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

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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; (31)
Prerequisite(s):
One of Mathematics 301, 302, 212 or Mathematics 2 (offered by Continuing Education).

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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; (42)
Prerequisite(s):
Admission to the Energy Engineering Program.

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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; (31)
Prerequisite(s):
One of Mathematics 301, 212 or Mathematics 2 (offered by Continuing Education).
Antirequisite(s):
Credit for Mathematics 211 and 213 will not be allowed.

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Mathematics
212

Mathematical Ways of Thinking


A mathematical exploration course focused on investigating fundamental mathematical concepts and their applications. The course is designed to prepare students for further study in mathematics and to meet the mathematical requirements for their program of study. Learners will build conceptual understanding together with technical skill in applying the ideas covered in the course to solve problems. This course pairs elements from high school mathematics and introductory university mathematics with Indigenous Ways of Knowing.
Course Hours:
3 units; (31)
Prerequisite(s):
Mathematics 201 or 302 and admission to an Indigenous Pathway program.
Notes:
Not included in the Field of Mathematics.

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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; (31)
Prerequisite(s):
A grade of 90 per cent or higher in both Mathematics 301 and Mathematics 31 or consent of the Department.
Antirequisite(s):
Credit for Mathematics 213 and 211 will not be allowed.

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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; (41)
Prerequisite(s):
One of Mathematics 301, 212 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.

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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; (31)
Prerequisite(s):
One of Mathematics 301, 212 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.

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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; (31)
Prerequisite(s):
3 units from Mathematics 249, 265 or 275.
Antirequisite(s):
Credit for Mathematics 267 and 277 will not be allowed.

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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; (31T1)
Prerequisite(s):
Mathematics 211 or 213.

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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; (31T1)
Prerequisite(s):
A grade of 90 per cent or higher in both Mathematics 301 and 31 or consent of the Department.

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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; (31T1.5)
Prerequisite(s):
One of Mathematics 301, 212 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.

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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; (31T1.5)
Prerequisite(s):
Mathematics 275; and Mathematics 211 or 213.
Antirequisite(s):
Credit for Mathematics 277 and 267 will not be allowed.

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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; (31T)
Prerequisite(s):
Mathematics 211 or 213; and Mathematics 271 or 273.
Notes:
This course will be cotaught by scholars from the Faculty of Science and Werklund School of Education.
Also known as:
(Education 305)

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Mathematics
307

Complex Analysis I


An initial treatment of complex analytic functions in a single variable. Topics include differentiation, CauchyRiemann 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; (31T)
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.

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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 ranknullity theorem; the GramSchmidt algorithm and orthogonal diagonalization; singular value decomposition and other applications.
Course Hours:
3 units; (31T)
Prerequisite(s):
Mathematics 211 or 213.
Antirequisite(s):
Credit for Mathematics 311 and 313 will not be allowed.

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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; (31T)
Prerequisite(s):
Mathematics 213.
Antirequisite(s):
Credit for Mathematics 311 and 313 will not be allowed.

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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; (31T)
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)

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Mathematics
318

Introduction to Cryptography


The basics of cryptography, with emphasis on attaining welldefined and practical notions of security. Symmetric and publickey cryptosystems; oneway 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 prooforiented homework problems; additional application programming exercises will be available for extra credit.
Course Hours:
3 units; (30)
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)

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Mathematics
319

Transformation Geometry


Geometric transformations in the Euclidean plane: Symmetry, Frieze, and Wallpaper groups.
Course Hours:
3 units; (31T)
Prerequisite(s):
Mathematics 211 or 213; and Mathematics 271 or 273.
Also known as:
(formerly Pure Mathematics 319)

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Mathematics
322

Curves and Surfaces


The fundamentals of 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 GaussBonnet theorem.
Course Hours:
3 units; (30)
Prerequisite(s):
Mathematics 271 or 273; and Mathematics 367; and Mathematics 375 or 376.
Also known as:
(formerly Pure Mathematics 423)

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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; (31T)
Prerequisite(s):
Mathematics 311 or 313; and Mathematics 367.
Also known as:
(formerly Applied Mathematics 425)

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Mathematics
327

Number Theory


Divisibility and the Euclidean algorithm, modular arithmetic and congruences, quadratic reciprocity, arithmetic functions, distribution of primes.
Course Hours:
3 units; (30)
Prerequisite(s):
Mathematics 271 or 273.
Also known as:
(formerly Pure Mathematics 427)

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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; (42T)
Prerequisite(s):
Mathematics 209 and admission to the Energy Engineering program.

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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; (31T)
Prerequisite(s):
Mathematics 267 or 277; and Mathematics 271 or 273.
Antirequisite(s):
Credit for Mathematics 335 and 355 will not be allowed.

