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## Department of Mathematics

K10512 Shrum Science Centre, (604) 2913331 Tel, (604) 2914947 Fax, www.math.sfu.ca

- Chair
A.H. Lachlan BA, MA, PhD (Camb), FRSC

- Graduate Program Chair
M.R. Trummer Dipl Math, DrScMath (ETH Zurich)

- Faculty and Areas of Research
see "Department of Mathematics". for a complete list of faculty.

B.R. Alspach* - graph theory, discrete mathematics

J.L. Berggren - history of mathematics, algebra

J.M. Borwein - analysis, computation

P.B. Borwein - analysis, computation, number theory

T.C. Brown - algebra, combinatorics

I. Chen - number theory, arithmetic geometry

K-K.S. Choi - number theory, algebra

R. Choksi - calculus of variations, partial differential equations, and applications to material science

A. Das* - variational techniques; interior solutions in general relativity

G.A.C. Graham - viscoelastic solid mechanics

P. Hell - computational discrete mathematics

M.C.A. Kropinski - numerical solutions of non-linear differential equations; fluid dynamics

A.H. Lachlan - mathematical logic

A.S. Lewis - analysis, optimization

M.B. Monagan - symbolic computation, algebra

D. Muraki - asymptotic analysis and modelling for the physical sciences, non-linear waves and dynamics, atmospheric fluid dynamics

E. Pechlaner - relativistic continuum mechanics; approximation methods, self-similarity

K. Promislow - partial differential equations, non-linear waves, invariant manifolds

R.D. Russell - numerical analysis; numerical solution of differential equations, dynamical systems

S. Ruuth - scientific computing, differential equations, dynamics of interfaces

C.Y. Shen - electronmagnetic scattering; large scale scientific computing

M.R. Trummer - numerical analysis; differential equations, integral equations

- Advisors
Ms. M. Fankboner BA (Occidental), MSc (S Fraser), K10511 Shrum Science Centre, 778.782 4849

## Admission

see "Graduate General Regulations". for admission requirements. Applicants are normally required to submit aptitude section scores and an appropriate advanced section of the graduate record exams of the Educational Testing Service. Applicants whose first language is not English will be asked to submit TOEFL results.

## Co-operative Education Program

The department has introduced co-op education into its graduate program to allow students to gain work experience outside the academic sphere. Students who are currently enrolled in one of the department's MSc or PhD programs may apply to the department's graduate co-operative education committee.

## Applied and Computational Mathematics

## Admission

see "Graduate General Regulations". for admission requirements.

Applicants are normally required to submit scores in the aptitude section and an appropriate advanced section of the Graduate Record Examinations of the Educational Testing Service.

Applicants with backgrounds in areas other than mathematics, (for example, a bachelor's degree or its equivalent in engineering or physics) may be considered suitably prepared for these programs.

## MSc Program Requirements

A candidate for the MSc will normally be required to obtain a total of 28 credit hours beyond courses taken for the bachelor's degree. These 28 hours will consist of at least four courses chosen from the list of core courses below with at least one course from each of the pairs APMA 900,901; APMA 920, 922; APMA 930, 935; a further eight credit hours at the graduate level; and a further four credit hours which may be at the graduate level or at the 400 undergraduate level. The six core courses are

APMA 900-4 Advanced Mathematical Methods I

APMA 901-4 Advanced Mathematical Methods II

APMA 920-4 Numerical Linear Algebra

APMA 922-4 Numerical Solution of Partial Differential Equations

APMA 935-4 Mechanics of Solids

In addition to this course requirement (normally completed in five semesters), the student completes a project which involves a significant computational component and submits and successfully defends a project report. This project should be completed within about one semester.

## PhD Program Requirements

A PhD candidate must obtain at least a further eight graduate level credit hours beyond the MSc requirements. Candidates who are admitted to the PhD program without an MSc are required to obtain credit or transfer credit for an amount of course work equivalent to that obtained by students with a MSc.

PhD candidates normally pass an oral candidacy exam given by the supervisory committee before the end of the fourth full time semester. The exam consists of a proposed thesis topic defence and supervisory committee questions on related proposed research topics. The exam follows submission of a written PhD research proposal and is graded pass/fail. Those with a fail take a second exam within six months. A student failing twice will normally withdraw.

A PhD candidate must submit and defend a thesis based on his/her original work that embodies a significant contribution to mathematical knowledge.

## Applied and Computational Mathematics Graduate Courses

Note: course descriptions for MATH 800-899 appear in the Mathematics and Statistics section while those for STAT 801-890 can be found in the Statistics Program section. the courses listed below replace courses labelled MATH. Except for selected topics courses, students with credit for a MATH labelled course may not take the corresponding APMA labelled course for further credit.

