Pure Mathematics PMAT
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, 517, 521, and 545. Check with the divisional office to plan for the upcoming cycle of offered courses.
Senior Courses
Pure Mathematics 315 H(3-1T)
Abstract Algebra
Integers: division algorithm, prime factorization. Groups: permutations, Lagrange's theorem. Rings: congruences, polynomials.
Pure Mathematics 319 H(3-2T)
Transformation Geometry
Geometric transformations in the Euclidean plane. Frieze patterns. Wallpaper patterns. Tessellations.
Prerequisites: Mathematics 211 or 221 and one other 200-level Mathematics course.
Pure Mathematics 329 H(3-1T)
Introduction to Cryptography
Description and analysis of cryptographic methods used in the authentication and protection of data. Classical cryptosystems and cryptanalysis, information theory and perfect security, the Data Encryption Standard (DES) and Public-key cryptosystems.
Prerequisites: Mathematics 271.
Note: Credit for both Pure Mathematics 329 and 321 will not be allowed.
Pure Mathematics 371 H(3-1T)
Combinatorial Mathematics
Counting, graph theory, combinatorial optimization.
Prerequisites: Mathematics 271.
Pure Mathematics 415 H(3-1T)
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.
Prerequisites: Mathematics 271 or 311 or 353 or Pure Mathematics 315 or consent of the Division.
Pure Mathematics 419 H(3-0)
(Statistics 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.
Prerequisites: Mathematics 311, and Mathematics 321 or any Statistics course, or consent of the Division.
Pure Mathematics 421 H(3-1T)
Introduction to Complex Analysis
Complex numbers. Analytic functions. Complex integration and Cauchy's theorem. Maximum modulus theorem. Power series. Residue theorem.
Prerequisites: Mathematics 349 and 353; or consent of the Division.
Note: Credit for both Pure Mathematics 421 and 521 will not be allowed.
Pure Mathematics 423 H(3-0)
Differential Geometry
Fundamentals of the Gaussian theory of surfaces. Introduction to Riemannian geometry. Some topological aspects of surfaces.
Prerequisites: Mathematics 353 or consent of the Division.
Pure Mathematics 425 H(3-1T)
Geometry
Introduction to some of the following geometries: Discrete geometry, finite geometry, hyperbolic geometry, projective geometry, synthetic geometry.
Prerequisites: Pure Mathematics 315 or consent of the Division.
Pure Mathematics 427 H(3-1T)
Number Theory
Induction principles. Division Algorithm. Prime factorization theorem. Congruences. Arithmetic functions. Diophantine equations. Continued fractions.
Prerequisites: Pure Mathematics 315 or consent of the Division.
Pure Mathematics 429 H(3-0)
Cryptography - The Design of Ciphers
Review of basic algorithms and complexity. Symmetric key cryptography. Discrete log based cryptography. One-way functions and Hash functions. Knapsack. Introduction to primality testing. Factoring. Other topics may include elliptic curves, zero-knowledge, and quantum cryptography.
Prerequisites: Pure Mathematics 321 or 329.
Corequisites: Prerequisite or Corequisite: Pure Mathematics 427.
Pure Mathematics 431 H(3-1T)
Groups, Rings and Fields
Factor groups and rings, polynomial rings, field extensions, finite fields, Sylow theorems, solvable groups. Additional topics.
Prerequisites: Mathematics 311 and Pure Mathematics 315 or consent of the Division.
Pure Mathematics 435 H(3-1T)
Analysis I
Numbers, functions; sequences; limits and continuity; theory of differentiation and integration of functions of a single variable.
Prerequisites: Mathematics 253 or 263 or Applied Mathematics 219 or consent of the Division.
Note: This course is offered in the Fall Session and would normally be taken in the third year. Potential honours students are urged to consider taking this course in second year. Please consult the appropriate Division Chair.
Pure Mathematics 445 H(3-1T)
Analysis II
Series; sequences and series of functions, uniform convergence; basic topology in Euclidean spaces; analysis with functions of several variables; implicit and inverse function theorems.
Prerequisites: Mathematics 353 and Pure Mathematics 435, or consent of the Division.
Corequisites: Mathematics 311.
Pure Mathematics 471 H(3-0)
Discrete Optimization
Block designs and extremal set theory, efficiency of algorithms, complexity theory, trees and sorting, graphs and transversals of families of sets, networks and the max-flow min-cut theorem, dynamic programming, recursion.
Prerequisites: Pure Mathematics 371.
Pure Mathematics 501 H(3-0)
Integration Theory
Abstract measure theory, basic integration theorems, Fubini's theorem, Radon-Nikodym theorem, further topics.
Prerequisites: Pure Mathematics 545 or consent of the Division.
Note: Credit for both Pure Mathematics 501 and 601 will not be allowed.
Pure Mathematics 503 H(3-0)
Topics in Pure Mathematics
This course is offered under various subtitles. Consult Department for details.
Prerequisites: Consent of the Division.
MAY BE REPEATED FOR CREDIT
Pure Mathematics 505 H(3-0)
Topology I
Metric spaces. Introduction to general topology.
Prerequisites: Pure Mathematics 435 or consent of the Division.
Pure Mathematics 511 H(3-0)
Rings and Modules
Ring theory, and structure of modules. Application to Abelian groups and linear algebra. Additional topics.
Prerequisites: One of Pure Mathematics 431, Applied Mathematics 441; Mathematics 411 or consent of the Division.
