
Instruction offered by members of the Department of Mathematics and Statistics in the Faculty of Science.
Department Head  T. Bisztriczky
Note: For listings of related courses, see Actuarial Science, Applied Mathematics, Mathematics, and Statistics.
Note: The following courses, although offered on a regular basis, are not offered every year: Pure Mathematics 371, 415, 419, 423, 425, 427, 501, 505, 511, 521, and 545. Check with the divisional office to plan for the upcoming cycle of offered courses.


Pure Mathematics
315

Abstract Algebra


Integers: division algorithm, prime factorization. Groups: permutations, Lagrange's theorem. Rings: congruences, polynomials.
Course Hours:
H(31T)
Prerequisite(s):
One of Mathematics 211 or 213 or 221.
Notes:
Mathematics 271 or 273 is strongly recommended as preparation for this course.

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Pure Mathematics
319

Transformation Geometry


Geometric transformations in the Euclidean plane. Frieze patterns. Wallpaper patterns. Tessellations.
Course Hours:
H(31T)
Prerequisite(s):
One of Mathematics 211 or 213 or 221 and one other 200level course labelled Applied Mathematics, Mathematics or Pure Mathematics, not including Mathematics 205.
Notes:
Mathematics 271 or 273 is strongly recommended as preparation.

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Pure Mathematics
415

Set Theory


Axioms for set theory, the axiom of choice and equivalents, cardinal and ordinal arithmetics, induction and recursion on wellfounded sets, infinitary combinatorics, applications.
Course Hours:
H(30)
Prerequisite(s):
Mathematics 271 or 273 or 311 or 353 or 381 or Pure Mathematics 315, or consent of the Division.

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Pure Mathematics
418

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:
H(32T)
Prerequisite(s):
One of Mathematics 271, 273, or Pure Mathematics 315.
Antirequisite(s):
Credit for both Pure Mathematics 418 and any of Pure Mathematics 329, Computer Science 418, 429, or 557 will not be allowed.
Notes:
Lectures may run concurrently with Computer Science 418.

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Pure Mathematics
419

Information Theory and Error Control Codes


Information sources, entropy, channel capacity, development of Shannon's theorems, development of a variety of codes including error correcting and detecting codes.
Course Hours:
H(30)
Prerequisite(s):
Mathematics 311, and Mathematics 321 or any Statistics course, or consent of the Division.
Also known as:
(Statistics 419)

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Pure Mathematics
421

Introduction to Complex Analysis


Complex numbers. Analytic functions. Complex integration and Cauchy's theorem. Maximum modulus theorem. Power series. Residue theorem.
Course Hours:
H(31T)
Prerequisite(s):
Both Mathematics 349 and 353; or both Mathematics 283 and 381.
Antirequisite(s):
Not open to students with credit in Pure Mathematics 521.

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Pure Mathematics
423

Differential Geometry


Fundamentals of the Gaussian theory of surfaces. Introduction to Riemannian geometry. Some topological aspects of surfaces.
Course Hours:
H(30)
Prerequisite(s):
Mathematics 353 or 381, or consent of the Division.

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Pure Mathematics
425

Geometry


Introduction to some of the following geometries: Discrete geometry, finite geometry, hyperbolic geometry, projective geometry, synthetic geometry.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 315 or consent of the Division.

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Pure Mathematics
427

Number Theory


Induction principles. Division Algorithm. Prime factorization theorem. Congruences. Arithmetic functions. Diophantine equations. Continued fractions.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 315 or consent of the Division.

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Pure 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:
H(30)
Prerequisite(s):
Pure Mathematics 329 or 418.

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Pure Mathematics
431

Groups, Rings and Fields


Factor groups and rings, polynomial rings, field extensions, finite fields, Sylow theorems, solvable groups. Additional topics.
Course Hours:
H(30)
Prerequisite(s):
Mathematics 311 and Pure Mathematics 315 or consent of the Division.

