Last updated: June 23, 2008 @ 09:21AM

Mathematics and Statistics

Master of Science in Mathematics and Statistics

Fields of Specialization
Mathematics
Statistics

Interim Dean
A Joffre Mercier
Faculty of Mathematics and Science

Associate Dean
Rick Cheel
Faculty of Mathematics and Science

Graduate Faculty

Professor Emeritus
Howard Bell (Mathematics)

Professors
Stephen Anco (Mathematics), Hichem Ben-El-Mechaiekh (Mathematics), Mei Ling Huang (Mathematics), Ronald A. Kerman (Mathematics), Yuanlin Li (Mathematics), Jan Vrbik (Mathematics), Thomas Wolf (Mathematics)

Associate Professors
Henryk Fuks (Mathematics), Omar Kihel (Mathematics), Wai Kong (John) Yuen (Mathematics)

Assistant Professors
Chantal Buteau (Mathematics), Babak Farzad (Mathematics), Xiaojian Xu (Mathematics)

Adjunct Professors
Vladimir Sokolov (Landau Institute)

Graduate Program Director
Henryk Fuks
hfuks@brocku.ca

General Inquiries
mathstatgrad@brocku.ca

Administrative Assistants
Margaret Thomson, Josephine McDonnell
Mackenzie Chown J415
905-688-5550, extension 3300
mathstratgrad@brocku.ca
http://www.brocku.ca/mathematics/mscprog/index.php

Program Description
The MSc program aims to provide students with an intensive advanced education in areas of Mathematics and Statistics in preparation for further graduate studies or the job market. Students will choose a specializationin either Mathematics or Statistics.

The Mathematics specialization provides students with advanced training in areas of active research and current applicability in algebra and number theory, computer algebra algorithms, dynamical systems, partial differential equations, functional analysis, mathematical music theory, solitons and integrable systems, topology, and (as a bridge with the Statistics specialization) probability theory and stochastic processes. The Statistics specialization provides students with solid training in advanced statistical analysis and in computational methods and applications to stochastic models.

The program offers two options: a thesis option (intended normally for students planning to pursue further graduate studies) and a project option (intended normally for those planning to join the job market).

Fields of Specialization
Participating faculty are engaged in active research in the following areas of specialization:

Mathematics
·   Cellular automata, discrete dynamical systems and complex networks
·   Computational methods for solving algebraic and differential systems
·   Cryptography
·   Fourier and wavelets analysis
·   Functions spaces and wavelets applied to partial differential equations
·   Graph Theory and Algorithmic Game Theory
·   Group and ring theory
·   High performance parallel computing
·   Mathematical music theory
·   Mathematical physics and General Relativity
·   Nonlinear functional analysis and applications to optimization, game theory, mathematical economics, and differential systems
·   Probability and measure theory
·   Solitons and integrability of partial differential equations
·   Symmetry analysis and computer algebra applied to nonlinear differential equations

Statistics
·   Statistical Inference Methods and Applications
·   Computational methods and applications to stochastic models
·   Convergence and Efficiency of Markov Chain Monte Carlo Algorithms

Admission Requirements
Successful completion of an Honours Bachelor's degree, or equivalent, in Mathematics, or Statistics, or a related field, with an overall average of not less than B+. Agreement from a faculty advisor to supervise the student is also required for admission to the program.

The Graduate Admissions Committee will review all applications and recommend admission for a limited number of suitable candidates.

Those lacking sufficient background preparation may be required to complete a qualifying term or year to upgrade their applications. Completion of a qualifying term or year does not guarantee acceptance into the program.

Part-time study is available.

Degree Requirements
The program requirements in Mathematics and Statistics include successful completion of core and specialization courses, and a thesis (MATH 5F90) or a project (MATH 5P99).

Students in the thesis option of either specialization are required to complete four MATH half-credit courses: two core MATH courses, one MATH specialization course, and a graduate seminar. They must also write a thesis that demonstrates a capacity for independent work of acceptable scientific calibre.