- APMA 900-4 Advanced Mathematical Methods I
Hilbert spaces. Calculus of variations. Sturm-Liouville problems and special functions. Green's functions in one dimension. Integral equations. Prerequisite: MATH 314 or equivalent. Students with credit for MATH 900 may not take APMA 900 for further credit. Recommended: MATH 419.

- APMA 901-4 Advanced Mathematical Methods II
First order partial differential equations. Characteristics. Eigenfunction expansions and integral transforms. Discontinuities and singularities; weak solutions. Green's functions. Variational methods. Prerequisite: MATH 314 or equivalent. Students with credit for MATH 901 may not take APMA 901 for further credit. Recommended: MATH 418.

- APMA 902-4 Applied Complex Analysis
Review of complex power series and contour integration. Conformal mapping, Schwartz-Christoffel transformation. Special functions. Asymptotic expansions. Integral transform. Prerequisite: MATH 322 or equivalent. Students with credit for MATH 836 or 902 may not take APMA 902 for further credit.

- APMA 905-4 Applied Functional Analysis
Infinite dimensional vector spaces, convergence, generalized Fourier series. Operator Theory; the Fredholm alternative. Application to integral equations and Sturm-Liouville systems. Spectral theory. Prerequisite: MATH 900 or permission of the instructor. Students with credit for MATH 905 may not take APMA 905 for further credit.

- APMA 910-4 Ordinary Differential Equations
The solutions and properties of ordinary differential equations and systems of ordinary differential equations in the real and complex domains. Prerequisite: MATH 415 or equivalent. Students with credit for MATH 842 or 910 may not take APMA 910 for further credit.

- APMA 912-4 Partial Differential Equations
An advanced course on partial differential equations. Topics covered usually will include quasi-linear first order systems and hyperbolic, parabolic and elliptic second-order equations. Prerequisite: MATH 901 or permission of the instructor. Students with credit for MATH 845 or 912 may not take APMA 912 for further credit.

- APMA 920-4 Numerical Linear Algebra
Direct and iterative methods for the numerical solution of linear systems, factorization techniques, linear least squares problems, eigenvalue problems. Techniques for parallel architectures. Students with credit for MATH 850 or 920 may not take APMA 920 for further credit.

- APMA 921-4 Numerical Solution of Ordinary Differential Equations
Study of the practical numerical methods for solving initial and boundary value problems for ordinary differential equations. Students with credit for MATH 851 or 921 may not take APMA 921 for further credit.

- APMA 922-4 Numerical Solution of Partial Differential Equations
Analysis and application of numerical methods for solving partial differential equations. Finite difference methods, spectral methods, multigrid methods. Students with credit for MATH 852 or 922 may not take APMA 922 for further credit.

- APMA 923-4 Numerical Methods in Continuous Optimization
Numerical solution of systems of nonlinear equations, and unconstrained optimization problems. Newton's method, Quasi-Newton methods, secant methods, and conjugate gradient algorithms. Students with credit for MATH 853 or 923 may not take APMA 923 for further credit.

- APMA 929-4 Selected Topics in Numerical Analysis
Study of a specialized area of numerical analysis such as computational fluid dynamics, approximation theory, integral equations, integral transforms, computational complex analysis, special functions, numerical quadrature and multiple integrals, constrained optimization, finite elements methods, sparse matrix techniques, or parallel algorithms in scientific computing.

- APMA 930-4 Fluid Dynamics
Basic equations and theorems of fluid mechanics. Incompressible flow. Compressible flow. Effects of viscosity. Prerequisite: MATH 361 or equivalent. Students with credit for MATH 930 may not take APMA 930 for further credit. Recommended: MATH 462.

- APMA 934-4 Selected Topics in Fluid Dynamics
Study of a specialized area of fluid dynamics such as hydrodynamic stability, multiphase flow, non-Newtonian fluids, computational fluid dynamics, boundary-layer theory, magnetic fluids and plasmas, bio- and geo-fluid mechanics, gas dynamics. Prerequisite: APMA 930 or permission of the instructor.

- APMA 935-4 Mechanics of Solids
Analysis of stress and strain. Conservation laws. Elastic and plastic material behavior. Two and three dimensional elasticity. Variational principles. Wave propagation. Prerequisite: MATH 361 or equivalent. Students with credit for MATH 883 or 935 may not take APMA 935 for further credit. Recommended: MATH 468.

