Site Navigation
Welcome
Important Notice and Disclaimer
Academic Schedule
Types of Credentials and Sub-Degree Nomenclature
Undergraduate Degrees with a Major
Combined Degrees
Minor Programs
Student and Campus Services
Admissions
Academic Regulations
Experiential Learning
Tuition and General Fees
Student Financial Support
Architecture, Planning and Landscape, School of
Faculty of Arts
Cumming School of Medicine
Faculty of Graduate Studies
Haskayne School of Business
Faculty of Kinesiology
Faculty of Law
Faculty of Nursing
Qatar Faculty
Schulich School of Engineering
Faculty of Science
Faculty of Social Work
Faculty of Veterinary Medicine
Werklund School of Education
Embedded Certificates
Continuing Education
COURSES OF INSTRUCTION
How to Use
Courses of Instruction by Faculty
Course Descriptions
A
B
C
D
E
F
G
H
I
J, K
L
M
Management Studies MGST
Manufacturing Engineering ENMF
Marine Science MRSC
Marketing MKTG
Mathematics MATH
Mechanical Engineering ENME
Medical Graduate Education MDGE
Medical Physics MDPH
Medical Precision Health MDPR
Medical Science MDSC
Medicine MDCN
Museum and Heritage Studies MHST
Music MUSI
Music Education MUED
Music Performance MUPF
N, O
P
R
S
T, U
V, W, Z
About the University of Calgary
Who's Who
Glossary of Terms
Contact Us
Archives
Summary of Changes for the 2021/22 Calendar
University of Calgary Calendar 2021-2022 COURSES OF INSTRUCTION Course Descriptions M Mathematics MATH
Mathematics MATH

For more information about these courses see the Department of Mathematics and Statistics math.ucalgary.ca.

Notes:

