Master of Science

in Mathematics and Statisticsat Brock University

What we offer

Tradition of individual excellence in publications andresearch. In terms of research output, the Department iscurrently in the top three in its category among Ontariouniversities as ranked by Canadian University Publications.

One-to-one interaction with faculty members (currently wehave ∼20 graduate students and 15 faculty members)

Competitive financial support

Location in the heart of wine country, close to NiagaraFalls, 120 km from Toronto and 50 km from Buffalo (US)

Why come to Brock

One of the goals of our program is to ensure that studentssucceed after graduation, either by getting accepted to aPh.D. program or by obtaining a satisfactory employment.

Our graduates who pursue careers in industry enjoy veryhigh employment rate. Many work for companies in fieldsas diverse as insurance, finance, medical and technologicalimaging, software development, banking, marketing, sales,etc. Examples of their job titles are statistical analyst,senior data analyst, R&D manager, risk modelling analyst,software developer, general manager, market analyst,actuarial analyst, academic sales representative, assistantmanager, etc.

Nearly 25% of our M.Sc. graduates pursue Ph.D. in otherinstitutions after graduation.

Structure of M.Sc. program

Two streams to choose from: thesis stream and projectstream

Project stream: six courses and research project

Thesis stream: four courses and thesis.

In spring/summer term students work on research projector thesis under the supervision of faculty advisor

Fields of specialization

We offer two fields of specialization, Mathematics andStatistics.

Statistics: typical duration 1.5 year, normally projectstream

Mathematics: typical duration 2 years, project or thesis

Admission requirements

Minimal admission requirements to M.Sc. program inMathematics and Statistics:

Honours Bachelor’s degree (or foreign equivalent) inmathematics, statistics, or related field

Overall average not less than B+ (75%)

Foreign students: English proficiency (see http:

//brocku.ca/nextstep/international-students/

english-language-proficiency/ for details)

Students who do not meet these formal requirements butbelieve that their background may nevertheless be sufficient toqualify for admission are also encouraged to apply.

Admission procedure

Admission is conducted entirely online:https://brocku.ca/programs/graduate/msc-math/Application deadline: February 1 for the September entry (ourmain entry point). We also accept applications for January entry,the deadline is November 30.Applications arriving after the deadline will be considered whenreceived until the program is full. You will need:

three letters of reference

transcript(s)

statement of interest

information form, on which you need to indicate potentialadvisor and field(s) of specialization you are interested in

application fee

for international students: proof of English proficiency

Choice of potential specialization and advisor

At the time of application, you have to fill out Mathematics and Statistics InformationForm (download at https://brocku.ca/programs/graduate/msc-math/) whichincludes field like this:

In order to help you to choose potential areas of interest and names of potentialsupervisors, the following slides describe research areas in our Department, with briefdescription of interests of individual faculty members who are available for supervisinggraduate students.

Specialization areas

Specialization areas in Mathematics:

Cellular automata, discrete dynamical systems and complex networks (HenrykFuks)

Computational methods for solving algebraic and differential systems (ThomasWolf)

Combinatorial number theory (Yuanlin Li)

Cryptography (Omar Kihel)

Groups, rings and group rings (Yuanlin Li)

High performance parallel computing (Thomas Wolf)

Mathematical music theory (Chantal Buteau)

Mathematical physics and General Relativity (Stephen Anco, Alexander Odesski,Thomas Wolf, Jan Vrbik)

Mathematics education (Chantal Buteau)

Nonlinear functional analysis and applications (Hichem Ben-El-Mechaiekh)

Number theory (Omar Kihel)

Solitons and integrability of partial differential equations (Stephen Anco,Alexander Odesski, Thomas Wolf)

Symmetry analysis and computer algebra applied to nonlinear differentialequations (Stephen Anco, Thomas Wolf)

Specialization areas – continued

Specialization areas in Statistics:

Advanced statistical analysis (Xiaojian Xu)

Computational methods and applications to stochasticmodels (Jan Vrbik)

Computational statistics with applications to neuroscience(William Marshall)

High Dimensional Data Analysis, Shrinkage Estimation,Asymptotic Theory and applications, Statistical QualityControl, Biostatistics (S. Ejaz Ahmed)

Statistical inference theory and methods (Mei Ling Huang)

For more details, visit web pages of individual faculty membersand view the following slides.

S. Ejaz Ahmed

High Dimensional Data Analysis,

Shrinkage Estimation, Asymp-

totic Theory and applications,

Statistical Quality Control, Bio-

statistics.

