Mathematics and Statistics

Master of Science in Mathematics and Statistics

Fields of Specialization

Mathematics

Statistics

Dean

Peter Berg

Faculty of Mathematics and Science

Associate Dean

Melanie Pilkington

Faculty of Mathematics and Science

Core Faculty

Professors

S. Ejaz Ahmed, Stephen Anco, Chantal Buteau, Hichem Ben-El-Mechaiekh, Henryk Fuks, Omar Kihel, Yuanlin Li, Alexander Odesskii, Jan Vrbik, Thomas Wolf, Xiaojian Xu

Associate Professor

William Marshall

Assistant Professor

Tianyu Guan, Dongchen Li, Pouria Ramazi

Participating Faculty

Adjunct Professor

Vladimir Sokolov (Landau Institute)

Professors Emeriti

Howard Bell, Mei Ling Huang, Ronald Kerman, Eric Muller

Graduate Program Director

Stephen Anco

sanco@brocku.ca

Graduate Administrative Coordinator

Elena Genkin

905-688-5550, extension 3115

Mackenzie Chown D473

fmsgradoffice@brocku.ca

Administrative Assistant

Alan Liu

905-688-5550, extension 3300

Mackenzie Chown J415

mathstats@brocku.ca

brocku.ca/mathematics-science/mathematics/graduate-programs/

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 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.

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). The paper for the project option 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 their Supervisor when planning a program of study and choosing courses and must receive approval from the Graduate Program Director.

Field(s) of Specialization

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

Mathematics

Statistics

Admission Requirements

Degree Requirements

Mathematics

Thesis Option:

	MATH 5F90 (Thesis)
	Two Courses from List A: MATH 5P10, MATH 5P35, MATH 5P50, MATH 5P66
	Two Courses from List B: MATH 5P20, MATH 5P30, MATH 5P40, MATH 5P60
	One Additional 0.5 Credit MATH or STAT 5(alpha)XX course

Major Research Paper Option:

	MATH 5P99 (MRP)
	Two Courses from List A: MATH 5P10, MATH 5P35, MATH 5P50, MATH 5P66
	Two Courses from List B: MATH 5P20, MATH 5P30, MATH 5P40, MATH 5P60
	Two Additional 0.5 Credit MATH or STAT 5(alpha)XX courses

Statistics

Thesis Option:

	STAT 5F90 (Thesis)
	Two Core Courses from: STAT 5P81, STAT 5P82, STAT 5P83, STAT 5P84, STAT 5P85, STAT 5P86
	Two Courses from: STAT 5P87, STAT 5P88, STAT 5P89, STAT 5P95

Major Research Paper Option:

	STAT 5P99 (MRP)
	Four Core Courses from: STAT 5P81, STAT 5P82, STAT 5P83, STAT 5P84, STAT 5P85, STAT 5P86
	One Course from: STAT 5P87, STAT 5P88, STAT 5P89, STAT 5P95
	One Additional 0.5 Credit STAT or MATH 5(alpha)XX course

Courses by Specialization:

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 and Nonlinear Wave Equations

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 5P75 Theory of Financial Mathematics

MATH 5P92 Cryptography

MATH 5P96 Technology and Mathematics Education

Statistics

Core courses:

STAT 5P81 Sampling Theory

STAT 5P82 Nonparametric Statistics

STAT 5P83 Linear Models

STAT 5P84 Time Series Analysis and Stochastic Processes

STAT 5P85 Mathematical Statistical Inference

STAT 5P86 Multivariate Statistics

Non-core courses:

STAT 5P87 Computational Statistics

STAT 5P88 Advanced Statistics

STAT 5P89 Bayesian and Causal Bayesian Networks

STAT 5P95 MSc Statistics Seminar

Facilities

Course Descriptions

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

Students must 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.

MATHEMATICS COURSES

*MATH 5F90

MSc Thesis

(also offered as STAT 5F90)

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

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

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

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

MATH 5P75

Theory of Financial Mathematics

Mathematical models arising in modern investment practices. Compound interest, annuities, the time-value of money, Markowitz portfolio theory, efficient frontier, random walks, Brownian processes, future contracts, European and American options, and put-call parity. Introduction to Black-Scholes.

Note: Taught in conjunction with MATH 3P75

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 5P96

Technology and Mathematics Education

Critical appraisal of mathematics learning and teaching with digital technology, such as computer algebra systems and dynamic geometry software. Topics may include contemporary research concerning impact on curriculum; use in assessment; implementation in the classroom; mathematics computer games; use in modeling, probability, and statistics; data science and computational thinking; and the design of a rich technology-based mathematical activity.

Note: taught in conjunction with MATH 4P96.

*MATH 5P99

Major Research Paper

(also offered as STAT 5P99)

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

Investigation of a specific area or group of related topics in mathematics or statistics.

STATISTICS COURSES

#STAT 5F90

MSc Thesis

(also offered as MATH 5F90)

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.

STAT 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 STAT 4P81

STAT 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 STAT 4P82.

STAT 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): STAT 3P86 (or equivalent) or permission of the instructor.

STAT 5P84

Time Series Analysis and Stochastic Processes

Multivariate, marginal and conditional Normal distributions; partial correlation coefficient. Probability density function of sample correlation coefficient. Autoregressive and Moving Average models; stability analysis; serial correlation; forecasting; maximum likelihood estimation. Spectral theory; spectrum of ARMA models; aliasing. Spectral analysis of time-series data; periodogram; data filtering; spectrum smooting by spectral or lag window; using a kernel. Estimating serial correlation; large-sample theory.

STAT 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.

STAT 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): STAT 3P86 (or equivalent) or permission of the instructor.

STAT 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): STAT 3P82 and STAT 3P86 (or their equivalences), or permission of the instructor.

STAT 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.

#STAT 5P89

Bayesian and Causal Bayesian Networks

(also offered as STAT 4P89)

Representation (factorizing joint probabilistic distributions, exploiting probabilistic independence properties, d-separation, I-Map, naive Bayes); parameter learning (maximum likelihood estimation, Bayesian parameter estimation); structure learning (constrained-based approaches, score-based approaches, Bayesian model averaging); learning with incomplete data (parameter and structure learning, learning with hidden variables); causal Bayesian networks (intervention, SCMs, inference, learning). Use of R or Python.

Prerequisite(s): STAT 2P81, STAT 3P85, or the permission of the instructor.

STAT 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.

#STAT 5P99

Major Research Paper

(also offered as MATH 5P99)

Survey paper on a topic chosen in consultation with a supervisor from one of the research areas of specialization.