  • For listings of related courses, see Actuarial Science and Statistics.
  • Effective Fall 2018, all Applied Mathematics and Pure Mathematics courses have been renamed as Mathematics with a change in course number in some cases. Please refer to the descriptions of the individual Mathematics courses for details. Students enrolled in any program that requires any Applied Mathematics or Pure Mathematics course should use the corresponding Mathematics course as replacement.
Mathematics 177       Further Topics from Mathematics 277
An overview of the basic notions in multivariate calculus: vector functions and differentiation, curves and parametrization, functions of several variables, partial differentiation, differentiability, implicit functions, extreme values.
Course Hours:
0.75 units; (16 hours)
Prerequisite(s):
Mathematics 267; and Mathematics 211 or 213.
Notes:
This course covers topics to allow students with credit in Mathematics 267 to be permitted to register in Mathematics 375.
NOT INCLUDED IN GPA
back to top
Junior Courses
Mathematics 205       Mathematical Explorations
A mathematics appreciation course. Topics selected by the instructor to provide a contemporary mathematical perspective and experiences in mathematical thinking. May include historical material on the development of classical mathematical ideas as well as the evolution of recent mathematics.
Course Hours:
3 units; (3-1)
Prerequisite(s):
Mathematics 30-1, 30-2, or Mathematics 2 (offered by Continuing Education).
Notes:
Not included in the Field of Mathematics.
back to top
Mathematics 209       Applied and Computational Linear Algebra for Energy Engineers
An introduction to systems of linear equations, vectors in Euclidean space, matrix algebra, linear transformations, eigenvalues and engenvectors. Geometrical applications and computing techniques will be emphasized. Students will complete a project using mathematical software.
Course Hours:
3 units; (4-2)
Prerequisite(s):
Admission to the Energy Engineering Program.
back to top
Mathematics 211       Linear Methods I
An introduction to systems of linear equations, vectors in Euclidean space and matrix algebra. Additional topics include linear transformations, determinants, complex numbers, eigenvalues, and applications.
Course Hours:
3 units; (3-1)
Prerequisite(s):
Mathematics 30-1 or Mathematics 2 (offered by Continuing Education).
Antirequisite(s):
Credit for Mathematics 211 and 213 will not be allowed.
back to top
Mathematics 213       Linear Algebra I
A rigorous introduction to the theory of vector spaces, with an emphasis on proof writing and abstract reasoning. Topics include fields, subspaces, bases and dimension, linear transformations, determinants, eigenvalues and eigenvectors.
Course Hours:
3 units; (3-1)
Prerequisite(s):
A grade of 90 per cent or higher in both Mathematics 30-1 and Mathematics 31 or consent of the Department.
Antirequisite(s):
Credit for Mathematics 213 and 211 will not be allowed.
back to top
Mathematics 249       Introductory Calculus
An introduction to single variable calculus. Limits, derivatives and integrals of algebraic, exponential, logarithmic and trigonometric functions play a central role. Additional topics include applications of differentiation; the fundamental theorem of calculus, improper integrals and applications of integration.
Course Hours:
3 units; (4-1)
Prerequisite(s):
Mathematics 30-1 or Mathematics 2 (offered by Continuing Education).
Antirequisite(s):
Not open to students with 50 per cent or higher in Mathematics 31 or a grade of "C" or higher in Mathematics 3 offered through University of Calgary Continuing Education, except with special departmental permission. Credit for Mathematics 249 and either 265 or 275 will not be allowed.
back to top
Mathematics 265       University Calculus I
An introduction to single variable calculus intended for students with credit in high school calculus. Limits, derivatives, and integrals of algebraic, exponential, logarithmic and trigonometric functions play a central role. Additional topics include applications of differentiation; the fundamental theorem of calculus, improper integrals and applications of integration. Differential calculus in several variables will also be introduced. 
Course Hours:
3 units; (3-1)
Prerequisite(s):
Mathematics 30-1 or Mathematics 2 (offered by Continuing Education); and Mathematics 31 or Mathematics 3 (offered by Continuing Education).
Antirequisite(s):
Credit for Mathematics 265 and either 249 or 275 will not be allowed.
back to top
Mathematics 267       University Calculus II
A concluding treatment of single variable calculus and an introduction to calculus in several variables. Single variable calculus: techniques of integration, sequences, series, convergence tests, and Taylor series. Calculus of several variables: partial differentiation, multiple integration, parametric equations, and applications
Course Hours:
3 units; (3-1)
Prerequisite(s):
3 units from Mathematics 249, 265 or 275.
Antirequisite(s):
Credit for Mathematics 267 and 277 will not be allowed.
back to top
Mathematics 271       Discrete Mathematics
An introduction to proof techniques and abstract mathematical reasoning: sets, relations and functions; mathematical induction; integers, primes, divisibility and modular arithmetic; counting and combinatorics; elements of probability, discrete random variables and Bayes’ theorem.
Course Hours:
3 units; (3-1T-1)
Prerequisite(s):
Mathematics 211 or 213.
back to top
Mathematics 273       Numbers and Proofs
A rigorous introduction to proof techniques and abstract mathematical reasoning with an emphasis on number systems: functions, sets and relations; the integers, prime numbers, divisibility and modular arithmetic; induction and recursion; real numbers; Cauchy sequences and completeness; complex numbers.
Course Hours:
3 units; (3-1T-1)
Prerequisite(s):
A grade of 90 per cent or higher in both Mathematics 30-1 and 31 or consent of the Department.
back to top
Mathematics 275       Calculus for Engineers and Scientists
An extensive treatment of differential and integral calculus in a single variable, with an emphasis on applications. Differentiation: derivative laws, the mean value theorem, optimization, curve sketching and other applications. Integral calculus: the fundamental theorem of calculus, techniques of integration, improper integrals, and areas of planar regions. Infinite series: power series, Taylor’s theorem and Taylor series.