My area of expertise includes statistical inference, high dimensionaldata analysis Shrinkage estimation, statistical quality control, andasymptotic theory and its application. The high dimensional dataanalysis is a hot topic for the statistical research due to continuedrapid advancement of modern technology that is allowing scientiststo collect data of increasingly unprecedented size and complexity.Examples include epigenomic data, genomic data, proteomic data,high-resolution image data, high frequency financial data, functionaland longitudinal data, and network data, among others. Simulta-neous variable selection and estimation is one of the key statisticalproblems in analyzing such complex data. This joint variable selec-tion and estimation problem is one of the most actively researchedtopics in the current statistical literature. More recently, regular-ization, or penalized, methods are becoming increasingly popularand many new developments have been established. The shrinkageestimation strategy is playing in important role in this arena.Currently, I am working on the following problems including:

Shrinkage Estimation for High Dimensional Data Analysis

Difference Based Shrinkage Analysis in High DimensionalPartially Linear Regression

Improved Estimation Strategies in Generalized Linear Models

Shrinkage Estimation and Variable Selection in MultipleRegression Models with Random Coefficient AutoregressiveErrors.

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Stephen Anco

Nonlinear differential equations, integra-

bility and solitons, mathematical physics

and analysis.

My research lies in several areas of nonlinear differential equations, integrability and solitons,mathematical physics and analysis. Some problems I am currently working on include:

new exact solutions of radial nonlinear Schrodinger equations and wave equationsin n dimensions

“hidden” conservation laws of fluid flow equations and related potential systems,

integrable group-invariant soliton equations and their derivation from curve flowsin geometric manifolds,

symmetry and conservation law structure of wave maps and Schrodinger maps,

symmetries and conservation laws in curved spacetime for Maxwell’selectromagnetic field equations, gravity wave equations, and other fundamentalphysical field equations,

exact monopole, plane wave, Witten-ansatz solutions in a nonlinear generalizationof Yang-Mills/wave map equations,

novel nonlinear generalizations (deformations) of Yang-Mills equations for gaugefields, and Einstein’s equations for gravitational fields

In addition I am coauthoring two books with G. Bluman in the area of symmetry methodsand differential equations, in the Applied Mathematical Sciences series of Springer-Verlag.The first book provides an introduction to symmetry methods for both ordinary and partialdifferential equations, as well as a comprehensive treatment of first integral methods forordinary differential equations. The second book will cover conservation laws (local andnonlocal) and potential systems for partial differential equations, and Bluman’s nonclassicalmethod of finding exact solutions. I also have an active interest in symbolic computationusing Maple and some of my research in symmetry and conservation law analysis makesuse of this software and involves development of algorithmic computational methods.

Hichem Ben-El-Mechaiekh

Nonlinear Analysis

My research interests are in topological meth-ods in nonlinear analysis with focus on set-valuedanalysis and its applications to fixed point the-ory, mathematical economics, game theory andoptimization. I am particularly interested in thesolvability of nonlinear inclusions where classicalhypotheses of convexity fail. Methods include ablend of topology, functional analysis, and non-smooth analysis.

Chantal Buteau

Mathematical music theory.

Mathematics education.

My research field is the Mathematical Music Theory. I am particularly inter-ested in modeling motivic (melodic) structure and analysis of musical com-positions through a topological approach. The motivic analysis of a musiccomposition consists of identifying the short melody, called a motif, thatunites the composition through its strict repetitions, the so-called imitations,and its variations and transformations which are heard throughout the wholecomposition. Mainly using group theory, linear algebra and general topologyconcepts, we construct a (T0-) topological structure corresponding to themotivic hierarchy of a composition. Our program (JAVA) Melos can analysemusic compositions such as Schumann’s Dreamery from Kinderszenen Myongoing interdisciplinary research mainly concerns:

Concrete applications to a music corpus;

A categorical extension of our model including e.g. continuousfunctions between 2 motivic spaces, products of different spaces,natural transformations (gestalt spaces);

Visualisation of Melos’ multiple outputs in order to show and hear,and to explore mathematics and music results.

Regarding mathematics education I’m interested in developing tools usingmusic for the exploration of mathematics concepts.

Henryk Fuks

Spatially-extended discrete dynam-

ical systems. Cellular automata.

Complex networks.

My research interests fall into three main cathegories:

Theory: “Solving” of cellular automata (CA). Additive invariants inCA. Phase transitions in discrete dynamics. Discrete models ofcomputation. Maximal entropy approximation. Orbits of Bernoullimeasures in CA.

