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

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):One of Mathematics 211 or 213 or 221. 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.

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):One of Mathematics 213 or 221. 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.

Geometric transformations in the Euclidean plane. Frieze patterns. Wallpaper patterns. Tessellations. Course Hours:H(3-1T) Prerequisite(s):One of Mathematics 211 or 213 or 221 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.

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 353 or 381 or Pure Mathematics 315 or 317, or consent of the Division.

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

Information sources, entropy, channel capacity, Shannon's theorems, coding theory, error-correcting codes. Course Hours:H(3-0) Prerequisite(s):Mathematics 311, and Mathematics 321 or any Statistics course, or consent of the Division. Also known as:(Statistics 419)

Complex numbers. Analytic functions. Complex integration and Cauchy's theorem. Maximum modulus theorem. Power series. Residue theorem. Course Hours:H(3-1T) Prerequisite(s):Both Mathematics 349 and 353; or both Mathematics 283 and 381. Antirequisite(s):Not open to students with credit in Pure Mathematics 521.

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

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 Division.

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 Division.

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.

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 and Pure Mathematics 315 or 317 or consent of the Division.

Logic, sets, functions; real numbers, completeness, sequences; continuity and compactness; differentiation; integration; sequence and series of functions. Course Hours:H(3-1T) Prerequisite(s):Mathematics 253 or 263 or 283 or Applied Mathematics 219, or consent of the Division. Antirequisite(s):Credit for both Pure Mathematics 435 and 455 will not be allowed.

Euclidean space, basic topology; differentiation of transformations, Implicit Function Theorem; multiple integration, integrals over curves and surfaces; differential forms, Stokes' Theorem. Course Hours:H(3-0) Prerequisite(s):Mathematics 353 or 381; and Mathematics 311; and Pure Mathematics 435 or 455, or consent of the Division. Antirequisite(s):Not open to students with credit in Pure Mathematics 545.

Real and complex numbers, topology of metric spaces, sequences and series, continuity, differentiation, Riemann-Stieltjes integration. Rigorous approach throughout. Course Hours:H(3-1T) Prerequisite(s):Mathematics 283 or 263; or a grade of B+ or better in Mathematics 253 or Applied Mathematics 219. Antirequisite(s):Credit for both Pure Mathematics 435 and 455 will not be allowed.

Counting techniques, generating functions, inclusion-exclusion, introduction to graph theory and the theory of relational structures. Course Hours:H(3-1T) Prerequisite(s):Mathematics 271 or 273; and Mathematics 249 or 251 or 281 or Applied Mathematics 217. Antirequisite(s):Credit for both Pure Mathematics 471 and 371 will not be allowed. Also known as:(formerly Pure Mathematics 371)

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 CREDITNOT INCLUDED IN GPA

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.

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)

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.

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)

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.