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Mathematics
361

Linear Methods III


Canonical forms. Inner product spaces, invariant subspaces and spectral theory. Quadratic forms.
Course Hours:
3 units; (31T)
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)

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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; (31T)
Prerequisite(s):
Mathematics 267 or 277; and Mathematics 211 or 213.
Antirequisite(s):
Credit for Mathematics 367 and 377 will not be allowed.

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Mathematics
371

Combinatorics and Graph Theory


Counting techniques, generating functions, inclusionexclusion, introduction to graph theory.
Course Hours:
3 units; (30)
Prerequisite(s):
Mathematics 271 or 273; and 3 units from Mathematics 249, 265 or 275.
Also known as:
(formerly Pure Mathematics 471)

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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; (31.5T)
Prerequisite(s):
Mathematics 277; or both 267 and 177.
Antirequisite(s):
Credit for Mathematics 375 and 376 will not be allowed.

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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; (31T)
Prerequisite(s):
Mathematics 211; and 267 or 277.
Antirequisite(s):
Credit for Mathematics 376 and 375 will not be allowed.
Also known as:
(formerly Applied Mathematics 311)

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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; (31T)
Prerequisite(s):
Statistics 321.
Also known as:
(formerly Applied Mathematics 481)

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Mathematics
401

Special Topics


Higher level topics which can be repeated for credit.
Course Hours:
3 units; (30)
Prerequisite(s):
Consent of the Department.
MAY BE REPEATED FOR CREDIT

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Mathematics
413

Introduction to Partial Differential Equations


First order partial differential equations, SturmLiouville 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; (31T)
Prerequisite(s):
Mathematics 375 or 376; and Mathematics 367.
Also known as:
(formerly Applied Mathematics 413)

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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, identitybased cryptography, and quantum cryptography.
Course Hours:
3 units; (30)
Prerequisite(s):
Mathematics 315; and Mathematics 318 or Computer Science 418.
Also known as:
(formerly Pure Mathematics 429)

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Mathematics
431

Algebra II


An intermediate course in the theory of groups and fields. Group theory: group actions, Sylow theorems, solvable, nilpotent and pgroups, 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; (30)
Prerequisite(s):
Mathematics 315; and Mathematics 311 or 313.
Also known as:
(formerly Pure Mathematics 431)

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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 StoneWeierstrass and ArzelàAscoli; differentiability on Euclidean spaces and the implicit function theorem.
Course Hours:
3 units; (30)
Prerequisite(s):
Mathematics 367; and Mathematics 311 or 313; and Mathematics 335 or 355.
Antirequisite(s):
Credit for Mathematics 445 and 447 will not be allowed.

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Mathematics
476

Differential Equations II


Existence and uniqueness theorems, comparison and oscillation theorems, Green's functions, SturmLiouville problems, systems of equations, phase portraits, stability.
Course Hours:
3 units; (31T)
Prerequisite(s):
Mathematics 375 or 376; and Mathematics 367; and Mathematics 335 or 355.
Also known as:
(formerly Applied Mathematics 411)

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Mathematics
493

Numerical Analysis II


Numerical solution of ordinary differential equations, single and multistep methods, numerical solution of boundary value problems, numerical solution of partial differential equations, stability analysis.
Course Hours:
3 units; (31T)
Prerequisite(s):
Mathematics 375 or 376; and Mathematics 413; and 3 units from Mathematics 391, Computer Science 491 or Geophysics 419.
Also known as:
(formerly Applied Mathematics 493)

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Mathematics
501

Measure and Integration


Abstract measure theory, basic integration theorems, Fubini's theorem, RadonNikodym theorem, Lp Spaces, Riesz representation theorems.
Course Hours:
3 units; (30)
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.

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Mathematics
502

Topics in Mathematics


Study in advanced topics that may include reading projects, seminars, lectures, term papers and other aspects of training in theory or research methodology.
Course Hours:
3 units; (30)
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

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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; (30)
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)

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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; homtensor 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; (30)
Prerequisite(s):
Mathematics 431; and Mathematics 313 or 361; and 3 units of Mathematics in the Field of Mathematics at the 400 level or higher.
Antirequisite(s):
Credit for Mathematics 511 and 607 will not be allowed.
Also known as:
(formerly Pure Mathematics 511)

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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; (30)
Prerequisite(s):
Mathematics 431; and Mathematics 335 or 355.
Also known as:
(formerly Pure Mathematics 415)

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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; (1.53)
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.

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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; (1.53)
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.

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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 selfmaps of the unit disk.
Course Hours:
3 units; (30)
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.

