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 Graduate Studies Calendar 2011-2012 Courses of Instruction Course Descriptions P Pure Mathematics PMAT
Pure Mathematics PMAT
PMAT 501 503 505 511 521 527 529 545 571 601 603 607 611 621 627 629 649 669 671
 

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 501       Integration Theory
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       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
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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 consent of the Division.
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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.
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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(3-0)
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(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.
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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 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.
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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(3-0)
Prerequisite(s): Pure Mathematics 455; or a grade of B+ or better in Pure Mathematics 435.
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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.
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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, 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       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
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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.
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Pure Mathematics 611       Algebra IV
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.
Notes: Lectures may run concurrently with Pure Mathematics 511.
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Pure Mathematics 621       Research Seminar
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       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.
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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 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 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.
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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)
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Pure Mathematics 671       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 671 and 571 will not be allowed.
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