Course Hours:
3 units; (3-1T-1.5)
Prerequisite(s):
Mathematics 30-1 or Mathematics 2 (offered by Continuing Education); and Mathematics 31 or Mathematics 3 (offered by Continuing Education).
Antirequisite(s):
Credit for Mathematics 275 and either 249 or 265 will not be allowed. 
back to top
Mathematics 277       Multivariable Calculus for Engineers and Scientists
An introduction to calculus of several real variables: curves and parametrizations, partial differentiation, the chain rule, implicit functions; integration in two and three variables and applications; optimization and Lagrange multipliers.
Course Hours:
3 units; (3-1T-1.5)
Prerequisite(s):
Mathematics 275; and Mathematics 211 or 213.
Antirequisite(s):
Credit for Mathematics 277 and 267 will not be allowed.
back to top
Senior Courses
Mathematics 305       Inside Mathematics
An exploration of the usually tacit elements of mathematical concepts and processes, the course focuses on strategies for unpacking concepts and for sustained engagement in inquiry.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 211 or 213; and Mathematics 271 or 273.
Notes:
This course will be co-taught by scholars from the Faculty of Science and Werklund School of Education.
Also known as:
(Education 305)
back to top
Mathematics 307       Complex Analysis I
An initial treatment of complex analytic functions in a single variable. Topics include differentiation, Cauchy-Riemann equations, line integration, Cauchy’s theorem and Cauchy’s integral formula, Taylor’s theorem, the residue theorem, and applications to definite integrals.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 211 or 213; and Mathematics 267 or 277; and Mathematics 271 or 273.
Antirequisite(s):
Credit for Mathematics 307 and 421 will not be allowed.
back to top
Mathematics 311       Linear Methods II
An introductory course in the theory of abstract vector spaces: linear independence, spanning sets, basis and dimension; linear transformations and the rank-nullity theorem; the Gram-Schmidt algorithm and orthogonal diagonalization; singular value decomposition and other applications.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 211 or 213.
Antirequisite(s):
Credit for Mathematics 311 and 313 will not be allowed.
back to top
Mathematics 313       Linear Algebra II
The theory of linear operators acting on finite dimensional vector spaces: invariant subspaces, diagonalization and triangulation; canonical forms; inner product spaces and orthogonalization; spectral theory; singular value decomposition and other applications.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 213.
Antirequisite(s):
Credit for Mathematics 311 and 313 will not be allowed.
back to top
Mathematics 315       Algebra I
A broad overview of the elementary theory of groups, rings and fields. Group theory: cyclic, symmetric, alternating, dihedral and classical matrix groups, cosets and Lagrange’s theorem, group homomorphisms, normal subgroups, quotient groups and the isomorphism theorem. Rings and fields: the integers modulo n, polynomial rings, ring homomorphisms, ideals, quotient rings the isomorphism theorem, unique factorization domains, principal ideal domains, Euclidean domains and the construction of finite fields.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 271 or 273.
Antirequisite(s):
Credit for Mathematics 315 and Pure Mathematics 317 will not be allowed.
Also known as:
(formerly Pure Mathematics 315)
back to top
Mathematics 318       Introduction to Cryptography
The basics of cryptography, with emphasis on attaining well-defined and practical notions of security. Symmetric and public-key cryptosystems; one-way and trapdoor functions; mechanisms for data integrity; digital signatures; key management; applications to the design of cryptographic systems. Assessment will primarily focus on mathematical theory and proof-oriented homework problems; additional application programming exercises will be available for extra credit.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 211 or 213; and Mathematics 271 or 273.
Antirequisite(s):
Credit for Mathematics 318 and any of Pure Mathematics 329, Computer Science 418, 429, or 557 will not be allowed.
Also known as:
(formerly Pure Mathematics 418)
back to top
Mathematics 319       Transformation Geometry
Geometric transformations in the Euclidean plane: Symmetry, Frieze, and Wallpaper groups.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 211 or 213; and Mathematics 271 or 273.
Also known as:
(formerly Pure Mathematics 319)
back to top
Mathematics 322       Differential Geometry
The fundamentals of differential geometry primarily with a focus on the theory of curves and surfaces in three dimensional space. The theory of curves studies global properties of curves such as the four vertex theorem. The theory of surfaces introduces the fundamental quadratic forms of a surface, intrinsic and extrinsic geometry of surfaces, and the Gauss-Bonnet theorem.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 271 or 273; and Mathematics 367 or 377; and Mathematics 375 or 376.
Also known as:
(formerly Pure Mathematics 423)
back to top
Mathematics 325       Introduction to Optimization
An example driven overview of optimization problems: quadratic forms, minimum energy and distance, least squares, generalized inverses, location and classification of critical points, variational treatment of eigenvalues, Lagrange multipliers and linear programming.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 311 or 313; and 3 units from Mathematics 367, 377 or 331.
Also known as:
(formerly Applied Mathematics 425)
back to top
Mathematics 327       Number Theory
Divisibility and the Euclidean algorithm, modular arithmetic and congruences, quadratic reciprocity, arithmetic functions, distribution of primes.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 271 or 273.
Also known as:
(formerly Pure Mathematics 427)
back to top
Mathematics 331       Advanced Calculus for Energy Engineering
A broad introduction to ordinary differential equations, multivariable calculus and elements of vector calculus. Differential equations: linear ordinary differential equations, and systems of linear ordinary differential equations. Calculus of several variables: partial differentiation, the chain rule, double and triple integrals. Vector Calculus: vector fields, line integrals, and flux integrals.

