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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.
Note: Commencing in Fall 2014, Mathematics 265, 267, 367, Mathematics 275, 277, 375 and 377 will replace respectively Mathematics 251, 253, 353, Applied Mathematics 217, 219, 307 and 309 and will serve as prerequisites for appropriate courses. In some special cases, Mathematics 267 will replace Mathematics 349 or 353. For these and other deviations from the general rule, see individual course entries for details. Mathematics 267 supplemented by Mathematics 177 will be accepted as equivalent to Mathematics 277.
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Applied Mathematics
307
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Differential Equations for Engineers
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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(3-1.5T)
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.
Notes:
This course will not be offered after Winter 2015.
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Applied Mathematics
309
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Vector Calculus for Engineers
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Functions of several variables, chain rule and differentials. Vector calculus, line, surface and volume integrals, Green's, Gauss' and Stokes' theorems. Students will complete a project using a computer algebra system.
Course Hours:
H(3-1.5T)
Prerequisite(s):
Applied Mathematics 219; or Mathematics 114 and one of Mathematics 253 or 283.
Antirequisite(s):
Credit for more than one of Applied Mathematics 309 and Mathematics 353 or 381 will not be allowed.
Notes:
This course will not be offered after Winter 2015.
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Applied Mathematics
311
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Differential Equations I
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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(3-1T)
Prerequisite(s):
Mathematics 253 or 267 or 277 or 283 or Applied Mathematics 219.
Antirequisite(s):
Credit for more than one of Applied Mathematics 311 and Mathematics 375 or Applied Mathematics 307 will not be allowed.
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Applied Mathematics
411
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Differential Equations II
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Existence and uniqueness theorems, comparison and oscillation theorems, Green's functions, Sturm-Liouville problems, systems of equations, phase portraits, stability.
Course Hours:
H(3-1T)
Prerequisite(s):
One of Mathematics 375, Applied Mathematics 311 or 307, and one of Mathematics 331, 353, 367, 377, 381, Applied Mathematics 309, or consent of the Department.
Notes:
It is recommended that students complete Mathematics 335 or 355 or Pure Mathematics 435 or 455 before taking this course.
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Applied Mathematics
413
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Introduction to Partial Differential Equations
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Orthogonal sets of functions, Fourier series, solution of potential equation, heat equation and wave equation. Numerical solution of partial differential equations.
Course Hours:
H(3-1T)
Prerequisite(s):
One of Mathematics 353, 367, 375, 381, Applied Mathematics 307 or 309 or 311, Mathematics 331; or consent of the Department.
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
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Mathematical Methods
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Mathematical analysis of linear systems. Fourier and Laplace transforms, applications and numerical methods. Functions of a complex variable and applications.
Course Hours:
H(3-1T)
Prerequisite(s):
One of Applied Mathematics 311, 307, Mathematics 331, 353, 367, 375, 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
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Introduction to Optimization
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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(3-1T)
Prerequisite(s):
Mathematics 311 or 313; and one of Mathematics 331 or 353 or 367 or 377 or 381 or Applied Mathematics 309.
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Applied Mathematics
481
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Introduction to Mathematical Finance
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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(3-1T)
Prerequisite(s):
Mathematics 321 or Statistics 321 or consent of the Department.
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Applied Mathematics
501
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Seminar in Applied Mathematics
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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(3-0)
Prerequisite(s):
Consent of the Department.
MAY BE REPEATED FOR CREDIT
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Applied Mathematics
503
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The Mathematics of Wavelets, Signal and Image Processing
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Continuous and discrete Fourier transforms, the Fast Fourier Transform, wavelet transforms, multiresolution analysis and orthogonal wavelet bases, and applications.
Course Hours:
H(3-0)
Prerequisite(s):
Applied Mathematics 491 or Computer Science 491.
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Applied Mathematics
505
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Calculus on Manifolds
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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(3-0)
Prerequisite(s):
Pure Mathematics 445 or 545; and one of Mathematics 375 or Applied Mathematics 311 or 307; or consent of the Department.
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Applied Mathematics
507
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Introduction to Relativity Theory
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Mathematical theories of space and time. Special Relativity. Electro-dynamics. General Relativity.
Course Hours:
H(3-0)
Prerequisite(s):
Applied Mathematics 505 or consent of the Department.
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Applied Mathematics
509
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Analytical Dynamics
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Symplectic geometry, Hamilton's equation, Hamilton-Jacobi theory, constraints and reduction.
Course Hours:
H(3-0)
Prerequisite(s):
Applied Mathematics 505 or consent of the Department.