Modeling: Growth of complex networks. Models of granular andtraffic flow. Models of language acquisition. Complex graphs asmodels of vocabulary of human languages. Discrete models ofdiffusion and spread.

Software: Agent-based simulations of complex systems. Efficientalgorithms for simulation of cellular automata and lattice gases.

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Mei Ling Huang

Statistical inference. Comput-

ing and simulation methods in

statistics.

My research interests are in exploration of new efficient, optimal methodsof statistical inference for distribution function, quantile and regression, anddevelopment of computing and simulation methods with applications to sur-vival analysis, network and stochastic models. I am working on the followingtopics.

Nonparametric distribution, quantile and regression estimation andtesting are important research directions with many applications. Ihave been studying several methods in this field. Study weightedempirical distribution function to develop more efficient estimationand testing methods. For example, explore more efficient non-kernelquantile estimation methods. Study properties of these estimatorsand tests: consistency, rate of convergence, efficiencies.Computational methods and simulation methods also aredeveloping. Develop new prediction methods for stochasticprocesses. For example, use sample path of martingales and Markovprocesses. Apply these methods to economics, quality control,queueing networks, insurance and biostatistics.

Studies of truncated and censored data have important applicationsin biostatistics, industrial engineering and other fields. The topicsare: Search efficient estimation methods for truncated data of typesof heavy tail distribution for example, simulating and estimatingwaiting time of using Internet or other stochastic models by usingPareto distribution. Develop efficient estimation methods in survivalanalysis and its applications. For example, predicting recovery timesof cancer patients, estimating the value at risk of stocks.

Omar Kihel

Algebraic Number Theory,

Elliptic Curves, Diophantine

Equations. Permutation

polynomials over Finite Fields

and Galois Theory.

My research lies in finite field functions andtheir applications to coding theory and cryptog-raphy; existence of primitive polynomials overfinite fields; exponential sums over finite fields.

Yuanlin Li

Groups, rings and group rings.

Combinatorial number theory.

My research interest is in the areas of groups, rings, group rings and combina-torial number theory. The group ring of a group G over a commutative ringK is the ring KG of all formal finite sums: α =

∑ag g , and is an attractive

object of study. Here group theory, ring theory, commutative algebra andnumber theory come together in a fruitful way, and moreover the study ofgroup rings has important applications in coding theory. My recent researchwork has thrown light on structures of group rings and their unit groups. Iam also interested in studying homological properties of modules and rings.In addition, I investigate the interplay between rings and their graphs (suchas zero-divisor and annihilating ideal graphs). A few years ago, I started anew exciting research initiative and extended my research interest into theadditive number theory by investigating a few combinatorial problems (e.g.zero-sum problems) in that filed. Some of my on-going research projects arelisted below:

Zassenhaus conjectures and related problems.

The normalizer problem and Coleman automorphisms.

Generators of large subgroups of (central) unit groups of group rings.

Index of a sequence of a finite cyclic group.

The Erdos-Ginzburg-Ziv Theorem and its improvment.

Morphic groups and related problems.

Zero-divisor (annihilating ideal) graphs of (group) rings.

Morphic and reversible group rings.

Combinatorial problems in group theory and ring theory.

Injectivity of modules and related topics.

William Marshall

Computational Statistics with

Applications to Neuroscience

My primary research interest is computational statistics with applications toneuroscience. My current work falls into three main categories:

(A) Causal Networks – The brain can be viewed abstractly as acomplex network of interacting elements (whether as neurons,mini-columns, etc.). This aspect of my research focuses ondeveloping and exploring information theoretic and causal measuresof complexity (e.g., integrated information) to apply tocausal/Bayesian networks. I’m specifically interested in measureswhich can be applied across spatiotemporal scales.

(B) Measures of connectivity – Neuroimaging methods (e.g.,electroencephalography, functional magnetic resonance imaging,calcium imaging) generate large amounts of data. I am interested indeveloping measures of functional connectivity between recordingareas (brain regions). To do this, I utilize signal processingtechniques and statistical learning methods. Developing suchmethods requires substantial domain specific knowledge of theimaging modality. This aspect of my research can be viewed as anempirical analogue of (A).

(C) Linear Mixed Effect (LME) Models – The nature ofneuroscience experiments is such that there are often severalmeasurements taken from a single experimental unit (e.g., multiplesynapses from a single dendrite and multiple dendrites from a singlemouse). Linear mixed effect models are a powerful and flexible toolfor such “repeated measures” experiment designs. This aspect ofmy research aims to develop model fitting and inference methods forLME models, with specific thought to neuroscience experiments.

