University of Calgary : Pure Mathematics PMAT

Pure Mathematics PMAT

PMAT 315 317 319 415 418 423 425 427 429 431 471 503 505 511 513 527 529 571 603 607 611 613 621 627 629 649 669 671

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.

Note: The following courses, although offered on a regular basis, are not offered every year: Pure Mathematics 415, 423, 425, 427, 505 and 511. Check with the Department office to plan for the upcoming cycle of offered courses.

Note: Commencing in Fall 2014, Mathematics 265, 267, 367, Mathematics 275, 277, 375 and 377 will replace respectively Mathematics 251, 253, 353, Applied Mathematics 217, 219, 307 and 309 and will serve as prerequisites for appropriate courses. In some special cases, Mathematics 267 will replace Mathematics 349 or 353. For these and other deviations from the general rule, see individual course entries for details. Mathematics 267 supplemented by Mathematics 177 will be accepted as equivalent to Mathematics 277.

Senior Courses

Pure Mathematics 315

Algebra I

Basic ring theory: rings and fields, the integers modulon, Polynomial rings, polynomials over the integers and rationals, homomorphisms, ideals and quotients, principal ideal domains, adjoining the root of an irreducible polynomial; basic group theory: groups, examples including cyclic, symmetric, alternating and dihedral groups, subgroups, cosets and Lagrange’s theorem, normal subgroups and quotients, group homomorphisms, the isomorphism theorems, further topics as time permits, e.g., group actions, Cayley’s theorem.
Course Hours: H(3-1T)
Prerequisite(s): Mathematics 211 or 213.
Antirequisite(s): Credit for both Pure Mathematics 315 and 317 will not be allowed.
Notes: Mathematics 271 or 273 is strongly recommended as preparation for this course.

Pure Mathematics 317

Honours Algebra I

Basic ring theory: rings and fields, the integers modulo n, polynomial rings, polynomials over the integers and rationals, homomorphisms, ideals and quotients, principal ideal domains, adjoining the root of an irreducible polynomial; basic group theory: groups, examples including cyclic, symmetric, alternating and dihedral groups, subgroups, cosets and Lagrange’s theorem, normal subgroups and quotients, group homomorphisms, the isomorphism theorems, further topics as time permits, e.g., group actions, Cayley’s theorem.
Course Hours: H(3-1T)
Prerequisite(s): Mathematics 213.
Antirequisite(s): Credit for both Pure Mathematics 317 and 315 will not be allowed.
Notes: Mathematics 271 or 273 is strongly recommended as preparation for this course.

Pure Mathematics 319

Transformation Geometry

Geometric transformations in the Euclidean plane. Frieze patterns. Wallpaper patterns. Tessellations.
Course Hours: H(3-1T)
Prerequisite(s): Mathematics 211 or 213 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.

Pure Mathematics 415

Foundations

Set theory, mathematical logic, category theory, according to interests of students and instructor.
Course Hours: H(3-0)
Prerequisite(s): Mathematics 271 or 273 or 311 or 313 or 353 or 367 or 377 or 381 or Pure Mathematics 315 or 317; or consent of the Department.

Pure Mathematics 418

Introduction to Cryptography

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(3-0)
Prerequisite(s): One of Mathematics 271 or 273 or Pure Mathematics 315 or 317.
Antirequisite(s): Credit for both Pure Mathematics 418 and any of Pure Mathematics 329, Computer Science 418, 429, or 557 will not be allowed.

Pure Mathematics 423

Differential Geometry

Curvature, connections, parallel transport, Gauss-Bonnet theorem.
Course Hours: H(3-0)
Prerequisite(s): Mathematics 353 or 367 or 377 or 381; or consent of the Department.

Pure Mathematics 425

Geometry

Euclidean, convex, discrete, synthetic, projective or hyperbolic geometry, according to interests of the instructor.
Course Hours: H(3-0)
Prerequisite(s): Pure Mathematics 315 or 317; or consent of the Department.

Pure Mathematics 427

Number Theory

Divisibility and the Euclidean algorithm, modular arithmetic and congruences, quadratic reciprocity, arithmetic functions, distribution of primes.
Course Hours: H(3-0)
Prerequisite(s): Pure Mathematics 315 or 317; or consent of the Department.

Pure Mathematics 429

Cryptography – Design and Analysis of Cryptosystems

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 315 or 317; and one of Pure Mathematics 329, 418, Computer Science 418.

Pure Mathematics 431

Algebra II

Group theory: Sylow theorems, solvable, nilpotent and p-groups, simplicity of alternating groups and PSL(n,q), structure theory of finite abelian groups; field theory: gilds, algebraic and transcendental extensions, separability and normality, Galois theory, insolvability of the general quintic equation, computation of Galois groups over the rationals.
Course Hours: H(3-0)
Prerequisite(s): Mathematics 311 or 313 and Pure Mathematics 315 or 317; or consent of the Department.

Pure Mathematics 471

Combinatorics and Graph Theory

Counting techniques, generating functions, inclusion-exclusion, introduction to graph theory and the theory of relational structures.
Course Hours: H(3-0)
Prerequisite(s): Mathematics 271 or 273; and one of Mathematics 249, 251, 265, 275, 281 or Applied Mathematics 217.
Antirequisite(s): Credit for both Pure Mathematics 471 and 371 will not be allowed.

Pure Mathematics 503

Topics in Mathematics

According to interests of students and instructor.
Course Hours: H(3-0)
Prerequisite(s): Consent of the Department.
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 Department.

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 Department.
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.

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.

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.

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 Department.

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 Department. 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)
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 Department.
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 Department.
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: H(3-0)
Prerequisite(s): Pure Mathematics 429 or consent of the Department.
Antirequisite(s): Credit for both Pure Mathematics 529 and 649 will not be allowed.

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 Department.
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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