Students in the project option are required to complete six MATH half-credit courses. For students in Mathematics: four core MATH courses, one MATH specialization course, and a graduate seminar. For students in Statistics: four core MATH courses, one MATH specialization course, and an additional MATH core or specialization course. In addition, each student must complete a project under the supervision of a faculty member. The project is more practical in nature and must demonstrate a capacity for synthesis and understanding of concepts and techniques related to a specific topic.

Each student will consult with his or her Supervisor when planning a program of study and choosing courses and must receive approval from the Graduate Program Director.

Core courses for Mathematics include:
·   MATH 5P10 Modern Algebra
·   MATH 5P20 Computational Methods for Algebraic and Differential Systems
·   MATH 5P30 Dynamical Systems
·   MATH 5P35 Graph Theory
·   MATH 5P40 Functional Analysis
·   MATH 5P50 Algebraic Number Theory
·   MATH 5P60 Partial Differential Equations
·   MATH 5P70 Topology
·   MATH 5P87 Probability and Measure Theory
·   MATH 5P94 MSc Mathematics Seminar
Specialization courses for Mathematics include:
·   MATH 5P05 Introduction to Wavelets
·   MATH 5P09 Solitons and Nonlinear Wave Equations
·   MATH 5P11 Group Rings
·   MATH 5P21 Heuristic and Parallel Techniques for Algebraic and Differential Systems
·   MATH 5P31 Mathematical Models of Complex Systems
·   MATH 5P32 Mathematical Foundations of Statistical Physics
·   MATH 5P36 Algorithmic Game Theory
·   MATH 5P41 Nonlinear Functional Analysis I
·   MATH 5P42 Nonlinear Functional Analysis II
·   MATH 5P44 Wavelet Bases in Functions Spaces With Applications
·   MATH 5P61 Symmetry Analysis and Conservation Law Methods
·   MATH 5P63 Integrability and Formal Geometry of PDEs
·   MATH 5P64 Geometric Topics in Mathematical Physics
·   MATH 5P71 Advanced Topology
·   MATH 5P72 Mathematical Music Theory
·   MATH 5P84 Time Series Analysis and Stochastic Processes
·   MATH 5P92 Cryptography and Number Theory
Core courses for Statistics include:
·   MATH 5P81 Sampling Theory
·   MATH 5P82 Nonparametric Statistics
·   MATH 5P83 Mathematical Statistical Inference
·   MATH 5P84 Time Series Analysis and Stochastic Processes
·   MATH 5P85 Mathematical Statistical Inference
·   MATH 5P86 Multivariate Statistics
Specialization courses for Statistics include:
·   MATH 5P87 Probability and Measure Theory
·   MATH 5P88 Advanced Statistics
·   MATH 5P95 MSc Statistics Seminar
The MSc program is designed to normally be completed in six terms or twenty-four months. However, completion in twelve months is possible in Statistics specialization.

Facilities
Each graduate student will be provided with personal desk space and a desktop PC linked to the university network system. In addition, graduate students will have access to the Mathematics computer lab classroom equipped with 33 PCs as well as to computer labs located in the vicinity of the Mathematics Department. Software includes Maple 8 and 10, MatLab, Visual Studio, SAS, S-PLUS, SPSS and Minitab, computer language compilers (C, C++, Fortran), Tex/Latex. Students will have access to advanced computing facilities that include
·   a number of dedicated local servers running special software
·   a departmental 6-node cluster
·   a 42-node parallel (Beowulf) cluster of 3 GHz Pentium IV computers
·   20 computer clusters (with more than 8000 CPUs in total) in the Ontario
High Performance Computing consortium SHARCNET, ranging from highly connected clusters to SMP (Shared Memory Processor) computers with large memory and 70TB of disk space.

