
Instruction offered by members of the Department of Mathematics and Statistics in the Faculty of Science.
Department Head – M. Lamoureux
Note: For listings of related courses, see Actuarial Science, Mathematics, Pure Mathematics and Statistics.


Applied Mathematics
217

Calculus for Engineers and Scientists


Functions, limits, continuity, derivatives, Mean Value Theorem, integrals, Fundamental Theorem of Calculus, applications to the physical sciences.
Course Hours:
H(31T1.5)
Prerequisite(s):
A grade of 70 per cent or higher in Pure Mathematics 30 or Mathematics 301 and credit in Mathematics 31; or admission to the Faculty of Engineering including credit in Pure Mathematics 30 or Mathematics 301 and Mathematics 31.
Antirequisite(s):
Credit for more than one of Applied Mathematics 217 and Mathematics 249, 251, or 281 will not be allowed.

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Applied Mathematics
307

Differential Equations for Engineers


Definition, existence and uniqueness of solutions, first and second order equations with applications, series solutions about regular points and singular points, special functions. Laplace transform, systems of equations.
Course Hours:
H(31.5T)
Prerequisite(s):
Mathematics 211 or 213; and Applied Mathematics 219 or Mathematics 253 or 283.
Antirequisite(s):
Credit for both Applied Mathematics 307 and 311 will not be allowed.

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Applied Mathematics
311

Differential Equations I


Classification of ordinary differential equations, first and second order equations with applications, series solutions about regular points and singular points, special functions, Laplace transform.
Course Hours:
H(31T)
Prerequisite(s):
Mathematics 253 or 283 or Applied Mathematics 219.
Antirequisite(s):
Credit for both Applied Mathematics 311 and 307 will not be allowed.

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Applied Mathematics
411

Differential Equations II


Existence and uniqueness theorems, comparison and oscillation theorems, Green's functions, SturmLiouville problems, systems of equations, phase portraits, stability.
Course Hours:
H(31T)
Prerequisite(s):
One of Applied Mathematics 311 or 307, and one of Mathematics 331, 353, 381, Applied Mathematics 309, or consent of the Division.
Notes:
It is recommended that students complete Pure Mathematics 435 or 455 before taking this course.

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Applied Mathematics
413

Introduction to Partial Differential Equations


Orthogonal sets of functions, Fourier series, solution of potential equation, heat equation and wave equation. Numerical solution of partial differential equations.
Course Hours:
H(31T)
Prerequisite(s):
One of Mathematics 353, 381, Applied Mathematics 307 or 309 or 311, Mathematics 331; or consent of the Division.
Antirequisite(s):
Credit for both Applied Mathematics 413 and 415 will not be allowed for the Actuarial Science, Applied Mathematics, General Mathematics, Pure Mathematics, and Statistics programs.

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Applied Mathematics
415

Mathematical Methods


Mathematical analysis of linear systems. Fourier and Laplace transforms, applications and numerical methods. Functions of a complex variable and applications.
Course Hours:
H(31T)
Prerequisite(s):
One of Applied Mathematics 311, 307, Mathematics 331, 353, 381, or Applied Mathematics 309.
Antirequisite(s):
Credit will be not be allowed for more than one of Applied Mathematics 415 and 413 for the Actuarial Science, Applied Mathematics, General Mathematics, Pure Mathematics, and Statistics programs.
Notes:
Credit in an introductory Computer Science course prior to taking Applied Mathematics 415 is highly recommended.

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Applied Mathematics
425

Introduction to Optimization


Examples of optimization problems. Quadratic forms, minimum energy and distance. Least squares, generalized inverse. Location and classification of critical points. Variational treatment of eigenvalues. Lagrange multipliers. Linear programming.
Course Hours:
H(31T)
Prerequisite(s):
Mathematics 311 or 313; and Mathematics 353 or 381 or Applied Mathematics 309 or Mathematics 331.