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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; (30)
Prerequisite(s):
Mathematics 431; and Mathematics 445 or 447.

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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; (30)
Prerequisite(s):
Mathematics 327 and 431.
Antirequisite(s):
Credit for Mathematics 527 and 643 will not be allowed.
Also known as:
(formerly Pure Mathematics 527)

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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; (30)
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.

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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:
3 units; (30)
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 681 will not be allowed.
Also known as:
(formerly Applied Mathematics 581)

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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:
3 units; (30)
Prerequisite(s):
Mathematics 383, 413 and 493.
Antirequisite(s):
Credit for Mathematics 583 and 683 will not be allowed.
Also known as:
(formerly Applied Mathematics 583)

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Mathematics
585

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 FeynmanKac formula, Kolmogorov backward/forward equations, Dynkin’s formula.
Course Hours:
3 units; (30)
Prerequisite(s):
Statistics 323.
Antirequisite(s):
Credit for Mathematics 585 and Statistics 407 will not be allowed.

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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; (3S0)
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

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Mathematics
601

Measure and Integration


Abstract measure theory, basic integration theorems, Fubini's theorem, RadonNikodym theorem, Lp spaces, Riesz representation theorem.
Course Hours:
3 units; (30)
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.

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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; (30)
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.

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Mathematics
605

Differential Equations III


Systems of ordinary differential equations. Existence and uniqueness. Introduction to partial differential equations.
Course Hours:
3 units; (30)
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.

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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; homtensor 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; (30)
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 Mathematics 511, Pure Mathematics 511, 607 or 611 will not be allowed.

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Mathematics
617

Functional Analysis


Introduction to Hilbert and Banach spaces, linear operators, weak topologies, and the operator spectrum.
Course Hours:
3 units; (30)
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.

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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 selfmaps of the unit disk.
Course Hours:
3 units; (30)
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.

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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; (30)
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.

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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; (30)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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Mathematics
631

Discrete Mathematics


Discrete Geometry: Euclidean, spherical and hyperbolic nspaces, 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; (30)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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Mathematics
635

Geometry of Numbers


The interplay of the grouptheoretic 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; (30)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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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; (30)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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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; (30)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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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; (30)
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.

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Mathematics
647

Modular Forms


Modular forms and automorphic representations and their Lfunctions. 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 Lfunctions attached to modular forms and the modularity theorem.
An Introduction to Automorphic Representations: introduction to the Langlands Programme. A study of partial Lfunctions 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; (30)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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Mathematics
651

Topics in Applied Mathematics


Topics will be chosen according to the interest of the instructors and students.
Course Hours:
3 units; (30)
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

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Mathematics
653

Topics in Pure Mathematics


Topics will be chosen according to the interest of the instructors and students.
Course Hours:
3 units; (30)
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

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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; (30)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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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 LaxMilgram’s theorem, variational formulation of boundary value problems, finite element methods, Sobolev spaces, distributions, and pseudodifferential operators.
663.01. Interior Point Methods
663.03. Theoretical Numerical Analysis
663.05. Differential Equations
Course Hours:
3 units; (30)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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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, ChoiJamiolkowski isomorphism, the theory of majorization and entanglement, the PeresHorodecki 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 HolevoSchumacherWestmoreland theorem.
Course Hours:
3 units; (30)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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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 twodimensional 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; (30)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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Mathematics
681

Stochastic Calculus for Finance


Martingales in discrete and continuous time, riskneutral valuations, discrete and continuoustime (B,S)security markets, the CoxRossRubinstein formula, Wiener and Poisson processes, Itô’s formula, stochastic differential equations, Girsanov’s theorem, the 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:
3 units; (30)
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)

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Mathematics
683

Computational Finance


Basic computational techniques required for expertise quantitative finance. Topics include basic econometric techniques (model calibration), treebased methods, finitedifference methods, Fourier methods, Monte Carlo simulation and quasiMonte Carlo methods.
Course Hours:
3 units; (30)
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)

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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 FeynmanKac formula, Kolmogorov backward/forward equations, Dynkin’s formula.
Course Hours:
3 units; (30)
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.

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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 LévyKhintchine formula, the LévyItô decomposition, subordinators, LP as timechanged Brownian motions, and also dealing with semigroups and generators of LP, the Itô formula for LP, the Girsanov theorem, stochastic differential equations driven by LP, the FeynmanKac 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; (30)
Prerequisite(s):
Mathematics 681 and admission to a graduate program in Mathematics and Statistics.

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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, quasiMonte Carlo methods, computing ‘greeks', valuation of pathdependent and earlyexercise 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; (30)
Prerequisite(s):
Mathematics 681 and 683 and admission to a graduate program in Mathematics and Statistics.

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