Course Descriptions
Note: Not all courses are offered in every session. Students must consult with the Graduate Program Director regarding course offerings and course selection and must have their course selections approved by the Graduate Program Director each term. Refer to the Timetable for scheduling information:
http://www.brocku.ca/registrar/guides/grad/timetable/terms.php
MATH 5F90
MSc Thesis
A research project involving the preparation of a thesis which will demonstrate a capacity for independent work. The research shall be carried out under the supervision of a faculty member.

MATH 5P05
Introduction to Wavelets
An introduction to wavelets in the context of Fourier Analysis. Topics include inner product spaces, Fourier series, Fourier transform, Haar wavelet analysis, multiresolution analysis: linear and cubic spline wavelets, Daubechies wavelets, convergence theorems, numerical Fourier and wavelet analysis, wavelet-Glaerkin numerical solution of ordinary differential equations.
Note: taught in conjunction with MATH 4P05.

MATH 5P09
Solitons and Nonlinear Wave Equations
Introduction to solitons: travelling waves, nonlinear wave and evolution equations (Korteweg de Vries, Bousinesq, nonlinear Schrodinger, sine-Gordon), soliton solutions and their interaction properties, Lax pairs and construction of single and multi soliton solutions.
Note: taught in conjunction with MATH 4P09.

MATH 5P10
Modern Algebra
Advanced group theory and ring theory, such as group actions, p-groups and Sylow subgroups, solvable and nilpotent groups, FC-groups, free groups, finiteness conditions in rings, semisimplicity, the Wedderburn-Artin theorem, the Jacobson radical, rings of algebraic integers.

MATH 5P11
Group Rings
An introduction to group rings. Group rings and their unit groups, augmentation ideals, algebraic elements, several important types of units, isomorphism problem, free groups of units.

MATH 5P20
Computational Methods for Algebraic and Differential Systems
Computer algebra applications of solving polynomial systems of algebraic and differential systems of equations are covered, including the necessary algebraic background. Polynomials and ideals,Groebner bases, affine varieties, solving by elimination, Groebner basis conversion, solving equations by resultants, differential algebra, differential Groebner bases.

MATH 5P21
Heuristic and Parallel Techniques for Algebraic and Differential Systems
Heuristic methods in simplifying systems, parallel computer algebra applied to very non-linear problems, safety aspects of large computations, human-machine interface issues for handling large systems.

MATH 5P30
Dynamical Systems
Introduction to dynamical systems and their applications in mathematical modelling. Linear flows, local theory of nonlinear flows, linearization theorems, stable manifold theorem. Global theory: limit sets and attractors, Poincare´-Bendixson theorem. Structural stability and bifurcations of vector fields. Low dimensional phenomena in discrete dynamics. Chaotic dynamics: routes to chaos, characterization of chaos and strange attractors.

MATH 5P31
Mathematical Models of Complex Systems
Bottom-up approach to mathematical modeling. Cellular automata and agent-based models: rules, approximate methods, kinetic growth phenomena, site-exchange automata. Networks: graphs, random networks, small-world networks, scale-free networks, dynamics of network models. Additional topics may include power-law distributions in complex systems, self-organized criticality, phase transitions, and critical exponents.

MATH 5P32
Mathematical Foundations of Statistical Physics
The phase space of a mechanical system, theorems of Liouville and Birkhoff, ergodic problem; statistical mechanics as probability theory with constraints; the concept of temperature in thermal and non-thermal systems; phase transitions and critical behavior, spin models, scaling; renormalization group theory; phase transitions in percolation models, calculations of critical exponents, open problems.

MATH 5P35
Graph Theory
Basic definitions, paths and cycles, connectivity, trees and forests, bipartite graphs, Eulerian graphs; Matchings in bipartite graphs and in general graphs; Planar graphs, Euler's formula and Kuratowski's theorem. Graph colourings, Brooks' and Vizing's theorem and colouring of planar graphs; Network flows, Min-Max Theorem.

MATH 5P36
Algorithmic Game Theory
Basic definitions, games, strategies, costs and payoffs, equilibria, cooperative games; Complexity of finding Nash equilibria; Mechanism design; Combinatorial auctions; Profit maximization in mechanism design; Cost sharing; Online mechanisms; Inefficiency of equilibria; Selfish routing; Network formation games; Potential function method; The price of anarchy; Sponsored search auctions.