Alexander Odesskii

MathematicalPhysics.

My main research interests are in MathematicalPhysics in the sense of Mathematics inspired byideas that come from Theoretical Physics. Moreprecisely, I am interested in: algebraic and ge-ometric structures which come from quantumfield theory, statistical mechanics and the theoryof integrable systems.

Jan Vrbik

Celestial Mechanics.

Probability and Statistics.

Research interests:

Approximating sampling distributions of various estimators (MLE in

particular) by Edgeworth Series (an extension of the basic Normal

approximation) which is capable of achieving high accuracy even

with relatively small sample. Currently, this involves:

investigating theoretical properties of such series (such as

the nature of its asymptotic convergence, applicability to

discrete distributions, etc.),

constructing accurate confidence regions of distribution’s

parameters (with preference for ML approach),

extending the technique to correlated samples (so far

restricted to autoregressive models).

Mote Carlo simulation in Quantum Chemistry (computing propertiesof small molecules in particular).

Constructing analytic solution to perturbed Kepler problem,focusing on resonances and the onset of chaos.

Thomas Wolf

Differential equations, Com-

puter algebra, General Relativ-

ity, Computer Go.

My research interests include differential equations and integra-bility, computer algebra and classical General Relativity.Work in computer algebra concerns algorithms to simplify andsolve overdetermined systems of equations (linear/non-linear),(algebraic, ODEs, PDEs). These algorithms and implementa-tions are applied in higher level programs for the determination ofsymmetries, conservation laws or other properties of differentialequations. Applications include the classification of integrablesystems of various types.Attempts to increase the efficiency of related programs lead toa study of the parallelization of my algorithms and programs.For the last 10 years I was the Brock site leader of the SHARC-NET consortium and serve currently on the board of BISC, theBrock Institute of Scientific Computing.A hobby of mine concerns the mathematical analysis and com-puterization of the Asian game of Go. Recent work includesthe static analysis of positions interpreted as discrete dynamicalsystems and the mathematics of semeai and seki positions.

Xiaojian Xu

Statistical Models and Inference;

Optimal Regression Designs; Ro-

bust Methods; Multivariate Analy-

sis; Accelerated Life testing

My research lies in several areas of experimental designs, robust inferences,and survey sampling. Some problems that I am currently working on include:

Constructing exact designs that provide optimal solutions for avariety of inferences.

Analysis for robustness of experimental designs against differentmodel violations.

Optimal planning for accelerated life testing experiments.

Optimal designs for mixed models.

Robust designs for nonlinear models.

Optimal methods for statistical inferences in indirect sampling.

Publications co-authored by our graduate students (chronologically)

H. Fuks and M. Krzeminski. Core decomposition spectra of large graphs and their applications in modelling. In E. Wamkeue,editor, Proceedings of the 19th IASTED International Conference on Modelling and Simulation, pp. 620–143. Acta Press, 2008.

H. Fuks and M. Krzeminski. Topological structure of dictionary graphs. J. Phys. A: Math. Theor., 42:art. no. 375101, 2009.

Y. Li and Y. Tan. On b 5-groups. Ars Combin., 2009. Accepted.

H. Fuks and A. Skelton. Response curves for cellular automata in one and two dimensions – an example of rigorous calculations.Int. J. of Natural Computing Research, 1:85–99, 2010.

Y. Li, C. Plyley, P. Yuan, and X. Zeng. Minimal Zero Sum sequences of Length Four over Finite Cyclic Groups. J. NumberTheory, 130:2033–2048, 2010.

Y. Li and Y. Tan. On B(4, k) groups. J. Algebra Appl., 9:27–42, 2010.

Huang, M. L. and Zhao, K., “On Estimation of the Truncated Pareto Distribution”, Advances and Applications in Statistics,Volume 16, No. 1, pp.83-102, 2010.

S. Anco, N. Tchegoum Ngatat, and M. Willoughby. Interaction properties of complex mKdV solitons. Physica D, 240:1378–1394,2011.

H Fuks and A. Skelton. Orbits of Bernoulli measure in asynchronous cellular automata. Dis. Math. Theor. Comp. Science,AP:95–112, 2011.

H. Fuks and A. Skelton. Response curves and preimage sequences of two-dimensional cellular automata. In Proceedings of the2011 International Conference on Scientific Computing: CSC-2011, pages 165–171. CSERA Press, 2011.

