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Fields of Specialization Mathematics Statistics Dean Ian D. Brindle Faculty of Mathematics & Science Associate Dean Greg Finn Faculty of Mathematics & Science Graduate Faculty Professor Emeritus Howard Bell (Mathematics) Professors Hichem Ben-El-Mechaiekh (Mathematics), Mei Ling Huang (Mathematics), Ronald A. Kerman (Mathematics), Jan Vrbik (Mathematics), Thomas Wolf (Mathematics) Associate Professors Stephen Anco (Mathematics), Henryk Fuk (Mathematics), Omar Kihel (Mathematics), Yuanlin Li (Mathematics) Assistant Professors Chantal Buteau (Mathematics), Xiaojian Xu (Mathematics), Wai Kong (John) Yuen (Mathematics) Adjunct Professors Vladimir Sokolov (Landau Institute) Graduate Program Director Stephen Anco General Inquiries E-mail: mathstatgrad@brocku.ca Administrative Assistants Margaret Thomson, Josephine McDonnell Mackenzie Chown J415 905-688-5550, extension 3300 |
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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 concentration in either Mathematics or Statistics. The Mathematics concentration 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 Concentration) probability theory and stochastic processes. The Statistics concentration 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). |
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Participating faculty are engaged in active research in the following areas of specialization: |
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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. |
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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 are required to complete four half-credit courses: two core concentration courses, one 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 half-credit courses. For students in Mathematics, four core concentration courses, one specialization course, and a graduate seminar; for students in Statistics, four core concentration courses, one specialization course, and an additional 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 the Mathematics concentration include:
Specialization courses for the Mathematics concentration include:
Core courses for the Statistics concentration include:
Specialization courses for the Statistics concentration include:
The normal duration of the M.Sc. program is twenty-four months. However, completion in twelve months is possible in the Statistics concentration. |
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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
High Performance Computing consortium SHARCNET, ranging from highly connected clusters to SMP (Shared Memory Processor) computers with large memory and 70TB of disk space. |
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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 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. 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. 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. 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. 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. 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. 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. 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 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. 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. 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. 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 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). 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. 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. 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). 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. 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. 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. 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. 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. 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. 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 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. 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. MSc Mathematics Seminar Independent study and presentation of major research papers in areas of specialization. MSc Statistics Seminar Independent study and presentation of major research papers in areas of specialization. MSc Project Students will complete a survey paper on a topic chosen in consultation with a supervisor from one of the research areas of specialization. |
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2007-2008 Graduate Calendar
Last updated: January 22, 2008 @ 01:03PM