MATH 5P40
Functional Analysis
The basic theory of Hilbert spaces, including the Projection Theorem, the Riesz Representation Theorem and the weak topology; weak derivatives, Sobolev spaces and the Sobolev Imbedding Theorem; the variational formulation of boundary value problems for ordinary and partial differential equations, the Lax-Milgram Lemma and its applications; the finite element method.

MATH 5P41
Nonlinear Functional Analysis I
An introduction to the theory of linear monotone operators and their applications to linear differential equations. Topics include variational problems; the Ritz method; the Galerkin method for differential and integral equations; Hilbert space methods and linear elliptic, parabolic, and hyperbolic differential equations.

MATH 5P42
Nonlinear Functional Analysis II
An introduction to the theory of nonlinear monotone operators and their applications to nonlinear differential equations. Topics include monotone and pseudo-monotone operators, applications to quasi-linear elliptic differential equations, noncoercive equations, nonlinear Fredholm aternative, maximal accretive operators, nonexpansive semi-groups and first order evolution equations, maximal monotone operators and applications to integral equations and to first and second order evolution equations.

MATH 5P44
Wavelet Bases in Functions Spaces With Applications
Wavelet bases in Sobolev and Besov spaces and their applications to the numerical solution of PDEs and statistical estimation. Topics include an overview of Lebesgue integration, Lp-spaces, weak differentiability and Sobolev spaces, Besov spaces, wavelet expansions in Sobolev and Besov spaces, Galerkin wavelet methods for the resolution of elliptic problems in bounded domains, density estimation.

MATH 5P50
Algebraic Number Theory
An introduction to algebraic aspects of number theory. Topics include the general theory of factorization of ideals in Dedekind domains and number fields, Kummer's theory on lifting of prime ideals in extension fields, factorization of prime ideals in Galois extensions, local fields, the proof of Hensel's lemma, arithmetic of global fields.

MATH 5P60
Partial Differential Equations
Heat equation, wave equation, basic existence and uniqueness theory of parabolic and hyperbolic linear PDEs, fundamental solutions, introduction to weak solutions and Sobolev spaces, analysis of nonlinear evolution equations, exact solution techniques and formal geometric properties (symmetries and conservation laws).

MATH 5P61
Symmetry Analysis and Conservation Law Methods
Overview of computational methods and theory for symmetry and conservation law analysis of differential equations. Noether's theorem, characteristic form and determining equations for symmetries and conservation laws, computer algebra programs, applications to nonlinear ODEs and evolutionary PDEs.

MATH 5P63
Integrability and Formal Geometry of PDEs
Properties of integrable equations and soliton solutions, recursion operators, bi-Hamiltonian structures, connections with classical differential geometry, classification of integrable evolution equations, advanced symmetry and conservation law classification problems, applications to nonlinear PDEs in applied mathematics and mathematical physics.

MATH 5P64
Geometric Topics in Mathematical Physics
Classical aspects of Yang-Mills equations and nonlinear gauge fields. General Relativity theory and black hole spacetimes. Killing tensors and spinors, twistors. Geometrical algebraic approach to classical mechanics and special relativity using quaternions. Advanced aspects of mechanics (Poisson brackets, symplectic manifolds, Hamiltonian dynamics, Lie-Poisson structures).

MATH 5P70
Topology
An introduction to point set topology concepts and principles. Metric spaces; topological spaces; continuity, compactness; connectedness; countability and separation axioms; metrizability; completeness; Baire spaces.

MATH 5P71
Advanced Topology
An introduction to topological fixed point theory with applications to differential systems and game theory. Topics include the theorems of Brouwer, Borsuk, Schauder-Tychnoff, and Knaster-Kuratowski-Mazurkiewicz; fixed points and equilbria for set-valued maps; existence and qualitative properties of differential systems; Min-max theorems and Nash equilibria.

