Áù¾ÅÉ«ÌÃ : Pure Mathematics PMAT

Pure Mathematics PMAT

PMAT 503 505 511 513 527 529 571 603 607 611 613 621 627 629 649 669 671

Instruction offered by members of the Department of Mathematics and Statistics in the Faculty of Science.

Department Head - M. Lamoureux

Note: For listings of related courses, see Actuarial Science, Applied Mathematics, Mathematics, and Statistics.

Pure Mathematics 503

Topics in Mathematics

According to interests of students and instructor.
Course Hours: H(3-0)
Prerequisite(s): Consent of the Division.
MAY BE REPEATED FOR CREDIT

Pure Mathematics 505

Topology I

Basic point set topology: metric spaces, separation and countability axioms, connectedness and compactness, complete metric spaces, function spaces, homotopy.
Course Hours: H(3-0)
Prerequisite(s): Pure Mathematics 435 or 455 or Mathematics 335 or 355 or consent of the Division.

Pure Mathematics 511

Algebra III

Linear algebra: Modules, direct sums and free modules, tensor products, linear algebra over modules, finitely generated modules over PIDs, canonical forms, computing invariant factors from presentations; projective, injective and flat modules.
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.
Notes: Pure Mathematics 431 is recommended.

Pure Mathematics 513

Advanced Galois Theory

Existence of separable and algebraic closures of fields, infinite Galois extensions, profinite groups, Krull topology.
Course Hours: H(3-0)
Prerequisite(s): Pure Mathematics 431.
Antirequisite(s): Credit for both Pure Mathematics 513 and 613 will not be allowed.

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(3-0)
Prerequisite(s): Pure Mathematics 427 or 429.
Antirequisite(s): Credit for both Pure Mathematics 527 and 627 will not be allowed.
Notes: Lectures may run concurrently with Pure Mathematics 627.

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 post-quantum cryptography, and new developments in cryptography.
Course Hours: H(3-0)
Prerequisite(s): Pure Mathematics 429.
Antirequisite(s): Credit for both Pure Mathematics 529 and 649 will not be allowed.
Notes: Lectures may run concurrently with Pure Mathematics 649.

Pure Mathematics 571

Discrete Mathematics

Discrete aspects of convex optimization; computational and asymptotic methods; graph theory and the theory of relational structures; according to interests of students and instructor.
Course Hours: H(3-0)
Prerequisite(s): Pure Mathematics 471.
Antirequisite(s): Credit for both Pure Mathematics 571 and 671 will not be allowed.
Notes: Lectures may run concurrently with Pure Mathematics 671.

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 603

Conference Course in Pure Mathematics

This course is offered under various subtitles. Consult Department for details.
Course Hours: H(3-0)
MAY BE REPEATED FOR CREDIT

Pure Mathematics 607

Topology II

Fundamental groups: covering spaces, free products, the van Kampen theorem and applications; homology.
Course Hours: H(3-0)
Prerequisite(s): Pure Mathematics 505 or consent of the Division.

Pure Mathematics 611

Algebra III

Linear algebra: modules, direct sums and free modules, tensor products, linear algebra over modules, finitely generated modules over PIDs, canonical forms, computing invariant factors from presentations; projective, injective and flat modules.
Course Hours: H(3-0)
Prerequisite(s): Pure Mathematics 431 or Mathematics 411 or consent of the Division. Pure Mathematics 431 is recommended.
Antirequisite(s): Credit for both Pure Mathematics 511 and 611 will not be allowed.

Pure Mathematics 613

Advanced Galois Theory

Pure Mathematics 621

Research Seminar

A professional skills course, focusing on the development of technical proficiencies that are essential for students to succeed in their future careers 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: Q(2S-0)
Notes: Lectures may run concurrently with Applied Mathematics 621 and/or Statistics 621.
MAY BE REPEATED FOR CREDIT NOT INCLUDED IN GPA

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(3-0)
Prerequisite(s): Pure Mathematics 427 or 429, or consent of the Division.
Antirequisite(s): Credit for both Pure Mathematics 527 and 627 will not be allowed.

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 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)

Pure Mathematics 649

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 post-quantum cryptography, and new developments in cryptography.
Course Hours: H3-0
Prerequisite(s): Pure Mathematics 429 or consent of Division.
Antirequisite(s): Credit for both Pure Mathematics 529 and 649 will not be allowed.
Notes: Lectures may run concurrently with Pure Mathematics 529.

Pure Mathematics 669

Cryptography

An overview of the 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)

Pure Mathematics 671

Discrete Mathematics

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