- APMA 939-4 Selected Topics in Mechanics of Solids
Study of a specialized area of the mechanics of solids such as composite materials, micromechanics, fracture, plate and shell theory, creep, computational solid mechanics, wave propagation, contact mechanics. Prerequisite: APMA 935 or permission of the instructor.

- APMA 981-4 Selected Topics in Continuum Mechanics

- APMA 982-4 Selected Topics in Mathematical Physics

- APMA 990-4 Selected Topics in Applied Mathematics
## Mathematics

## MSc Program Requirements

A candidate normally obtains at least 20 credit hours beyond courses taken for the bachelor's degree. Of these, at least 12 are graduate courses or seminars, and the remaining eight may be from graduate courses or seminars or 400 division undergraduate courses. The student must also submit a satisfactory thesis and will attend an oral examination based on that thesis and related topics.

Note: APMA 900-990 (see "Department of Biological Sciences".) and STAT 800-890 (see "Department of Statistics and Actuarial Science".) may be used to satisfy requirements for the master of science degree.

## PhD Program Requirements

A candidate will generally obtain at least 28 credit hours beyond those for the bachelor's degree. Of these, at least 16 are graduate courses or seminars and the remaining 12 may be graduate courses, seminars or 400 level undergraduate courses. Students with an MSc in mathematics or statistics are deemed to have earned 12 of the 16 hours and eight of the 12 undergraduate or graduate hours required. Course work in all cases will involve study in at least four different areas of mathematics and/or statistics.

Candidates will normally pass a two stage general exam. The first stage covers a broad range of senior undergraduate material. In the second, students present to their supervisory committee a written thesis proposal and then defend this at an open oral defence. The supervisory committee evaluates the thesis proposal and defence and either passes or fails the student. A candidate ordinarily cannot take either stage of the general examination more than twice. Both stages must be completed within six full time semesters of initial enrolment in the PhD program.

The supervisory committee may require proficiency in reading mathematical papers in either French, German or Russian.

Students must submit and successfully defend a thesis which embodies a significant contribution to mathematical knowledge.

see "Graduate General Regulations". for further information and regulations.

Note: APMA 900-990 (page 369) and STAT 800-890 (page 380) may be satisfy PhD requirements.

## Mathematics Graduate Courses

- MATH 601-4 Discovering Mathematics I
Arithmetic and Geometry form the core of the elementary school curriculum. The fundamental concepts in both these areas of mathematics will be approached through exploratory exercises and problems as well as in projects. The students will work both singly and in groups to explore the ideas of mathematics. The presentations will be non-theoretical. Prerequisite: acceptance into the master's program in mathematics education or permission of the department. Graduate students in Department of Mathematics cannot take this course to satisfy their degree requirements.

- MATH 602-4 Discovering Mathematics II
Discrete mathematics is used in computer communications, scheduling and transportation problems. Statistics is encountered by each of us every day in the newspapers and on television as medical findings, sporting results and economic strategies are discussed. These are two of the most accessible areas of modern applied mathematics and many problems and the ideas behind their solution can be understood and appreciated by students with only a modest mathematical background. Several topics in these areas and their relationship to real world problems will be explored. The exploration will be done through a series of projects with students often working in teams and making presentations of their discoveries. The presentation will be non-theoretical. Prerequisite: MATH 601 and acceptance into the master's program in mathematics education or permission of the department. Graduate students in Department of Mathematics cannot take this course to satisfy their degree requirements.

- MATH 603-4 Foundations of Mathematics
Crises in mathematics, their historical and philosophical background and their resolution. Prerequisite: acceptance into the MSc program in mathematics education or permission of the department. Graduate students in the Department of Mathematics cannot take this course to satisfy their degree requirements.

- MATH 604-4 Geometry
Euclidean and non-Euclidean geometries. Klein's erlangen program. Prerequisite: entrance into the MSc in mathematics education program or permission of the department. Graduate students in the Department of Mathematics cannot take this course to satisfy their degree requirements.

- MATH 605-4 Mathematical Modeling
Introduction to mathematical modeling using algebraic, geometric techniques along with techniques using calculus. Prerequisite: acceptance into the MSc program in mathematics education and one year of university level calculus. Graduate students in the Department of Mathematics cannot take this course to satisfy their degree requirements.

- MATH 800-4 Pure Mathematics: Selected Topics
- MATH 806-4 Mathematical Logic II
First-order theories. Some syntactical theorems concerning provability, such as the equivalence and equality theorems; the completeness theorem and some of its consequences for equivalence of syntactical and semantical notions, and introduction to model theory; incompleteness of formal arithmetic.