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Applied Mathematics
581
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Stochastic Calculus for Finance
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Martingales in discrete and continuous time, risk-neutral valuations, discrete- and continuous-time (B,S)-security markets, Cox-Ross-Rubinstein formula, Wiener and Poisson processes, Ito formula, stochastic differential equations, Girsanov’s theorem, Black-Scholes and Merton formulas, stopping times and American options, stochastic interest rates and their derivatives, energy and commodity models and derivatives, value-at-risk and risk management.
Course Hours:
H(3-0)
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
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Computational Finance
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Review of financial asset price and option valuation models; model calibration; tree-based methods; finite-difference methods; Monte Carlo simulation; Fourier methods.
Course Hours:
H(3-0)
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 Department is a prerequisite for all graduate courses in Applied Mathematics.
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Applied Mathematics
601
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Topics in Applied Mathematics
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Topics will be chosen according to the interests of instructors and students.
Course Hours:
H(3-0)
Prerequisite(s):
Consent of the Department.
MAY BE REPEATED FOR CREDIT
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Applied Mathematics
605
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Differential Equations III
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Linear systems, classification. Non-linear systems: Existence and uniqueness. Flow and one parameter groups of transformations. Stability theory. Hyperbolicity, Unstable/Stable/Center manifold theorems. Poincare-Bendixson.
Course Hours:
H(3-0)
Prerequisite(s):
Applied Mathematics 411 and Pure Mathematics 445 or 545 or equivalents.
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Applied Mathematics
613
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Partial Differential Equations II
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Fundamental solutions, integral equations, eigenvalue problems, non-linear problems.
Course Hours:
H(3-0)
Prerequisite(s):
Consent of the Department.
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Applied Mathematics
617
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Analysis IV
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Analysis in abstract spaces. Function spaces.
Course Hours:
H(3-0)
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
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Research Seminar
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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(2S-0)
MAY BE REPEATED FOR CREDIT
NOT INCLUDED IN GPA
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Applied Mathematics
643
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Perturbation Theory
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Perturbation problems for ordinary differential equations, matrices and more general operators. Applications. Methods will be motivated by discussion of physical problems.
Course Hours:
H(3-0)
Prerequisite(s):
Familiarity with complex variables, linear algebra and differential equations.
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Applied Mathematics
651
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Monte Carlo Methods for Quantitative Finance
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Fundamental concepts of Monte Carlo methods; review of quantitative finance; random number generation; simulating stochastic differential equations; variance reduction; quasi-Monte Carlo methods; computing sensitivities; early exercise options; Levy processes and other price models; applications to risk management.
Course Hours:
H(3-0)
Prerequisite(s):
Consent of the Department.
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Applied Mathematics
653
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Introduction to Levy Processes with Applications
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Infinite divisibility, Levy processes (LP), the Levy-Khintchine formula; examples of LP; Poisson integration, the Levy-Ito decomposition, subordinators; Markov processes, semi-groups and generators of LP; Ito-formula for LP, quadratic variation; LP as time-changed Brownian motion, change of measure (Girsanov theorem); stochastic differential equations driven by LP; Feynman-Kac formula and martingale problem for LP; applications of LP; simulation of LPs.
Course Hours:
H(3-0)
Prerequisite(s):
Consent of the Department.
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Applied Mathematics
671
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Numerical Linear Algebra
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Iterative and elimination methods for linear systems of equations, determination of eigenvalues, linear and convex programming.
Course Hours:
H(3-0)
Prerequisite(s):
Mathematics 411 and Applied Mathematics 491.
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Applied Mathematics
673
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Approximation Theory
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Existence, uniqueness of minimal solutions, Haar systems, characterization by alternation, Remez algorithm, monotone operators, spline approximation.
Course Hours:
H(3-0)
Prerequisite(s):
Applied Mathematics 491; and Mathematics 335 or 355 or Pure Mathematics 435 or 455.
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Applied Mathematics
677
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Numerical Solution of Partial Differential Equations
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Explicit and implicit methods for PDE, difference equations.
Course Hours:
H(3-0)
Prerequisite(s):
Applied Mathematics 311 and 491.
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Applied Mathematics
681
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Stochastic Calculus for Finance
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Martingales in discrete and continuous time, risk-neutral valuations, discrete- and continuous-time (B,S)-security markets, Cox-Ross-Rubinstein formula, Wiener and Poisson processes, Ito formula, stochastic differential equations, Girsanov’s theorem, Black-Scholes and Merton formulas, stopping times and American options, stochastic interest rates and their derivatives, energy and commodity models and derivatives, value-at-risk and risk management.
Course Hours:
H(3-0)
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
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Computational Finance
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Review of financial asset price and option valuation models; model calibration; tree-based methods; finite-difference methods; Monte Carlo simulation; Fourier methods.
Course Hours:
H(3-0)
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.
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COURSES OF INSTRUCTION