Last updated: July 7, 2015 @ 03:56PM

Mathematics and Statistics

Master of Science in Mathematics and Statistics

Fields of Specialization

S. Ejaz Ahmed
Faculty of Mathematics and Science

Associate Dean
Alan Castle
Faculty of Mathematics and Science

Core Faculty

S. Ejaz Ahmed (Mathematics and Statistics), Stephen Anco (Mathematics and Statistics), Hichem Ben-El-Mechaiekh (Mathematics and Statistics), Henryk Fuks (Mathematics and Statistics), Mei Ling Huang (Mathematics and Statistics), Omar Kihel (Mathematics and Statistics), Yuanlin Li (Mathematics and Statistics), Alexander Odesskii (Mathematics and Statistics), Jan Vrbik (Mathematics and Statistics), Thomas Wolf (Mathematics and Statistics)

Associate Professors
Chantal Buteau (Mathematics and Statistics), Babak Farzad (Mathematics and Statistics), Xiaojian Xu (Mathematics and Statistics), Wai Kong (John) Yuen (Mathematics and Statistics)

Participating Faculty

Associate Professor
Bill Ralph (Mathematics and Statistics)

Adjunct Professor
Vladimir Sokolov (Landau Institute)

Professors Emeriti
Howard Bell (Mathematics and Statistics), Ronald Kerman (Mathematics and Statistics), Eric Muller (Mathematics and Statistics

Graduate Program Director
Stephen Anco

Administrative Assistant
Margaret Thomson
905-688-5550, extension 3300
Mackenzie Chown J415

Program Description
The MSc program aims to provide students with an intensive advanced education in areas of Mathematics and Statistics in preparation for doctoral studies or the job market. Students will choose a specialization in 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, mathematics education, 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 doctoral studies) and a project option (intended normally for those planning to join the job market).

Field(s) of Specialization
Participating faculty are engaged in active research in the following areas of specialization:

Cellular automata, discrete dynamical systems and complex networks
Computational methods for solving algebraic and differential systems
Graph Theory and Algorithmic Game Theory
Group and ring theory
High performance parallel computing
Mathematical music theory
Mathematical physics and General Relativity
Mathematics education
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

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 four year Bachelor's degree, or equivalent, in Mathematics or Statistics, or a related field, with an average of not less than B+ average. 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 Major Research Paper (MATH 5P99).

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.

Students in the thesis option of either specialization are required to complete four MATH 5(alpha)00 half-credit courses, including at least two MATH 5(alpha)00 core MATH courses. Additionally, they must register in Math 5P90 and write a Thesis that demonstrates a capacity for independent work of acceptable scientific calibre.

Students in the major paper option of either specialization are required to complete six MATH 5(alpha)00 half-credit courses. Additionally, they must register for at least one term in MATH 5P99. For students in Mathematics, at least four of the courses must be core courses in mathematics. The remaining two can be any MATH 5(alpha)00 level half-credit courses in mathematics. For students in Statistics, courses must include four MATH 5(alpha) 00 level core courses in statistics and one MATH 5(alpha)00 level specialization course in statistics. The remaining course can be either a MATH 5(alpha)00 level core course in statistics or a MATH 5(alpha)00 level specialization course in statistics.

Students enrolled in MATH 5P99 must complete a Major Research Paper under the supervision of a faculty member. The paper will be based on a research project of a practical 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

Core courses for Statistics include:
MATH 5P81 Sampling Theory
MATH 5P82 Nonparametric Statistics
MATH 5P83 Linear Models
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

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 as well as to computer labs located in the vicinity of the Mathematics Department. Software includes a wide array of both commercial and open source applications for supporting research in mathematics and statistics.
Brock is also a full member of the SHARCNET consortium with access to all its high performance clusters of powerful workstations and vast storage resources.

Course Descriptions

Students must check to ensure that prerequisites are met. Students may be deregistered, at the request of the instructor, from any course for which prerequisites and/or restrictions have not been met.
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.

Solitons and Nonlinear Wave Equations
Introduction to solitons: Linear and nonlinear travelling waves. Nonlinear evolution equations (Korteweg de Vries, nonlinear Schrodinger, sine-Gordon). Soliton solutions and their interaction properties. Lax pairs, inverse scattering, zero-curvature equations and Backlund transformations, Hamiltonian structures, conservation laws.
Note: taught in conjunction with MATH 4P09.

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.

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.

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.

High Performanace Computing
Parallel computing architectures, new programming models, pilot parallel framework, parallel programming with MPI, thread-based parallelism, and a final project regarding the application of parallel computing to a mathematical problem.
Note: Students entering this course are expected to have a good grasp of basic procedural programming in a language such as C or FORTRAN.

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.

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.

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.

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.

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.

Nonlinear Functional Analysis
Topological fixed point theory with applications to dynamical systems and optimization. Topics include the theorems of Brouwer, Borsuk, Schauder-Tychnoff, and Kakutani as well as the Knaster-Kuratowski-Mazurkiewicz principle. Applications of these landmark results to the solvability and qualitative analysis of dynamical systems as well as convex and non-convex optimization are discussed.

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.

Partial Differential Equations
Review of linear and nonlinear equations in two variables. Existence and uniqueness theory, fundamental solutions, initial/boundary-value formulas for the heat equation, wave equation, Laplace equation in multi-dimensions. Exact solution techniques for 1st and 2nd order linear and nonlinear equations. Analysis of solutions, variational formulations, conservation laws, Noether's theorem.

Integrable Systems
Symmetries and conservation laws of differential equations. Basic examples of integrable differential equations. Connections with classical differential geometry. Pseudo-differential operators, Lax representations, and applications to integrability theory.

Differential Geometry and Mathematical Physics
Topics may include: Lagrangian and Hamiltonian mechanics, field theory, differential geometric structures, Lie groups and Lie algebras, G-bundles, manifolds, introduction to algebraic topology. Applications to theoretical physics.

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

Advanced Topology
Introduction to algebraic topology and homology theory. Simplicial homology. The Lefschetz-Hopf and the Borsuk-Hirch Theorems. The Brouwer and Leray-Schauder degrees and fixed points on absolute neighborhood retracts. Singular homology and Lefschetz fixed point theory. Finite codimensional Cech cohomology, Vietoris fractions and coincidence theorems.

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. Case studies.
Note: taught in conjunction with MATH 4P81.

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 use of SAS, Maple or other statistical packages, Case Studies.
Note: taught in conjunction with MATH 4P82.

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.

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.
Note: Math 5P84 has been approved by the VEE (Validation by Education Experience) Administration Committee of the Society of Actuaries. To receive VEE credit, candidates will need a grade of 70 or better.

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.

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 T-square 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.
Prerequisite(s): MATH 3P86 (or equivalent) or permission of the instructor.

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.

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.

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.

Mathematics Seminar
Independent study and presentation of major research papers in areas of specialization.
Note: this course will be evaluated as Credit/No-Credit.

Statistics Seminar
Independent study and presentation of major research papers in areas of specialization.
Note: this course will be evaluated as Credit/No-Credit.

Technology and Mathematics Education
Topics may include contemporary research concerning digital technologies, such as computer algebra systems and Web 2.0, in learning and teaching mathematics; design of educational tools using VB.NET, HTML, Geometer's Sketchpad, Maple, Flash, etc.; critical appraisal of interactive learning objects in mathematics education.
Note: taught in conjunction with MATH 4P96.

Major Research Paper
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.