Course Hours:
3 units; (4-2T)
Prerequisite(s):
Mathematics 209 and admission to the Energy Engineering program.
back to top
Mathematics 335       Analysis I
A rigorous treatment of the theory of functions of a single real variable: functions, countable and uncountable sets; the axioms and basic topology of the real numbers; convergence of sequences; limits of functions, continuity and uniform continuity; differentiability and the mean value theorem; the Riemann integral and the fundamental theorem of calculus; series and convergence tests.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 267 or 277; and Mathematics 271 or 273.
Antirequisite(s):
Credit for Mathematics 335 and 355 will not be allowed.
back to top
Mathematics 355       Enriched Analysis I
A rigorous treatment of the theory of functions of a single real variable: functions, countable and uncountable sets; the axioms and basic topology of the real numbers; convergence of sequences; limits of functions, continuity and uniform continuity; differentiability and the mean value theorem; the Riemann integral and the fundamental theorem of calculus; series and convergence tests. Mathematics 355 will contain more challenging and deeper assessment than Mathematics 335.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 267 or 277; and Mathematics 271 or 273.
Antirequisite(s):
Credit for Mathematics 355 and 335 will not be allowed.
back to top
Mathematics 361       Linear Spaces with Applications
Canonical forms. Inner product spaces, invariant subspaces and spectral theory. Quadratic forms.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 311 or a grade of "B+" or higher in Mathematics 213; and Mathematics 267 or 277.
Antirequisite(s):
Credit for Mathematics 361 and 313 will not be allowed.
Also known as:
(formerly Mathematics 411)
back to top
Mathematics 367       University Calculus III
An overview of differential calculus in several variables and vector calculus. Functions of several variables; limits, continuity, differentiability, partial differentiation, applications including optimization and Lagrange multipliers. Vector calculus: vector functions, line integrals and surface integrals, Green’s theorem, Stokes’ theorem, and the Divergence theorem.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 267 or 277; and Mathematics 211 or 213.
Antirequisite(s):
Credit for Mathematics 367 and either 331 or 377 will not be allowed.
back to top
Mathematics 371       Combinatorics and Graph Theory
Counting techniques, generating functions, inclusion-exclusion, introduction to graph theory.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 271 or 273; and 3 units from Mathematics 249, 265 or 275.
Also known as:
(formerly Pure Mathematics 471)
back to top
Mathematics 375       Differential Equations for Engineers and Scientists
Definition, existence and uniqueness of solutions; first order and higher order equations and applications; Homogeneous systems; Laplace transform; partial differential equations of mathematical physics.
Course Hours:
3 units; (3-1.5T)
Prerequisite(s):
Mathematics 277 or both Mathematics 267 and 177.
Antirequisite(s):
Credit for Mathematics 375 and either 376 or Applied Mathematics 311 will not be allowed.
back to top
Mathematics 376       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:
3 units; (3-1T)
Prerequisite(s):
Mathematics 267 or 277.
Antirequisite(s):
Credit for Mathematics 376 and 375 or Applied Mathematics 307 will not be allowed.
Also known as:
(formerly Applied Mathematics 311)
back to top
Mathematics 377       Vector Calculus for Engineers and Scientists
Review of calculus of functions of several variables. Vector fields, line integrals, independence of path, Green’s theorem; surface integrals, divergence theorem, Stokes’ theorem; applications; curvilinear co-ordinates; partial differential equations in mathematical physics.
Course Hours:
3 units; (3-1.5T)
Prerequisite(s):
Mathematics 375.
Antirequisite(s):
Credit for Mathematics 377 and either 331 or 367 will not be allowed.
back to top
Mathematics 383       Introduction to Mathematical Finance
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:
3 units; (3-1T)
Prerequisite(s):
Statistics 321.
Also known as:
(formerly Applied Mathematics 481)
back to top
Mathematics 391       Numerical Analysis I
Interpolation and approximation, numerical integration and differentiation, numerical methods for the solution of non-linear equations, systems of linear equations and the eigenvalue problem, introduction to a scientific computing software.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 211 or 213; and Mathematics 267 or 277; and 3 units from Computer Science 217, 231, 235 or Data Science 211.
Antirequisite(s):
Credit for Mathematics 391 and Computer Science 491 will not be allowed.
Also known as:
(formerly Applied Mathematics 491)
back to top
Mathematics 401       Special Topics
Higher level topics which can be repeated for credit.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Consent of the Department.
MAY BE REPEATED FOR CREDIT
back to top
Mathematics 403       Topics in Mathematics for Economics
Techniques of integration. Multiple integrals. Analysis of functions. Continuity. Compact sets. Convex sets. Separating hyperplanes. Lower and upper hemi-continuous correspondences. Fixed point theorems, Optimal control.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 211 or 213; and 3 units from Mathematics 267 or 277 or 6 units from Economics 387 and 389.
back to top
Mathematics 413       Introduction to Partial Differential Equations
First order partial differential equations, Sturm-Liouville systems, Fourier series, Double Fourier series, Fourier integrals, Applications to boundary value problems in bounded and unbounded domains, Bessel function with applications.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 331; or 6 units from Mathematics 375 or 376 and Mathematics 367 or 377.
Also known as:
(formerly Applied Mathematics 413)
back to top
Mathematics 429       Cryptography: Design and Analysis of Cryptosystems
Review of basic algorithms and complexity. Designing and attacking public key cryptosystems based on number theory. Basic techniques for primality testing, factoring and extracting discrete logarithms. Elliptic curve cryptography. Additional topics may include knapsack systems, zero knowledge, attacks on hash functions, identity-based cryptography, and quantum cryptography.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 315; and Mathematics 318 or Computer Science 418.
Also known as:
(formerly Pure Mathematics 429)
back to top
Mathematics 431       Algebra II
