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# Mathematics

## Program Requirements

## Mathematics Stream

The Master of Science degree requires a minimum of 30 graduate units consisting of a thesis (12 units) with 18 units of coursework or a project (6 units) with 24 units of coursework. All coursework is subject to supervisory committee and departmental graduate studies committee approval.

### Thesis Option

Students must complete a minimum of 12 units of coursework from the courses listed in Groups 1-5 below. The coursework must involve at least three different 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.

Section | Instructor | Day/Time | Location |
---|---|---|---|

G100 |
Nathan Ilten |
Tu, Th 10:30 AM – 12:20 PM |
AQ 5005, Burnaby |

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

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.

Section | Instructor | Day/Time | Location |
---|---|---|---|

G400 |
Zhaosong Lu |
We 3:30 PM – 5:20 PM Fr 2:30 PM – 4:20 PM |
SUR 2980, Surrey SUR 2980, Surrey |

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.

#### Group 5

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.

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

Section | Instructor | Day/Time | Location |
---|---|---|---|

G100 |
Weiran Sun |
Tu 2:30 PM – 4:20 PM Th 2:30 PM – 4:20 PM |
WMC 2532, Burnaby AQ 5030, Burnaby |

and

6 units of any graduate course

and a thesis

### Project Course Option

Students must complete a minimum of 12 units of coursework from the courses listed in Groups 1-5 above. The coursework must involve at least three different groups

and

12 units of any graduate course

and a project

A project leading to research in mathematics completed under the supervision of a faculty member. The project will consist of a written report and a public presentation. This course can only be used for credit towards the MSc project course option.

MATH 880 can be attempted at most twice.

## Operations Research Stream

The Master of Science degree requires a minimum of 30 graduate units consisting of a thesis (12 units) with 18 units of coursework.

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.

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.

Section | Instructor | Day/Time | Location |
---|---|---|---|

G400 |
Zhaosong Lu |
We 3:30 PM – 5:20 PM Fr 2:30 PM – 4:20 PM |
SUR 2980, Surrey SUR 2980, Surrey |

and

4 units of graduate courses numbered 800 or above

and

3 units of any graduate course.

At least one course must be from an area of mathematics or operations research outside the operations research core courses. All coursework is subject to supervisory committee and departmental graduate studies committee approval.

and a thesis

## Program Length

The estimated completion time for a Master of Science in Mathematics is two years.

## 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.