W. Gao, Y. Li, J. Peng, C. Plyley, and G. Wang. On the index of sequences over cyclic groups. Acta Arith., 148:119–134, 2011.

Y. Li and X. Pan. On B(5, k) groups. Bull. Austral. Math. Soc., 84:393–407, 2011.

Y. Li and Y. Tan. On b(4, 13) 2-groups. Comm. Algebra, 39(10):3769–3780, 2011.

X. Xu and L. Zhao. Robust designs for haar wavelet approximation models. Applied Stochastic Models in Business and Industry,27(5):531–550, 2011.

X. Xu, and X. Shang, 2011. Optimal and Robust Designs for Full and Reduced Fourier Regression Models. Proceedings ofInternational Conference on Applied Mathematics, Modeling and Computational Science. 289–292.

X. Xu, and A. Chen, 2011. Robust Designs for Three Commonly Used Nonlinear Models. Proceedings of International Conferenceon Applied Mathematics, Modeling and Computational Science. 313316.

X. Xu, and X. Shang, 2011. Optimal Designs for Fourier Regression Models. Proceedings of the Seventh International Conferenceon Mathematical Methods in Reliability, 889–895.

Publications co-authored by our graduate students – continued

S. C. Anco, S. MacNaughton, and T. Wolf. Conservation laws and symmetries of quasilinear radial wave equations inmulti-dimensions. J. Math. Phys., 53:art. no. 053703, 2012.

S. C. Anco, M. Mohiuddin, and T. Wolf. Traveling waves and conservation laws for complex mkdv-type equations. Appl. Math.Comput., 219:679–698, 2012.

H. Fuks and A. Skelton. Classification of two-dimensional binary cellular automata with respect to surjectivity. In Proceedings ofthe 2012 International Conference on Scientific Computing: CSC-2012, pages 51–57. CSERA Press, 2012.

X. Xu and S. Hunt, 2012. Robust Designs of Step-Stress Accelerated Life Testing Experiments for Reliability Prediction.Proceedings of the 1st ISM International Statistical Conference, 504-511.

S. Hunt and X. Xu. Optimal design for accelerated life testing with simple step-stress plans. International Journal of PerformabilityEngineering., 8(5):575–579, 2012.

F. Aliniaeifard, Y. L, and K. Nicholson. Morphic p-groups. J.Pure Appl. Algebra, 2013. Accepted.

S. Anco, A. S. Mia, and M. Willoughby. Complex MKdV solitons with time-varying phase. 2013. In preparation.

N. Marshall and C. Buteau. Learning by designing and experimenting with interactive, dynamic mathematics exploratory objects.Int. J. for Technology in Mathematics Education, 2013. accepted.

Huang, M. L., Coia, V. and Brill, P. H., “A Mixture Truncated Pareto Distribution”, JSM Proceedings, Statistical ComputingSection, American Statistical Association, pp. 2488-2498, 2013.

Huang, M. L., Yuen, W. K. and Zhang, M., “Efficient Methods on Confidence Intervals of Prediction Intervals”, Advances andApplications in Statistics, Volume 33, No. 1, pp.1-21, 2013.

Coia, V. and Huang, M. L., “A Sieve Model for Extreme Values”, Journal of Statistical Computation and Simulation, 2012,Published online, January 17, 2013. (23 pages)

H. Fuks, B. Farzad and Y. Cao, “A Model of Language Inflection Graphs”, Int. J. Mod. Phys. C Vol. 25, No. 6 (June 2014)1450013

Y. Li and Yilan Tan, “On B5−groups”, Ars Combinatoria, 114 (2014), 3-14.

Y. Li and Hongdi Huang, “On B(5,18) groups”, Comm. Algebra, Accepted Oct. 2014.

Y. Li and Hongdi Huang, “On B(4,14) non-2-groups”, Journal of Algebra and Its Applications, Accepted Oct. 2014.

Y. Li, Jianlong Cheng and Yanyan Gao, “Some ∗-clean group rings”, Algebra Colloq. 22 (1) (2015) 169-180.

Y. Li and F. Aliniaeifard, “Zero-divisor Graphs for Group Rings”, Comm. Algebra, 42 (11) 2014, 4790-4800.

Thank you!

We are looking forward to receiving your application.

For more information, please visithttps:

//brocku.ca/mathematics-science/mathematics/

or email us at mathstatgrad@brocku.ca