An intermediate course in the theory of groups and fields. Group theory: group actions, Sylow theorems, solvable, nilpotent and p-groups, simplicity of alternating groups and PSL(n,q). Field theory: algebraic and transcendental extensions, separability and normality, Galois theory, insolvability of the general quintic equation, and computation of Galois groups over the rationals.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 315; and Mathematics 311 or 313.
Also known as:
(formerly Pure Mathematics 431)
back to top
Mathematics 433       Mathematical Methods in Physics
Fourier analysis. Laplace Transforms. Partial differential equations. Complex analysis. Residue integrals. Extensive physical applications.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 211 or 213; and Mathematics 375 or 376; and Mathematics 367 or 377.
Also known as:
(formerly Applied Mathematics 433)
back to top
Mathematics 445       Analysis II
An intermediate course in the theory of metric spaces and the continuous functions that act on them: metric spaces and normed vector spaces; complete metric spaces and the Baire category theorem; continuous functions on compact metric spaces and uniform convergence; the contraction mapping principle and applications; theorems of Stone-Weierstrass and Arzelà-Ascoli; differentiability on Euclidean spaces and the implicit function theorem.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 367 or 377; and 3 units from Mathematics 311, 313, 361 or 411; and Mathematics 335 or 355.
Antirequisite(s):
Credit for Mathematics 445 and 447 will not be allowed.               
back to top
Mathematics 447       Enriched Analysis II
An intermediate course in the theory of metric spaces and the continuous functions that act on them: metric spaces and normed vector spaces; complete metric spaces and the Baire category theorem; continuous functions on compact metric spaces and uniform convergence; the contraction mapping principle and applications; theorems of Stone-Weierstrass and Arzelà-Ascoli; differentiability on Euclidean spaces and the implicit function theorem. Mathematics 447 will contain more challenging and deeper assessment than Mathematics 445.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 367 or 377; and Mathematics 313 or "B+" or higher in Mathematics 311; and Mathematics 355 or "B+" or higher in Mathematics 335.
Antirequisite(s):
Credit for Mathematics 447 and either Mathematics 445 or Pure Mathematics 545 will not be allowed.
back to top
Mathematics 476       Differential Equations II
Existence and uniqueness theorems, comparison and oscillation theorems, Green's functions, Sturm-Liouville problems, systems of equations, phase portraits, stability.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
Mathematics 375 or 376; and 3 units from Mathematics 367, 377 or 331; and Mathematics 335 or 355.
Also known as:
(formerly Applied Mathematics 411)
back to top
Mathematics 493       Numerical Analysis II
Numerical solution of ordinary differential equations, single and multi-step methods, numerical solution of boundary value problems, numerical solution of partial differential equations, stability analysis.
Course Hours:
3 units; (3-1T)
Prerequisite(s):
3 units from Mathematics 331, 375 or 376; and Mathematics 375 or 413; and 3 units from Mathematics 391, Computer Science 491 or Geophysics 419.
Also known as:
(formerly Applied Mathematics 493)
back to top
Mathematics 501       Measure and Integration
Abstract measure theory, basic integration theorems, Fubini's theorem, Radon-Nikodym theorem, Lp Spaces, Riesz representation theorems.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 445 or 447; and 3 units of Mathematics in the Field of Mathematics at the 400 level or higher.
Antirequisite(s):
Credit for Mathematics 501 and any one of Mathematics 601, Pure Mathematics 501 or 601 will not be allowed.
back to top
Mathematics 502       Topics in Mathematics
Topics will be chosen according to the interests of instructors and students.
Course Hours:
3 units; (3-0)
Prerequisite(s):
6 units of Mathematics courses in the Field of Mathematics at the 400 level or above.
Also known as:
(formerly Pure Mathematics 503)
MAY BE REPEATED FOR CREDIT
back to top
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:
3 units; (3-0)
Prerequisite(s):
3 units from Mathematics 391, Computer Science 491 or Geophysics 419; and 6 units of Mathematics in the Field of Mathematics at the 400 level or above.
Also known as:
(formerly Applied Mathematics 503)
back to top
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:
3 units; (3-0)
Prerequisite(s):
Mathematics 445 or 447; and Mathematics 375 or 376.
Also known as:
(formerly Applied Mathematics 505)
back to top
Mathematics 511       Algebra III
A sophisticated introduction to modules over rings, especially commutative rings with identity. Major topics include: snake lemma; free modules; tensor product; hom-tensor duality; finitely presented modules; invariant factors; free resolutions; and the classification of finitely generated modules over principal ideal domains. Adjoint functors play a large role. The course includes applications to linear algebra, including rational forms.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 431; and 3 units from Mathematics 313, 361 or 411; and 3 units of Mathematics in the Field of Mathematics at the 400 level or higher.
Antirequisite(s):
Credit for Mathematics 511 and either 607 or Pure Mathematics 611 will not be allowed.
Also known as:
(formerly Pure Mathematics 511)
back to top
Mathematics 515       Foundations
Set theory, mathematical logic, and category theory. Topics covered will vary based on interests of students and instructor.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 431; and Mathematics 335 or 355.
Also known as:
(formerly Pure Mathematics 415)
back to top
Mathematics 516       Senior Project
A capstone course intended for Mathematics students in the final year of study (excluding those in the honours program or the statistics concentration). Students will investigate scientific or social issues by applying mathematical methods acquired in previous mathematics courses. A final project will be submitted at the end of the term and its contents summarized in a presentation.
Course Hours:
3 units; (1.5-3)
Prerequisite(s):
6 units in the Field of Mathematics at the 400 level or above.
Antirequisite(s):
Credit for Mathematics 516 and 518 will not be allowed.
back to top
Mathematics 518       Honours Thesis
A capstone course intended for Honours Mathematics students in their final year of study. Students will produce and present a substantial thesis under the supervision of a faculty member. The emphasis is on how to address theoretical or real world scientific or social issues by applying the various mathematical methods acquired in the earlier years in a unified and appropriate way.
Course Hours:
3 units; (1.5-3)
Prerequisite(s):
6 units in the Field of Mathematics at the 400 level or above.