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Applied Mathematics
481

Introduction to Mathematical Finance


This course is an introduction to the fundamental concepts of mathematical finance in an elementary setting. Topics include: risk, return, no arbitrage principle; basic financial derivatives: options, forwards and future contracts; risk free assets, time value of money, zero coupon bonds; risky assets, binomial tree model, fundamental theorem of asset pricing; portfolio management and capital asset pricing model; no arbitrage pricing of financial derivatives; hedging.
Course Hours:
H(31T)
Prerequisite(s):
Mathematics 321 or consent of the division.

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Applied Mathematics
501

Seminar in Applied Mathematics


Topics will be chosen according to the interests of instructors and students and could include analysis of optimization algorithms, approximation theory, control theory, differential equations, mathematical physics.
Course Hours:
H(30)
Prerequisite(s):
Consent of the Division.
MAY BE REPEATED FOR CREDIT

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Applied Mathematics
503

The Mathematics of Wavelets, Signal and Image Processing


Continuous and discrete Fourier transforms, the Fast Fourier Transform, wavelet transforms, multiresolution analysis and orthogonal wavelet bases, and applications.
Course Hours:
H(30)
Prerequisite(s):
Applied Mathematics 491 or Computer Science 491.

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Applied Mathematics
505

Calculus on Manifolds


Integral and differential calculus on manifolds including tensor fields, covariant differentiation, Lie differentiation, differential forms, Frobenius' theorem, Stokes' theorem, flows of vector fields.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 445 or 545; and one of Applied Mathematics 311 or 307; or consent of the Division.

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Applied Mathematics
507

Introduction to Relativity Theory


Mathematical theories of space and time. Special Relativity. Electrodynamics. General Relativity.
Course Hours:
H(30)
Prerequisite(s):
Applied Mathematics 505 or consent of the Division.

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Applied Mathematics
509

Analytical Dynamics


Symplectic geometry, Hamilton's equation, HamiltonJacobi theory, constraints and reduction.
Course Hours:
H(30)
Prerequisite(s):
Applied Mathematics 505 or consent of the Division.

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Applied Mathematics
581

Stochastic Calculus for Finance


Martingales in discrete and continuous time, riskneutral valuations, discrete and continuoustime (B,S)security markets, CoxRossRubinstein formula, Wiener and Poisson processes, Ito formula, stochastic differential equations, Girsanov’s theorem, BlackScholes and Merton formulas, stopping times and American options, stochastic interest rates and their derivatives, energy and commodity models and derivatives, valueatrisk and risk management.
Course Hours:
H(30)
Prerequisite(s):
Applied Mathematics 481.
Antirequisite(s):
Credit for both Applied Mathematics 581 and 681 will not be allowed.

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Applied Mathematics
583

Computational Finance


Review of financial asset price and option valuation models; model calibration; treebased methods; finitedifference methods; Monte Carlo simulation; Fourier methods.
Course Hours:
H(30)
Prerequisite(s):
Applied Mathematics 481 and 491.
Antirequisite(s):
Credit for both Applied Mathematics 583 and 683 will not be allowed.

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Graduate Courses
In addition to the prerequisites listed below, consent of the Applied Mathematics Division is a prerequisite for all graduate courses in Applied Mathematics.

Applied Mathematics
601

Topics in Applied Mathematics


Topics will be chosen according to the interests of instructors and students.
Course Hours:
H(30)
Prerequisite(s):
Consent of the Division.
MAY BE REPEATED FOR CREDIT

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Applied Mathematics
605

Differential Equations III


Linear systems, classification. Nonlinear systems: Existence and uniqueness. Flow and one parameter groups of transformations. Stability theory. Hyperbolicity, Unstable/Stable/Center manifold theorems. PoincareBendixson.
Course Hours:
H(30)
Prerequisite(s):
Applied Mathematics 411 and Pure Mathematics 445 or 545 or equivalents.

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Applied Mathematics
613

Partial Differential Equations II


Fundamental solutions, integral equations, eigenvalue problems, nonlinear problems.
Course Hours:
H(30)
Prerequisite(s):
Consent of the Division.

