Last updated: March 19, 2020 @ 01:08PM

Mathematics and Statistics

Master of Science in Mathematics and Statistics

Fields of Specialization
Mathematics
Statistics

Dean
S. Ejaz Ahmed
Faculty of Mathematics and Science

Associate Dean
Cheryl McCormick
Faculty of Mathematics and Science

Core Faculty

Professors
S. Ejaz Ahmed (Mathematics and Statistics), Stephen Anco (Mathematics and Statistics), Chantal Buteau (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), Xiaojian Xu (Mathematics and Statistics)

Assistant Professor
William Marshall (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
Henryk Fukś
hfuks@brocku.ca

Graduate Administrative Coordinator
Elena Genkin
905-688-5550, extension 3115
Mackenzie Chown D473
fmsgradoffice@brocku.ca

Administrative Assistant
Jayne Zarecky
905-688-5550, extension 3300
Mackenzie Chown J415
jzarecky@brocku.ca
https://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 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, discrete mathematics and graph theory, dynamical systems, partial differential equations, functional analysis, mathematical music theory, mathematics education, mathematical physics, 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 both advanced and applied statistical areas including design of experiment, optimal design for regression, sampling theory, parametric and nonparametric statistical inferences, multivariate statistics, survival analysis and risk models, robust methods, and computational statistics.

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:

Mathematics
Algebraic number theory
Cellular automata, discrete dynamical systems and complex networks
Combinatorial and additive number theory
Computational methods for solving algebraic and differential systems
Cryptography
Geometric curve flows
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
Solitons and integrability of partial differential equations
Symmetry analysis and computer algebra applied to nonlinear differential equations

Statistics
Nonparametric Statistical Inference Theory and Methods
Extreme Value Theory and Applications
Quantile Regression Method and Applications
Experimental Design and Regression Theory and Methods
Applied Probability, Stochastic Models and Queueing Network
Survival Analysis and Risk Models
Probability Distribution Theory and Applications
Monte Carlo Simulations and Resampling Techniques
Multivariate Analysis
Accelerated Life Testing
Linear, Generalized Linear, and Nonlinear Models
Robust Statistics
Computational methods and applications to stochastic models
Convergence and Efficiency of Markov Chain Monte Carlo Algorithms
Optimal Design of Experiments

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 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.

Mathematics

Students in the thesis option of Mathematics are required to complete five MATH 5(alpha)00 half-credit courses, including at least two core courses from List A and at least two core courses from List B. Additionally, they must complete a thesis (MATH 5F90) that demonstrates a capacity for independent work of acceptable scientific calibre.

Students in the major research paper option of Mathematics are required to complete six MATH 5(alpha)00 half-credit courses, including at least two core courses from List A and at least two core courses from List B. Additionally, they must complete a major research paper (MATH 5P99). 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 relatd to a specific topic.

Statistics

Students in the thesis option of Statistics are required to complete four MATH 5(alpha)00 half-credit courses, including at least two core courses in statistics. Additionally, they must complete a thesis (MATH 5F90) that demonstrates a capacity for independent work of acceptable scientific calibre.

Students in the major research paper option of Statistics are required to complete six MATH 5(alpha)00 half-credit, including at least four core courses in statistics and one non-core course in statistics. Additionally, they must complete a major research paper (MATH 5P99). 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.

Mathematics

List A Core courses:
MATH 5P10 Groups, Rings, and Group Rings
MATH 5P35 Graph Theory
MATH 5P50 Algebraic Number Theory
MATH 5P66 Matrix Groups and Linear Representations

List B Core courses:
MATH 5P20 Computational Methods for Algebraic and Differential Systems
MATH 5P30 Dynamical Systems
MATH 5P40 Functional Analysis
MATH 5P60 Partial Differential Equations

Non-core courses:
MATH 5P09 Solitons
MATH 5P11 Advanced Algebraic Structures
MATH 5P31 Ergodicity, Entropy and Chaos
MATH 5P36 Algorithmic Game Theory
MATH 5P41 Nonlinear Functional Analysis
MATH 5P64 Differential Geometry and Mathematical Physics
MATH 5P70 Topology
MATH 5P92 Cryptography
MATH 5P96 Technology and Mathematics Education

Statistics

Core courses:
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

Non-core courses:
MATH 5P87 Computational Statistics
MATH 5P88 Advanced Statistics
MATH 5P95 MSc Statistics Seminar

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 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

Note that not all courses are offered in every session. Refer to the applicable timetable for details.