Antirequisite(s):
Credit for Mathematics 518 and 516 will not be allowed.
Notes:
A grade of "B" or higher is required for the Honours program. Students are advised to consult with the Undergraduate Director for information and advice before registration into the course. Students earning an Honours degree in Mathematics along with a concentration in Statistics must complete both Mathematics 518 and Statistics 517.
back to top
Mathematics 521       Complex Analysis II
A rigorous study of function of a single complex variable. Holomorphic function, Cauchy integral formula and its applications. Conformal mappings. Fractional linear transformations. Argument principle. Schwarz lemma. Conformal self-maps of the unit disk.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 307; and Mathematics 445 or 447; and 3 units of Mathematics in the Field of Mathematics at the 400 level or above.
back to top
Mathematics 525       Introduction to Algebraic Topology
An introduction to the algebraic invariants that distinguish topological spaces. Specifically, the course focuses on the fundamental group and its applications, and homology. Students will be introduced to the basics of homological algebra.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 431; and Mathematics 445 or 447.   
back to top
Mathematics 527       Computational Number Theory
An investigation of major problems in computational number theory, with emphasis on practical techniques and their computational complexity. Topics include basic integer arithmetic algorithms, finite fields, primality proving, factoring methods, algorithms in algebraic number fields.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 327 and 431.
Antirequisite(s):
Credit for Mathematics 527 and either Pure Mathematics 527 or 643 will not be allowed.
Also known as:
(formerly Pure Mathematics 527)
back to top
Mathematics 545       Analysis III
Sequences and series of functions; Lebesgue integration on the line, Fourier series and the Fourier transform, pointwise convergence theorems, distributions and generalized functions.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 445 or 447; and 3 units of Mathematics in the Field of Mathematics at the 400 level or above.
Antirequisite(s):
Credit for Mathematics 545 and 603 will not be allowed.
back to top
Mathematics 581       Stochastic Calculus for Finance
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:
3 units; (3-0)
Prerequisite(s):
Mathematics 383; and 6 units of Mathematics in the Field of Mathematics at the 400 level or above.
Antirequisite(s):
Credit for Mathematics 581 and Applied Mathematics 681 will not be allowed.
Also known as:
(formerly Applied Mathematics 581)
back to top
Mathematics 583       Computational Finance
Review of financial asset price and option valuation models; model calibration; tree-based methods; finite-difference methods; Monte Carlo simulation; Fourier methods.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 381, 413 and 493.
Antirequisite(s):
Credit for Mathematics 583 and Applied Mathematics 683 will not be allowed.
Also known as:
(formerly Applied Mathematics 583)
back to top
Graduate Courses
Mathematics 600       Research Seminar
A professional skills course, focusing on the development of technical proficiencies that are essential to succeed 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:
1.5 units; (3S-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
MAY BE REPEATED FOR CREDIT
NOT INCLUDED IN GPA
back to top
Mathematics 601       Measure and Integration
Abstract measure theory, basic integration theorems, Fubini's theorem, Radon-Nikodym theorem, Lp spaces, Riesz representation theorem.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 601 and either Mathematics 501 or Pure Mathematics 501 will not be allowed.
back to top
Mathematics 603       Analysis III
Sequences and series of functions; Lebesgue integration on the line, Fourier series and the Fourier transform, pointwise convergence theorems, distributions and generalized functions.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.  
Antirequisite(s):
Credit for Mathematics 603 and either Mathematics 545 or Pure Mathematics 545 will not be allowed.
back to top
Mathematics 605       Differential Equations III
Systems of ordinary differential equations.  Existence and uniqueness. Introduction to partial differential equations.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 605 and Applied Mathematics 605 will not be allowed.
back to top
Mathematics 607       Algebra III
A sophisticated introduction to modules over rings, especially commutative rings with identity. Major topics include: snake lemma; free modules; tensor product; hom-tensor duality; finitely presented modules; invariant factors; free resolutions; and the classification of finitely generated modules over principal ideal domains. Adjoint functors play a large role. The course includes applications to linear algebra, including rational canonical form and Jordan canonical form.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 607 and any of Pure Mathematics 511, 607 or 611 will not be allowed.    
back to top
Mathematics 617       Functional Analysis
Introduction to Hilbert and Banach spaces, linear operators, weak topologies, and the operator spectrum.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 617 and Applied Mathematics 617 will not be allowed.      
back to top
Mathematics 621       Complex Analysis
A rigorous study of function of a single complex variable. Holomorphic function, Cauchy integral formula and its applications. Conformal mappings. Fractional linear transformations. Argument principle. Schwarz lemma. Conformal self-maps of the unit disk.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 621 and 521 will not be allowed.
back to top
Mathematics 625       Introduction to Algebraic Topology
Introduction to the algebraic invariants that distinguish topological spaces. Focuses on the fundamental group and its applications, and homology. Introduction to the basics of homological algebra.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 625 and either Mathematics 525 or Pure Mathematics 607 will not be allowed.        
back to top
Mathematics 627       Algebraic Geometry
The objective of this course is to provide an introduction to modern algebraic geometry sufficient to allow students to read research papers in their fields which use the language of schemes. Topics will include Spectra of rings; the Zariski topology; affine schemes; sheaves; ringed spaces; schemes; morphisms of finite type; arithmetic schemes; varieties; projective varieties; finite morphisms, unramified morphisms; etale morphisms.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
back to top
Mathematics 631       Discrete Mathematics