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Applied Mathematics
617

Analysis IV


Analysis in abstract spaces. Function spaces.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 545.
Antirequisite(s):
Credit for Applied Mathematics 617 and Pure Mathematics 617 will not be allowed.

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Applied Mathematics
621

Research Seminar


A professional skills course, focusing on the development of technical proficiencies that are essential for students to succeed in their future careers as practicing mathematicians in academia, government, or industry. The emphasis is on delivering professional presentations and using modern mathematical research tools. A high level of active student participation is required.
Course Hours:
Q(2S0)
MAY BE REPEATED FOR CREDIT
NOT INCLUDED IN GPA

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Applied Mathematics
643

Perturbation Theory


Perturbation problems for ordinary differential equations, matrices and more general operators. Applications. Methods will be motivated by discussion of physical problems.
Course Hours:
H(30)
Prerequisite(s):
Familiarity with complex variables, linear algebra and differential equations.

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Applied Mathematics
651

Monte Carlo Methods for Quantitative Finance


Fundamental concepts of Monte Carlo methods; review of quantitative finance; random number generation; simulating stochastic differential equations; variance reduction; quasiMonte Carlo methods; computing sensitivities; early exercise options; Levy processes and other price models; applications to risk management.
Course Hours:
H(30)
Prerequisite(s):
Consent of the Department.

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Applied Mathematics
653

Introduction to Levy Processes with Applications


Infinite divisibility, Levy processes (LP), the LevyKhintchine formula; examples of LP; Poisson integration, the LevyIto decomposition, subordinators; Markov processes, semigroups and generators of LP; Itoformula for LP, quadratic variation; LP as timechanged Brownian motion, change of measure (Girsanov theorem); stochastic differential equations driven by LP; FeynmanKac formula and martingale problem for LP; applications of LP; simulation of LPs.
Course Hours:
H(30)
Prerequisite(s):
Consent of the Department.

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Applied Mathematics
671

Numerical Linear Algebra


Iterative and elimination methods for linear systems of equations, determination of eigenvalues, linear and convex programming.
Course Hours:
H(30)
Prerequisite(s):
Mathematics 411 and Applied Mathematics 491.

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Applied Mathematics
673

Approximation Theory


Existence, uniqueness of minimal solutions, Haar systems, characterization by alternation, Remez algorithm, monotone operators, spline approximation.
Course Hours:
H(30)
Prerequisite(s):
Applied Mathematics 491; and Pure Mathematics 435 or 455.

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Applied Mathematics
677

Numerical Solution of Partial Differential Equations


Explicit and implicit methods for PDE, difference equations.
Course Hours:
H(30)
Prerequisite(s):
Applied Mathematics 311 and 491.

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Applied Mathematics
681

Stochastic Calculus for Finance


Martingales in discrete and continuous time, riskneutral valuations, discrete and continuoustime (B,S)security markets, CoxRossRubinstein formula, Wiener and Poisson processes, Ito formula, stochastic differential equations, Girsanov’s theorem, BlackScholes and Merton formulas, stopping times and American options, stochastic interest rates and their derivatives, energy and commodity models and derivatives, valueatrisk and risk management.
Course Hours:
H(30)
Prerequisite(s):
Applied Mathematics 481.
Antirequisite(s):
Credit for both Applied Mathematics 681 and 581 will not be allowed.

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Applied Mathematics
683

Computational Finance


Review of financial asset price and option valuation models; model calibration; treebased methods; finitedifference methods; Monte Carlo simulation; Fourier methods.
Course Hours:
H(30)
Prerequisite(s):
Applied Mathematics 481 and 491.
Antirequisite(s):
Credit for both Applied Mathematics 683 and 583 will not be allowed.

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In addition to the numbered and titled courses shown above, the department offers a selection of advanced level graduate courses specifically designed to meet the needs of individuals or small groups of students at the advanced doctoral level. These courses are numbered in the series 800.01 to 899.99. Such offerings are, of course, conditional upon the availability of staff resources.