Pure Mathematics 519 H(3-0)
Information Theory, Codes, and Cryptography
A continuation of Pure Mathematics 419. Topics include: Entropy, Symmetric encryption and perfect secrecy, Source and Channel coding, Huffman and Lempel-Ziv compression, Shift registers, Cyclic codes, Reed-Solomon decoding, M.D.S. codes and hash functions, Convolutional codes, Industrial applications.
Prerequisites: Pure Mathematics 419.
Pure Mathematics 521 H(3-0)
Complex Analysis
A rigorous study of functions of a single complex variable. Consequences of differentiability. Proof of the Cauchy integral theorem, applications.
Prerequisites: Pure Mathematics 435 or consent of the Division.
Note: Credit for both Pure Mathematics 521 and 421 will not be allowed.
Pure Mathematics 529 H(3-0)
Advanced Cryptography and Cryptanalysis
Probability and perfect secrecy. Provably secure cryptosystems. Prime generation and primality testing. Cryptanalysis of factoring-based cryptosystems. Discrete log based and elliptic curve cryptography and cryptanalysis. Other advanced topics may include hyperelliptic curve cryptography, other factoring methods and other primality tests.
Pure Mathematics 545 H(3-1T)
Analysis III
Metric spaces and function spaces; equi-continuity; trigonometric series and Fourier series; elements of Lebesgue integration.
Prerequisites: Pure Mathematics 445 or consent of the Division.
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 H(3-0)
Integration Theory
Abstract measure theory, basic integration theorems, Fubini's theorem, Radon-Nikodym theorem, further topics.
Prerequisites: Pure Mathematics 545 or consent of the Division.
Note: Credit for both Pure Mathematics 601 and 501 will not be allowed.
Pure Mathematics 603 H(3-0)
Conference Course in Pure Mathematics
This course is offered under various subtitles. Consult Department for details.
MAY BE REPEATED FOR CREDIT
Pure Mathematics 607 H(3-0)
Topology II
General topology, elementary combinatorial topology.
Prerequisites: Pure Mathematics 505 or consent of the Division.
Pure Mathematics 613 H(3-0)
Introduction to Field Theory
Field theory, Galois theory.
Prerequisites: Pure Mathematics 431 or consent of the Division.
Pure Mathematics 621 Q(2S-0)
Research Seminar
Reports on studies of the literature or of current research.
Note: 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
Pure Mathematics 627 H(3-0)
Topics in Computational Number Theory
Examines some difficult problems in number theory and discusses a few of the computational techniques that have been developed for solving them. Such problems include: modular exponentiation, primality testing, integer factoring, solution of polynomial congruences, quadratic partitions or primes, invariant computation in certain algebraic number fields, etc. Emphasis will be placed on practical techniques and their computational complexity.
Prerequisites: Pure Mathematics 427 or consent of the Division.
Pure Mathematics 629 H(3-0)
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 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.
Prerequisites: Pure Mathematics 315 or consent of the Division.
Pure Mathematics 631 H(3-0)
Algebraic Topology I
Elements of category theory and homological algebra. Various examples of homology and cohomology theories. Eilenberg-Steenrod axioms. Geometrical applications.
Pure Mathematics 633 H(3-0)
Algebraic Topology II
Cohomology operations, CW-complexes, introduction to homotopy theory.
Pure Mathematics 642 F(3-0)
Differentiable Manifolds
Definition of differentiable manifold. Vector fields and differentiable forms. Connections. Curvature. Relations between topology and geometry.
Pure Mathematics 669 H(3-0)
(Computer Science 669)
Cryptography
An introduction to the fundamentals of cryptographic systems, with emphasis on attaining well-defined notions of security. Public-key cryptosystems; examples, semantic security. One-way and trapdoor functions; hard-core predicates of functions; applications to the design of cryptosystems.
Prerequisites: Consent of the Division.
Note: Computer Science 413 and Mathematics 321 are recommended as preparation for this course.
Pure Mathematics 685 H(3-0)
Topics in Algebra
The following topics are available as decimalized courses: Algebraic Number Theory, Algebraic K-Theory, Representation Theory, Abelian Group Theory, Brauer Group Theory, Homological Algebra, Ring Theory, Associative Algebras, Commutative Algebra, Universal Algebra.
MAY BE REPEATED FOR CREDIT
Pure Mathematics 707 H(3-0)
Topics in Topology
The following topics are available as decimalized courses: Fibre Bundles, Characteristic Classes, K-Theory of Vector Bundles, Theory of Transformation Groups, Homotopy Theory.
MAY BE REPEATED FOR CREDIT
Pure Mathematics 717 H(3-0)
Topics in Analysis
The following topics are available as decimalized courses: Abstract Harmonic Analysis, Gelfand Theory for Banach Algebras and C*-Algebras, Invariant Means on Semigroups, Functional Analysis, Differential Operator Theory, Spectral Theory of Operators, Integral Transforms, Matrix Theory, Analysis of Manifolds, Dynamical Systems, Differential Equations.
MAY BE REPEATED FOR CREDIT
Pure Mathematics 727 H(3-0)
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.
Pure Mathematics 729 H(3-0)
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.
In addition to the numbered and titled courses shown above, the department offers a selection of advanced level graduate courses specifically designed to meet the needs of individuals or small groups of students at the advanced doctoral level. These courses are numbered in the series 800.01 to 899.99. Such offerings are, of course, conditional upon the availability of staff resources.