Discrete Geometry: Euclidean, spherical and hyperbolic n-spaces, trigonometry, isometries, convex sets, convex polytopes, (mixed) volume(s), classical discrete groups, tilings, isoperimetric inequalities, packings, coverings. Graph Theory: connectivity; trees; Euler trails and tours; Hamilton cycles and paths; matchings; edge colourings; vertex colourings; homomorphisms; plane and planar graphs; extremal graph theory and Ramsey theory.

631.01. Discrete Geometry

631.03. Graph Theory


Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
back to top
Mathematics 635       Geometry of Numbers
The interplay of the group-theoretic notion of lattice and the geometric concept of convex set, the lattices representing periodicity, the convex sets geometry. Topics include convex bodies and lattice points, the critical determinant, the covering constant and the inhomogeneous determinant of a set, Star bodies, methods related to the above, and homogeneous and inhomogeneous forms.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
back to top
Mathematics 637       Infinite Combinatorics
An excursion into the infinite world, from Ramsey Theory on the natural numbers, to applications in Number Theory and Banach Spaces, introduction to tools in Model Theory and Logic, fascinating homogeneous structures such as the rationals and the Rado graph, and possibly further explorations into the larger infinite world.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
back to top
Mathematics 641       Number Theory

Algebraic Number Theory: an introduction to number fields, rings of integers, ideals, unique factorization, the different and the discriminant. The main objective to the course will be to prove the finiteness of the class number and Dirichlet's Unit Theorem.
Analytic Number Theory: students will learn tools to aid in the study of the average behaviour of arithmetic functions, including the use of zeta functions, to prove results about the distribution of prime numbers.

641.01. Algebraic Number Theory

641.03. Analytic Number Theory


Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
back to top
Mathematics 643       Computational Number Theory
An investigation of major problems in computational number theory, with emphasis on practical techniques and their computational complexity. Topics include basic integer arithmetic algorithms, finite fields, primality proving, factoring methods, algorithms in algebraic number fields.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 643 and any of Mathematics 527, Pure Mathematics 527, or Pure Mathematics 627 will not be allowed.    
back to top
Mathematics 647       Modular Forms

Modular forms and automorphic representations and their L-functions. Modularity Theorem from two perspectives.

Classical Perspective on Modular Forms: introduction to modular curves as moduli spaces for elliptic curves and as differential forms on modular curves. A study of L-functions attached to modular forms and the modularity theorem.

An Introduction to Automorphic Representations: introduction to the Langlands Programme. A study of partial L-functions attached to automorphic representations and known instances of the Langlands Correspondence.

647.01. Classical Perspective on Modular Forms

647.03. An Introduction to Automorphic Representations


Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
back to top
Mathematics 651       Topics in Applied Mathematics
Topics will be chosen according to the interest of the instructors and students.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Also known as:
(formerly Applied Mathematics 603)
MAY BE REPEATED FOR CREDIT
back to top
Mathematics 653       Topics in Pure Mathematics
Topics will be chosen according to the interest of the instructors and students.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Also known as:
(formerly Pure Mathematics 603)
MAY BE REPEATED FOR CREDIT
back to top
Mathematics 661       Scientific Modelling and Computation I

The Convex Optimization: an introduction to modern convex optimization, including basics of convex analysis and duality, linear conic programming, robust optimization, and applications.

Scientific Computation: an introduction to both the methodological and the implementation components underlying the modern scientific computations with the natural emphasis on linear algebra, including modern computing architecture and its implications for the numerical algorithms.

Numerical Differential Equations: fundamentals of solving DEs numerically addressing the existence, stability and efficiency of such methods.

661.01. Convex Optimization

661.03. Scientific Computation

661.05. Numerical Differential Equations


Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
back to top
Mathematics 663       Applied Analysis

Interior Point Methods: exposes students to the modern IPM theory with some applications, to the extent that at the end of the course a student should be able to implement a basic IPM algorithm.

Theoretical Numerical Analysis: provides the theoretical underpinnings for the analysis of modern numerical methods, covering topics such as linear operators on normed spaces, approximation theory, nonlinear equations in Banach spaces, Fourier analysis, Sobolev spaces and weak formulations of elliptic boundary value problems, with applications to finite difference, finite element and wavelet methods.

Differential Equations: essential ideas relating to the analysis of differential equations from a functional analysis point of view. General topics include Hilbert spaces and the Lax-Milgram’s theorem, variational formulation of boundary value problems, finite element methods, Sobolev spaces, distributions, and pseudo-differential operators.

