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# Mathematics

The master of science (MSc) in mathematics initiates students to the exciting world of mathematical exploration and research. Students take courses in advanced topics and work with world-class research faculty to create original mathematics. Graduates of the program are qualified for work in industry, academia and government.

## Admission Requirements

Applicants must satisfy the University admission requirements as stated in Graduate General Regulations 1.3 in the SFU Calendar.

## Program Requirements

This program offers two streams: mathematics and operations research for a minimum of 30 units. All courses are subject to supervisory committee and departmental graduate studies committee approval.

### Mathematics Stream

Students must complete

a minimum of 12 units of course work from at least three different groups listed below

and an additional six graduate units of course work

and the requirements from either the thesis or project option

#### Thesis Option

and a thesis

#### Project Option

and an additional six graduate units

and a project

#### Groups

##### Group 1

A survey of graduate group and/or ring theory. Possible topics include generators and relations, composition series, Sylow theory, permutation groups, abelian groups, p-groups, nilpotent and solvable groups, aspects of simple groups, representation theory, group algebras, chain conditions, Jacobson radical, Chevalley-Jacobson density theorem, Wedderburn-Artin theorems.

An introduction to algebraic geometry with supporting commutative algebra. Possible topics include Hilbert basis theorem, Hilbert's Nullstellensatz, Groebner bases, ideal decomposition, local rings, dimension, tangent and cotangent spaces, differentials, varieties, morphisms, rational maps, non-singularity, intersections in projective space, cohomology theory, curves, surfaces, homological algebra.

##### Group 2

Algebraic graph theory, extremal graph theory, coloring problems, path and cycle structure of graphs, application of graphs, hypergraphs, and current research topics.

An introduction to the theory of incidence structures (finite geometries, block designs) and their relation to linear codes. Algebraic techniques - finite group actions, orbit enumeration, generation of orbit representatives. Exact and asymptotic enumeration of labelled and unlabelled structures.

##### Group 3

Review of Galois theory, integrality, rings of integers, traces, norms, discriminants, ideals, Dedekind domains, class groups, unit groups, Minkowski theory, ramification, cyclotomic fields, valuations, completions, applications.

Arithmetical functions, distribution of prime numbers, theory of Dirichlet characters, Dirichlet series, theory of Riemann Zeta functions and Dirichlet L-functions, exponential sums, character sums, Diophantine equations, Diophantine approximations, applications.

An introduction to the subject of modern cryptography. Classical methods for cryptography and how to break them, the data encryption standard (DES), the advanced encryption standard (AES), differential and linear cryptanalysis. RSA and EIGamal public key cryptosystems, digital signatures, secure hash functions and pseudo-random number generation. Algorithms for computing with long integers including the use of probabilistic algorithms. Elliptic curve cryptography. Post-quantum cryptography. Students with credit for either MACM 442 or MATH 742 may not take this course for further credit.

##### Group 4

Theory and algorithms of non-linear programming with an emphasis on modern computational considerations. Topics may include: optimality conditions for unconstrained and constrained optimization, gradient methods, conjugate direction methods, Newton method, quasi-Newton methods, penalty and barrier methods, augmented Langrangian methods and interior point methods.

Computing with long integers, polynomials, and mathematical formulae. Topics include computing polynomial greatest common divisors, the Fast Fourier Transform, Hensel's Lemma and p-adic methods, differentiation and simplification of formulae, polynomial factorization. Integration of rational functions and elementary functions, Liouville's principle, the Risch algorithm. Students will use a computer algebra system such as Maple for calculations and programming. Students who have credit for either MACM 401 or MATH 701 may not take this course for further credit.

Convex geometry, the simplex method and duality, pivot rules, degeneracy, decomposition and column generation methods, the complexity of linear programming and the ellipsoid algorithm, interior point methods for linear programming.

##### Group 5

Infinite dimensional vector spaces, convergence, generalized Fourier series. Operator Theory; the Fredholm alternative. Application to integral equations and Sturm-Liouville systems. Spectral theory.

An intensive study of Lebesque measure, integration and the Lebesque convergence theorems together with the treatment of such topics as absolute continuity, the fundamental theorem of calculus, the Lp-spaces, comparison of types of convergence in function spaces, the Baire category theorem.

### Operations Research Stream

Students must complete all of

Theory and algorithms of non-linear programming with an emphasis on modern computational considerations. Topics may include: optimality conditions for unconstrained and constrained optimization, gradient methods, conjugate direction methods, Newton method, quasi-Newton methods, penalty and barrier methods, augmented Langrangian methods and interior point methods.

Held jointly with MATH 408-3. See description for MATH 408-3. Students may not take a 700-division course if it is being offered in conjunction with a 400-division course which they have taken previously.

Convex geometry, the simplex method and duality, pivot rules, degeneracy, decomposition and column generation methods, the complexity of linear programming and the ellipsoid algorithm, interior point methods for linear programming.

and four units of graduate courses numbered 800 or above

and an additional three graduate units of course work*

and a thesis

*At least one course must be from an area of mathematics or operations research outside the operations research core courses.

### Accelerated Master's

SFU students accepted in the accelerated master's within the Department of Mathematics may apply a maximum of 10 graduate course units, taken while completing the bachelor's degree, towards the upper division electives of the bachelor's program and the requirements of the master's degree. For more information go to: https://www.sfu.ca/gradstudies/apply/programs/accelerated-masters.html.

## Program Length

Students are expected to complete the program requirements in six terms.

## Academic Requirements within the Graduate General Regulations

All graduate students must satisfy the academic requirements that are specified in the Graduate General Regulations, as well as the specific requirements for the program in which they are enrolled.