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Pure Mathematics
435

Analysis I


Logic, sets, functions; real numbers, completeness, sequences; continuity and compactness; differentiation; integration; sequence and series of functions.
Course Hours:
H(31T)
Prerequisite(s):
Mathematics 253 or 263 or 283 or Applied Mathematics 219, or consent of the Division.
Antirequisite(s):
Credit for both Pure Mathematics 435 and 455 will not be allowed.

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Pure Mathematics
445

Analysis II


Euclidean space, basic topology; differentiation of transformations, Implicit Function Theorem; multiple integration, integrals over curves and surfaces; differential forms, Stokes' Theorem.
Course Hours:
H(30)
Prerequisite(s):
Mathematics 353 or 381; and Mathematics 311; and Pure Mathematics 435 or 455, or consent of the Division.
Antirequisite(s):
Not open to students with credit in Pure Mathematics 545.

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Pure Mathematics
455

Honours Real Analysis I


Real and complex numbers, topology of metric spaces, sequences and series, continuity, differentiation, RiemannStieltjes integration. Rigorous approach throughout.
Course Hours:
H(31T)
Prerequisite(s):
Mathematics 283 or 263; or a grade of B+ or better in Mathematics 253 or Applied Mathematics 219.
Antirequisite(s):
Credit for both Pure Mathematics 435 and 455 will not be allowed.

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Pure Mathematics
501

Integration Theory


Abstract measure theory, basic integration theorems, Fubini's theorem, RadonNikodym theorem, further topics.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 545 or consent of the Division.
Antirequisite(s):
Credit for both Pure Mathematics 501 and 601 will not be allowed.

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Pure Mathematics
503

Topics in Pure Mathematics


This course is offered under various subtitles. Consult Department for details.
Course Hours:
H(30)
Prerequisite(s):
Consent of the Division.
MAY BE REPEATED FOR CREDIT

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Pure Mathematics
505

Topology I


Metric spaces. Introduction to general topology.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 435 or 455 or consent of the Division.

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Pure Mathematics
511

Rings and Modules


Ring theory, and structure of modules. Application to Abelian groups and linear algebra. Additional topics.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 431 or Mathematics 411, or consent of the Division.
Antirequisite(s):
Credit for both Pure Mathematics 511 and 611 will not be allowed.

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Pure Mathematics
521

Complex Analysis


A rigorous study of functions of a single complex variable. Consequences of differentiability. Proof of the Cauchy integral theorem, applications.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 435 or 455 or consent of the Division.

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Pure 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:
H(30)
Prerequisite(s):
Pure Mathematics 427 or 429.

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Pure Mathematics
529

Advanced Cryptography and Cryptanalysis


Cryptography based on quadratic residuacity. Advanced techniques for factoring and extracting discrete logarithms. Hyperelliptic curve cryptography. Pairings and their applications to cryptography. Code based and lattice based cryptography. Additional topics may include provable security, secret sharing, more postquantum cryptography, and new developments in cryptography.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 429.

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Pure Mathematics
545

Honours Real Analysis II


Sequences and series of functions; theory of Fourier analysis, functions of several variables: Inverse and Implicit Functions and Rank Theorems, integration of differential forms, Stokes' Theorem, Measure and Lebesgue integration.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 455; or a grade of B+ or better in Pure Mathematics 435.

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Graduate Courses
Note: Students are urged to make their decisions as early as possible as to which graduate courses they wish to take, since not all these courses will be offered in any given year.

Pure Mathematics
601

Integration Theory


Abstract measure theory, basic integration theorems, Fubini's theorem, RadonNikodym theorem, further topics.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 545 or consent of the Division.
Antirequisite(s):
Credit for both Pure Mathematics 601 and 501 will not be allowed.
Notes:
Lectures may run concurrently with Pure Mathematics 501.