- MATH 807-4 Mathematical Logic: Selected Topics
- MATH 808-4 Mathematical Logic III
Introduction to recursion theory. Church's Thesis, Godel-Rosser incompleteness theorem, undecidability. Kleen's normal form theorem and enumerations theorem, the recursion theorem. The arithmetic hierarchy, the analytical hierarchy. Degrees of unsolvability. Basic theorems. Additional topics, if time permits. Prerequisite: MATH 806.

- MATH 812-4 Algebra I
Theory of fields. Topics covered will include separable, normal, Galois, and transcendental extensions; finite fields and algebraically closed fields. Additional topics may include infinite Galois groups, valuation, Kummer extensions and Galois cohomology, further material in algebraic number theory.

- MATH 813-4 Algebra II
Group theory. Generators and relations, normalizers and centralizers, composition series. Permutation groups, Sylow theory, abelian groups. Other topics covered will be the theory of p-groups, nilpotent and solvable groups, and some aspects of simple groups.

- MATH 814-4 Algebra: Selected Topics
- MATH 815-4 Algebra III
Rings and modules. Commutative and noncommutative associate rings with ascending or descending chain condition. Jacobson radical Chevalley-Jacobson density theorem, Wedderburn-Artin theorems, Goldie theorems, with applications to matrix groups and group algebras. As time permits, homological and local methods.

- MATH 816-4 Algebra IV
Homology. Categories, functors, adjoint functors, homology, and cohomology of a complex. Universal coefficient theorem; Extn cohomology of groups; Schurs theorem. Tensor and torsion products. Global dimension of rings.

- MATH 820-4 Graph Theory
A first graduate course in graph theory dealing with some of the following: algebraic graph theory, extremal graph theory, coloring problems, applications of graphs, hypergraphs, and current research topics.

- MATH 821-4 Combinatorics
An introduction to the theory of block designs, finite geometries and related topics.

- MATH 825-4 Enumeration
Enumeration problems concerned with permutations, sequences, partitions, lattice walks and graphs, algebraic and analytic properties of generating functions, asymptotic analysis.

- MATH 826-4 Posets and Matroids
An introduction to the theory of posets, geometric lattices and matroids.

- MATH 827-4 Discrete Mathematics: Selected Topics
- MATH 831-4 Real Analysis I
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.

- MATH 832-4 Real Analysis II
This course normally covers abstract measure and integration, and material which collectively might be called an introduction to functional analysis (e.g. complete metric spaces, normal spaces, the Stone-Weierstrass theorem, linear functionals and the Hahn-Banach theorem). Other specialized topics in modern analysis. Prerequisite: MATH 831.

- MATH 833-4 Analysis: Selected Topics
- MATH 836-4 Complex Analysis I
Topics covered normally will include: Riemann surfaces, complex conjugate co-ordinates; the maximum principle, boundary value problems; conformal mappings, Schwartz-Christoffel formula; the symmetry principle, analytic continuation.

- MATH 837-4 Complex Analysis II
Topics covered will include some of the following: entire functions, normal families, Hilbert space of analytic functions; conformal mappings of special functions; Picard's theorem. Prerequisite: MATH 836.

- MATH 839-4 Topology I
A first graduate course in general topology, dealing with some of the following topics: set-theoretic preliminaries, topological spaces, filters and nets, connectedness notions, separation properties, countability properties, compactness properties, paracompactness, metrization, uniform spaces, function spaces.

- MATH 840-4 Topology II
A second graduate course in general topology dealing with additional topics among those listed for MATH 839. Prerequisite: MATH 839.

- MATH 841-4 Topology: Selected Topics
- MATH 890-0 Practicum I
First semester of work experience in a co-operative education program. (0-0-0)

- MATH 891-0 Practicum II
Second semester of work experience in a cooperative education program. (0-0-0)

- MATH 892-0 Practicum III
Third semester of work experience in the Cooperative Education Program. Prerequisite: MATH 891. (0-0-0)

- MATH 893-0 Practicum IV
Fourth semester of work experience in the Cooperative Education Program. Prerequisite: MATH 892. (0-0-0)

- MATH 895-4 Reading
- MATH 896-2 Introductory Seminar
- MATH 897-2 Advanced Seminar
- MATH 898-0 MSc Thesis
- MATH 899-0 PhD Thesis

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Index : searchable with the Find function in your web browser | Calendar.pdfs | Office of the Registrar / SFU |

Table of Contents : searchable with the Find function in your web browser | Course Database or Course Outlines (opens in new window) |
Financial Assistance |