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.
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 5P09
Solitons and Nonlinear Wave Equations
(also offered as PHYS 5P09)
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.

MATH 5P10
Groups, Rings, and Groups Rings
Advanced group theory and ring theory, such as group actions, p-groups and Sylow subgroups, solvable and nilpotent groups, finiteness conditions in rings, semisimplicity, the Wedderburn-Artin theorem. Introduction to group rings, such as unit groups, augmentation ideals, several important types of units, and the isomorphism problem.

MATH 5P11
Advanced Algebraic Structures
Topics may include: Algebraic coding theory; Combinatorial group theory; Advanced structures in ring theory.
Note: Taught in conjunction with either 4P11 or 4P13

MATH 5P20
Computational Methods for Algebraic and Differential Systems
(also offered as PHYS 5P20)
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
High Performance 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.

MATH 5P30
Dynamical Systems
(also offered as PHYS 5P68)
Introduction to dynamical systems and their applications in mathematical modeling. 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
Ergodicity, Entropy and Chaos
Introduction to ergodic theory, invariant measures, Birkhoff ergodic theorem. The first return formula, Kac's lemma, recurrence theorems. Entropy, coding maps, Shannon-McMillian Breiman theorem. Chaos, predictability, Lyapunov exponent, speed of divergence, Pesin theorem. Ergodicity, entropy and chaos in shift dynamical systems and cellular automata.

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
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.

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
(also offered as PHYS 5P60)
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.

MATH 5P64
Differential Geometry and Mathematical Physics
(also offered as PHYS 5P64)
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.

MATH 5P66
Matrix groups and linear representations
(also offered as PHYS 5P66)
Abelian groups, permutation groups, rotation groups. Representations of discrete and continuous groups by linear transformations (matrices). General properties and constructions of group representations. Representations of specific groups. Lie groups and Lie algebras. Applications in various areas of Mathematics, including invariant theory and group algebras, and Theoretical Physics, including crystallography and symmetries in quantum systems.

MATH 5P69
Introduction to Scientific Computing
(also offered as PHYS 5P10)
Survey of computational methods and techniques commonly used in condensed matter physics research; use of common subroutine libraries; symbolic computing systems; case studies from various areas of computational science; an independent- study term project. Use of graphing and visualization software. Numerical differentiation and integration. Use of special functions. Monte Carlo and molecular dynamics simulation of structure, energetic and thermodynamic properties of metallic, semiconducting and ionic solids and nanoparticles.

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 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. Case studies.
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 use of SAS, Maple or other statistical packages, Case Studies.
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.
Prerequisite(s): MATH 3P86 (or equivalent) or permission of the instructor.

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 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.

MATH 5P87
Computational Statistics
Classification: logistic regression, linear and quadratic discriminant analysis. Resampling methods: cross-validation and bootstrap. Linear model selection and regularization: subset selection, shrinkage methods, dimension reduction methods, considerations in high dimensions. Nonlinear regression: polynomial regression, regression splines, smoothing splines, local regression and generalized additive models. Tree-based methods: decision trees, bagging, random forests, and boosting. Support vector machines: maximal margin classifier, support vector classifier, support vector machines (SVMs), SVMs with more than two classes. Unsupervised learning: principal component analysis, clustering methods.
Prerequisite(s): MATH 3P82 and MATH 3P86 (or their equivalences), or permission of the instructor.

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 5P95
Statistics Seminar
Independent study and presentation of major research papers in areas of specialization.
Note: this course will be evaluated as Credit/No-Credit.

MATH 5P96
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.

MATH 5P99
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.