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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.
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Pure Mathematics
315
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Abstract Algebra
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Integers: division algorithm, prime factorization. Groups: permutations, Lagrange's theorem. Rings: congruences, polynomials.
Course Hours:
H(3-1T)
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
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Transformation Geometry
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Geometric transformations in the Euclidean plane. Frieze patterns. Wallpaper patterns. Tessellations.
Course Hours:
H(3-1T)
Prerequisite(s):
One of Mathematics 211 or 213 or 221 and one other 200-level 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
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Set Theory
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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(3-0)
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
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Introduction to Cryptography
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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:
H(3-2T)
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
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Information Theory and Error Control Codes
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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(3-0)
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
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Introduction to Complex Analysis
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Complex numbers. Analytic functions. Complex integration and Cauchy's theorem. Maximum modulus theorem. Power series. Residue theorem.
Course Hours:
H(3-1T)
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
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Differential Geometry
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Fundamentals of the Gaussian theory of surfaces. Introduction to Riemannian geometry. Some topological aspects of surfaces.
Course Hours:
H(3-0)
Prerequisite(s):
Mathematics 353 or 381, or consent of the Division.
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Pure Mathematics
425
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Geometry
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Introduction to some of the following geometries: Discrete geometry, finite geometry, hyperbolic geometry, projective geometry, synthetic geometry.
Course Hours:
H(3-0)
Prerequisite(s):
Pure Mathematics 315 or consent of the Division.
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Pure Mathematics
427
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Number Theory
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Induction principles. Division Algorithm. Prime factorization theorem. Congruences. Arithmetic functions. Diophantine equations. Continued fractions.
Course Hours:
H(3-0)
Prerequisite(s):
Pure Mathematics 315 or consent of the Division.
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Pure Mathematics
429
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Cryptography – Design and Analysis of Cryptosystems
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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(3-0)
Prerequisite(s):
Pure Mathematics 329 or 418.
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Pure Mathematics
431
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Groups, Rings and Fields
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Factor groups and rings, polynomial rings, field extensions, finite fields, Sylow theorems, solvable groups. Additional topics.
Course Hours:
H(3-0)
Prerequisite(s):
Mathematics 311 and Pure Mathematics 315 or consent of the Division.
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Pure Mathematics
435
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Analysis I
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Logic, sets, functions; real numbers, completeness, sequences; continuity and compactness; differentiation; integration; sequence and series of functions.
Course Hours:
H(3-1T)
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
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Analysis II
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Euclidean space, basic topology; differentiation of transformations, Implicit Function Theorem; multiple integration, integrals over curves and surfaces; differential forms, Stokes' Theorem.
Course Hours:
H(3-0)
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
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Honours Real Analysis I
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Real and complex numbers, topology of metric spaces, sequences and series, continuity, differentiation, Riemann-Stieltjes integration. Rigorous approach throughout.
Course Hours:
H(3-1T)
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
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Integration Theory
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Abstract measure theory, basic integration theorems, Fubini's theorem, Radon-Nikodym theorem, further topics.
Course Hours:
H(3-0)
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
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Topics in Pure Mathematics
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This course is offered under various subtitles. Consult Department for details.
Course Hours:
H(3-0)
Prerequisite(s):
Consent of the Division.
MAY BE REPEATED FOR CREDIT
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Pure Mathematics
505
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Topology I
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Metric spaces. Introduction to general topology.
Course Hours:
H(3-0)
Prerequisite(s):
Pure Mathematics 435 or 455 or consent of the Division.
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Pure Mathematics
511
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Rings and Modules
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Ring theory, and structure of modules. Application to Abelian groups and linear algebra. Additional topics.
Course Hours:
H(3-0)
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
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Complex Analysis
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A rigorous study of functions of a single complex variable. Consequences of differentiability. Proof of the Cauchy integral theorem, applications.
Course Hours:
H(3-0)
Prerequisite(s):
Pure Mathematics 435 or 455 or consent of the Division.
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Pure Mathematics
527
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Computational Number Theory
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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(3-0)
Prerequisite(s):
Pure Mathematics 427 or 429.
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Pure Mathematics
529
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Advanced Cryptography and Cryptanalysis
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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 post-quantum cryptography, and new developments in cryptography.
Course Hours:
H(3-0)
Prerequisite(s):
Pure Mathematics 429.
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Pure Mathematics
545
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Honours Real Analysis II
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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(3-0)
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.
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Pure Mathematics
601
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Integration Theory
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Abstract measure theory, basic integration theorems, Fubini's theorem, Radon-Nikodym theorem, further topics.
Course Hours:
H(3-0)
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
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Conference Course in Pure Mathematics
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This course is offered under various subtitles. Consult Department for details.
Course Hours:
H(3-0)
MAY BE REPEATED FOR CREDIT
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Pure Mathematics
607
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Topology II
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General topology, elementary combinatorial topology.
Course Hours:
H(3-0)
Prerequisite(s):
Pure Mathematics 505 or consent of the Division.
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Pure Mathematics
613
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Introduction to Field Theory
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Field theory, Galois theory.
Course Hours:
H(3-0)
Prerequisite(s):
Pure Mathematics 431 or consent of the Division.
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Pure Mathematics
615
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Topics in Logic
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Course Hours:
H(3-0)
MAY BE REPEATED FOR CREDIT
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Pure Mathematics
621
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Research Seminar
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Reports on studies of the literature or of current research.
Course Hours:
Q(2S-0)
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
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Computational Number Theory
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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(3-0)
Prerequisite(s):
Pure Mathematics 427 or 429, or consent of the Division.
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Pure Mathematics
629
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Elliptic Curves and Cryptography
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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 curve-based Diffie-Hellman key exchange, El Gamal encryption, and digital signatures, as well as the associated computational problems on which their security is based.
Course Hours:
H(3-0)
Prerequisite(s):
Pure Mathematics 315 or consent of the Division.
Also known as:
(Computer Science 629)
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Pure Mathematics
631
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Algebraic Topology I
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Elements of category theory and homological algebra. Various examples of homology and cohomology theories. Eilenberg-Steenrod axioms. Geometrical applications.
Course Hours:
H(3-0)
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Pure Mathematics
633
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Algebraic Topology II
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Cohomology operations, CW-complexes, introduction to homotopy theory.
Course Hours:
H(3-0)
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Pure Mathematics
649
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Modern Cryptography and Cryptoanalysis
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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 post-quantum cryptography, and new developments in cryptography.
Course Hours:
H3-0
Prerequisite(s):
Pure Mathematics 429 or consent of Division.
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Pure Mathematics
669
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Cryptography
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An overview of basic techniques in modern cryptography, with emphasis on fit-for-application primitives and protocols. Topics include symmetric and public-key cryptosystems; digital signatures; elliptic curve cryptography; key management; attack models and well-defined notions of security.
Course Hours:
H(3-0)
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
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Topics in Algebra
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The following topics are available as decimalized courses: Algebraic Number Theory, Algebraic K-Theory, Algebraic Geometry, Representation Theory, Abelian Group Theory, Brauer Group Theory, Homological Algebra, Ring Theory, Associative Algebras, Commutative Algebra, Universal Algebra.
Course Hours:
H(3-0)
MAY BE REPEATED FOR CREDIT
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Pure Mathematics
727
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Advanced Topics in Computational Number Theory
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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(3-0)
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Pure Mathematics
729
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Advanced Topics in Cryptography
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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(3-0)
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COURSES OF INSTRUCTION