663.01. Interior Point Methods

663.03. Theoretical Numerical Analysis

663.05. Differential Equations


Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
back to top
Mathematics 667       Introduction to Quantum Information

Focus on the mathematical treatment of a broad range of topics in quantum Shannon theory. Topics include quantum states, quantum channels, quantum measurements, completely positive maps, Neumarkís theorem, Stinespring dilation theorem, Choi-Jamiolkowski isomorphism, the theory of majorization and entanglement, the Peres-Horodecki criterion for separability, Shannon’s noiseless and noisy channel coding theorems, Lieb’s theorem and the strong subadditivity of the von Neumann entropy, Schumacher’s quantum noiseless channel coding theorem, and the Holevo-Schumacher-Westmoreland theorem.


Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
back to top
Mathematics 669       Scientific Modelling and Computation II

Wavelet Analysis: covers the design and implementation of wavelet methods for modern signal processing, particularly for one- and two-dimensional signals (audio and images).

Mathematical Biology: introduction to discrete models of mathematical biology, including difference equations, models of population dynamics and the like. Topics include stability of models describe by difference equations, continuous spatially homogeneous processes and spatially distributed models.

669.01. Wavelet Analysis

669.03. Mathematical Biology


Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
back to top
Mathematics 681       Stochastic Calculus for Finance
Martingales in discrete and continuous time, risk-neutral valuations, discrete- and continuous-time (B,S)-security markets, the Cox-Ross-Rubinstein formula, Wiener and Poisson processes, Itô’s formula, stochastic differential equations, Girsanov’s theorem, the 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:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 681 and any one of Mathematics 581, Applied Mathematics 681, or Applied Mathematics 581 will not be allowed.
Also known as:
(formerly Applied Mathematics 681)
back to top
Mathematics 683       Computational Finance
Basic computational techniques required for expertise quantitative finance. Topics include basic econometric techniques (model calibration), tree-based methods, finite-difference methods, Fourier methods, Monte Carlo simulation and quasi-Monte Carlo methods.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 683 and any one of Mathematics 583, Applied Mathematics 683, or Applied Mathematics 583 will not be allowed.    
Also known as:
(formerly Applied Mathematics 683)
back to top
Mathematics 685       Stochastic Processes
Stochastic processes are fundamental to the study of mathematical finance, but are also of vital importance in many other areas, from neuroscience to electrical engineering. Topics to be covered: Elements of stochastic processes, Markov chains and processes, Renewal processes, Martingales (discrete and continuous times), Brownian motion, Branching processes, Stationary processes, Diffusion processes, The Feynman-Kac formula, Kolmogorov backward/forward equations, Dynkin’s formula.
Course Hours:
3 units; (3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 685 and Statistics 761  will not be allowed.
back to top
Mathematics 691       Advanced Mathematical Finance I

Topics include specific areas of mathematical finance and build on Mathematics 681.

Lévy Processes (LP): fundamental concepts associated with LP such as infinite divisibility, the Lévy-Khintchine formula, the Lévy-Itô decomposition, subordinators, LP as time-changed Brownian motions, and also dealing with semi-groups and generators of LP, the Itô formula for LP, the Girsanov theorem, stochastic differential equations driven by LP, the Feynman-Kac formula, applications of LP and numerical simulation of LP.

Credit Risk: corporate bond markets, modelling the bankruptcy risk of a firm, and understanding how corporate bonds are priced.

Stochastic Optimal Control and Applications in Finance: An introduction to the theory of stochastic optimal control and applications in finance and economics. Dynamic programming approach to optimal controls, solutions to several classes of typical optimal control problems, and application of the general theory to some classical models in finance and economics.

691.01. Lévy Processes in Finance

691.03. Credit Risk

691.05 Stochastic Optimal Control and Applications in Finance


Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 681 and admission to a graduate program in Mathematics and Statistics.
back to top
Mathematics 693       Advanced Mathematical Finance II

Topics include specific areas of mathematical finance and build on Mathematics 681 and 683.

Monte Carlo Methods for Quantitative Finance: random number generation, simulation of stochastic differential equations, option valuation, variance reduction techniques, quasi-Monte Carlo methods, computing ‘greeks', valuation of path-dependent and early-exercise options; applications to risk management; Markov Chain Monte Carlo methods.

Energy, Commodity and Environmental Finance: energy and commodity markets; spot, futures, forwards and swap contracts; the theory of storage; stochastic models for energy prices; model calibration; emissions market modelling; weather derivatives; energy risk management; energy option valuation.

Advanced Topics in Mathematical Finance: An introduction to some of the main ideas in mathematical and computational finance through a guided reading of some seminal papers from the last 100 years, starting with Bachelier's 1900 thesis, and including papers by Samuelson, Markowitz, Black & Scholes, Merton, Hull & White, Schwartz, Glasserman, and others.

693.01 Monte Carlo Methods for Quantitative Finance

693.03 Energy, Commodity and Environmental Finance

693.05 Advanced Topics in Mathematical Finance


Course Hours:
3 units; (3-0)
Prerequisite(s):
Mathematics 681 and 683 and admission to a graduate program in Mathematics and Statistics.
back to top