MATH 5P72
Mathematical Music Theory
An introduction to mathematical music theory. Topics may include: category theory and local and global compositions; general topology and (music) metric and motive structures; group theory and rhythmic canons, group theory and (music) set theory; diophantine analysis and tone systems.

MATH 5P81
Sampling Theory
Theory of finite population sampling; simple random sampling; sampling proportion; estimation of sample size; Stratified sampling; optimal allocation of sample sizes; ratio estimators; regression estimators; systematic and cluster sampling; multi-stage sampling; error in surveys; computational techniques and computer packages, and related topics.
Note: taught in conjunction with MATH 4P81.

MATH 5P82
Nonparametric Statistics
Order statistics; rank tests and statistics; methods based on the binomial distribution; contingency tables; Kolmogorov-Smirnov statistics; nonparametric analysis of variance; nonparametric regression; comparisons with parametric methods; computational techniques and computer packages, related topic.
Note: taught in conjunction with MATH 4P82.

MATH 5P83
Linear Models
Classical linear model, generalized inverse matrix, distribution and quadratic forms, regression model, nested classification and classification with interaction. covariance analysis, variance components, binary data, polynomial data, log linear model, linear logit models, generalized linear model, conditional likelihoods, quasi-likelihoods, estimating equations, computational techniques and related topics.

MATH 5P84
Time Series Analysis and Stochastic Processes
Time series, trend, seasonality and error, theory of stationary processes, spectral theory, Box-Jenkins methods, theory of prediction, inference and forecasting. ARMA and ARIMA processes, vector time series models, state space models, Markov processes, renewal process, martingales, Brownian motion, diffusion processes, branching processes, queueing theory, stochastic models, computational techniques and related topics.

MATH 5P85
Mathematical Statistical Inference
Revision of probability theory, convergence of random variables, statistical models, sufficiency and ancillarity, point estimation, likelihood theory, optimal estimation, Bayesian methods, computational methods, minimum variance estimation, interval estimation and hypothesis testing, linear and generalized linear models, goodness-of-fit for discrete and continuous data, robustness, large sample theory, Bayesian inference.

MATH 5P86
Multivariate Statistics
Theory of multivariate statistics, matrix algebra and random vector, sample geometry and random sampling, multivariate normal distribution, inference about means, covariance matrix, generalized Hotelling's T2 distribution, sample covariance and sample generalized variance, Wishart distribution, general hypothesis testing, analysis of variance and linear regression model, principle components, factor analysis, covariance analysis, canonical correlation analysis, discrimination and classification, cluster analysis and related topics.

MATH 5P87
Probability and Measure Theory
An introduction to a rigorous treatment of probability theory using measure theory. Topics include probability measures, random variables, expectations, laws of large numbers, distributions and discrete Markov chains. Selected topics from weak convergence, characteristic functions and the Central Limit Theorem.

MATH 5P88
Advanced Statistics
Topics may vary year to year. Advanced methods and theory in statistical inference, survival analysis, risk analysis, sampling techniques, bootstrapping, Jackknife, generalized linear models, mixed models, modern computational statistics, quality control, life data modeling, biostatistics, multivariate analysis, time series analysis and related topics.

MATH 5P92
Cryptography and Number Theory
Topics may include RSA cryptosystems, ElGamal cryptosystem, algorithms for discrete logarithmic problem, elliptic curves, computing point multiples on elliptic curves, primality testing and factoring algorithms.
Note: taught in conjunction with MATH 4P92.

MATH 5P94
Mathematics Seminar
Independent study and presentation of major research papers in areas of specialization.

MATH 5P95
Statistics Seminar
Independent study and presentation of major research papers in areas of specialization.

MATH 5P99
Project
Students will complete a survey paper on a topic chosen in consultation with a supervisor from one of the research areas of specialization.

MATH 5V75-5V79
Selected Topics in Mathematics and Statistics
An investigation of a specific area or group of related topics in mathematics or statistics.