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Pure Mathematics
603

Conference Course in Pure Mathematics


This course is offered under various subtitles. Consult Department for details.
Course Hours:
H(30)
MAY BE REPEATED FOR CREDIT

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Pure Mathematics
607

Topology II


General topology, elementary combinatorial topology.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 505 or consent of the Division.

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Pure Mathematics
613

Introduction to Field Theory


Field theory, Galois theory.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 431 or consent of the Division.

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Pure Mathematics
615

Topics in Logic


Course Hours:
H(30)
MAY BE REPEATED FOR CREDIT

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Pure Mathematics
621

Research Seminar


Reports on studies of the literature or of current research.
Course Hours:
Q(2S0)
Notes:
All graduate students in Mathematics and Statistics are required to participate in one of Applied Mathematics 621, Pure Mathematics 621, Statistics 621 each semester.
MAY BE REPEATED FOR CREDIT
NOT INCLUDED IN GPA

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Pure Mathematics
627

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:
H(30)
Prerequisite(s):
Pure Mathematics 427 or 429, or consent of the Division.

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Pure Mathematics
629

Elliptic Curves and Cryptography


An introduction to elliptic curves over the rationals and finite fields. The focus is on both theoretical and computational aspects; subjects covered will include the study of endomorphism rings. Weil pairing, torsion points, group structure, and efficient implementation of point addition. Applications to cryptography will be discussed, including elliptic curvebased DiffieHellman key exchange, El Gamal encryption, and digital signatures, as well as the associated computational problems on which their security is based.
Course Hours:
H(30)
Prerequisite(s):
Pure Mathematics 315 or consent of the Division.
Also known as:
(Computer Science 629)

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Pure Mathematics
631

Algebraic Topology I


Elements of category theory and homological algebra. Various examples of homology and cohomology theories. EilenbergSteenrod axioms. Geometrical applications.
Course Hours:
H(30)

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Pure Mathematics
633

Algebraic Topology II


Cohomology operations, CWcomplexes, introduction to homotopy theory.
Course Hours:
H(30)

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Pure Mathematics
649

Modern Cryptography and Cryptoanalysis


Cryptography based on quadratic residuacity. Advanced techniques for factoring and extracting discrete logarithms. Hyperelliptic curve cryptography. Pairings and their applications to cryptography. Code based and lattice based cryptography. Additional topics may include provable security, secret sharing, more postquantum cryptography, and new developments in cryptography.
Course Hours:
H30
Prerequisite(s):
Pure Mathematics 429 or consent of Division.

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Pure Mathematics
669

Cryptography


An overview of basic techniques in modern cryptography, with emphasis on fitforapplication primitives and protocols. Topics include symmetric and publickey cryptosystems; digital signatures; elliptic curve cryptography; key management; attack models and welldefined notions of security.
Course Hours:
H(30)
Prerequisite(s):
Consent of the Division.
Notes:
Computer Science 413 and Mathematics 321 are recommended as preparation for this course. Students should not have taken any previous courses in cryptography.
Also known as:
(Computer Science 669)

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Pure Mathematics
685

Topics in Algebra


The following topics are available as decimalized courses: Algebraic Number Theory, Algebraic KTheory, Algebraic Geometry, Representation Theory, Abelian Group Theory, Brauer Group Theory, Homological Algebra, Ring Theory, Associative Algebras, Commutative Algebra, Universal Algebra.
Course Hours:
H(30)
MAY BE REPEATED FOR CREDIT

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Pure Mathematics
727

Advanced Topics in Computational Number Theory


Depending on student demand and interests this could cover topics concerning efficient computation in various number theoretic structures such as number rings, finite fields, algebraic number fields and algebraic curves.
Course Hours:
H(30)

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Pure Mathematics
729

Advanced Topics in Cryptography


Depending on student demand and interests this could cover topics in cryptography developed in diverse mathematical structures such as: finite fields, lattices, algebraic number fields and algebraic curves.
Course Hours:
H(30)

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