
Prerequisite(s): 
10.00 credits including MATH*2000 
Department(s): 
Department of Mathematics and
Statistics

MATH*4440 Case Studies in Mathematics and Statistics W (30) [0.50] 
This capstone course for the Mathematical Science major provides students with an opportunity to synthesize knowledge and
utilize problemsolving skills accumulated over the course of their studies. The course will focus on case studies drawn from
engineering, computer science, biology, life and physical sciences, medicine, and/or economics.

Prerequisite(s): 
At least 3.0 mathematics and/or statistics credits at the 3000 level or above. 
Restriction(s): 
Restricted to students in the Mathematical Science major. 
Department(s): 
Department of Mathematics and
Statistics

MATH*4600 Advanced Research Project in Mathematics F,W (06) [1.00] 
Each student in this course will undertake an individual research project in some area of mathematics, under the supervision
of a faculty member. A written report and a public presentation of the project will be required.

Prerequisite(s): 
1.00 credits in Mathematics at the 3000 level or above. 
Restriction(s): 
Approval of a supervisor and the course coordinator. 
Department(s): 
Department of Mathematics and
Statistics
