## 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, 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.

**
Prerequisites:
** Mathematics 211 or 221.

**
Note:
** Mathematics 271 or 273 is strongly recommended as preparation for this course.

### 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 course labelled Applied Mathematics, Mathematics or Pure Mathematics, not including Mathematics 205.

**
Note:
** Mathematics 271 or 273 is strongly recommended as preparation.

### 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 or 273.

**
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 or 273; and Mathematics 249 or 251 or 281 or Applied Mathematics 217.

### 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 273 or 311 or 353 or 381 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:
** Both Mathematics 349 and 353; or both Mathematics 283 and 381.

**
Note:
** Not open to students with credit in Pure Mathematics 521.

### 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 381 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 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

Logic, sets, functions; real numbers, completeness, sequences; continuity and compactness; differentiation; integration; sequence and series of functions.

**
Prerequisites:
** Mathematics 253 or 263 or 283 or Applied Mathematics 219 or consent of the Division.

**
Note:
** Credit for both Pure Mathematics 435 and 455 will not be allowed.

### Pure Mathematics 445 H(3-1T)

#### Analysis II

Euclidean space, basic topology; differentiation of transformations, Implicit Function Theorem; multiple integration, integrals over curves and surfaces; differential forms, Stokes' Theorem.

**
Prerequisites:
** Mathematics 353 or 381; and Pure Mathematics 435 or 455, or consent of the Division.

**
Corequisites:
** Mathematics 311.

**
Note:
** Not open to students with credit in Pure Mathematics 545.

### Pure Mathematics 455 H(3-1T)

#### Honours Real Analysis I

Real and complex numbers, topology of metric spaces, sequences and series, continuity, differentiation, Riemann-Stieltjes integration. Rigorous approach throughout.

**
Prerequisites:
** Mathematics 283 or 263; or a grade of B+ or better in Mathematics 253 or Applied Mathematics 219.

**
Note:
** Credit for both Pure Mathematics 435 and 455 will not be allowed.

### 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 455 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:
** Pure Mathematics 431 or Mathematics 411 or consent of the Division.

### 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 455 or consent of the Division.

### 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(3-0)

#### 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.

**
Prerequisites:
** Mathematics 455; or a grade of B+ or better in Pure Mathematics 445.

## 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 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 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.