Graduate Calendar 2006-2007

Ottawa-Carleton Institute of Mathematics and Statistics

Herzberg Building 4314
Telephone: 520-2152
Fax: 520-3536
E-mail: mathstat@carleton.ca
Web site: www.mathstat.carleton.ca

The Institute

Director of the Institute: Matthias Neufang
Associate Director: Vladimir Pestov

Students pursuing studies in pure mathematics, applied mathematics, probability and statistics at the graduate level leading to a M.Sc. or a Ph.D. degree do so in a joint program offered by the School of Mathematics and Statistics at Carleton University and the Department of Mathematics and Statistics at the University of Ottawa under the auspices of the Ottawa-Carleton Institute of Mathematics and Statistics. The Institute is responsible for supervising the programs, regulations, and student admissions, and for providing a framework for interaction between the two departments at the research level.

The list below of all members of the Institute along with their research interests can be used as a guide to possible supervisors.

In addition to the programs administered by the Institute, the School of Mathematics and Statistics at Carleton University offers several other programs.

In cooperation with the Department of Epidemiology and Community Medicine at the University of Ottawa, students may pursue a program leading to an M.Sc. with a Specialization in Biostatistics. For information, see the Ottawa-Carleton Collaborative Program in Biostatistic's section in this Calendar.

In cooperation with the Department of Systems and Computer Engineering and the School of Computer Science at Carleton University, students may pursue a program leading to an M.Sc. in Information and Systems Science. For information see the Information and Systems Science section of this Calendar.

The School of Mathematics and Statistics also offers a co-operative master's program in statistics in collaboration with the federal govern ment, emphasizing practical training through work experience, along with sound training in statistical inference and basic probability theory.

Requests for information and completed applications should be sent to the Director or Associate Director of the Institute.

Members of the Institute

The home department of each member of the Institute is indicated by (C) for the School of Mathematics and Statistics, Carleton University and (UO) for the Department of Mathematics and Statistics, University of Ottawa.

• Mayer Alvo, Nonparametric statistics, sequential analysis (UO)
• David Amundsen, Nonlinear wave equations, numerical analysis (C)
• Stephen Astels, Number theory (C)
• Yves Atchadé, Statistics (UO)
• Raluca Balan, Stochastic processes, probability theory, mathematical statistics (UO)
• Nick Barrowman, Biostatistics, applied statistics (C)
• Y. Billig, Algebra (C)
• R. Blute, Logic, Category theory (UO)
• Amitava Bose, Stochastic modeling, probability theory (C)
• Y. Bourgault, Numerical methods, mathematical modeling (UO)
• S. Boyd, Combinatorial optimization (UO)
• Inna Bumagin, Algebra (C)
• W.D. Burgess, Algebra, non-commutative rings (UO)
• Lucy Campbell, Geophysical fluid dynamics, partial differential equations (C)
• Charles Castonguay, Demography (UO)
• Kevin Cheung, Combinatorial optimization (C)
• Miklós Csörgó, Probability and statistics (C)
• A.R. Dabrowski, Dependence in probability and statistics, applications (UO)
• Daniel Daigle, Algebraic geometry, commutative algebra (UO)
• D.A. Dawson, Stochastic processes and probability theory (C)
• Benoit Dionne, Similarity and groups in bifurcation theory (UO)
• J.D. Dixon, Group theory, algebra computation (C)
• Vlastimil Dlab, Finite dimensional algebras, representation theory (C)
• Kokou Dossou, Numerical solution of partial differential equations of mathematical physics (C)
• S. Faridi, Commutative algebra, algebraic Combinatorics (UO)
• P. Farrell, Sampling, discrete data, applied statistics (C)
• Amy Felty, Logics and logical foundations of computing (UO)
• Che-Kao Fong, Operator theory (C)
• Eric Freeman, Number theory (C)
• Zhicheng Gao, Graph theory (C)
• Thierry Giordano, Operator algebras, ergodic theory (UO)
• D.E. Handelman, K-theory, operator algebras, ring theory (UO)
• B.G. Ivanoff, Probability, point processes, martingales (UO)
• Antal Jarai, Probability, mathematical physics and applied probability (C)
• W. Jaworski, Analysis, probability (C)
• Barry Jessup, Rational homotopy, lie algebra cohomology (UO)
• Alexander Kitaev, Isomonodromy deformations, Painleve equations (C)
• Daniel Krewski, Applied statistics in medicine (C)
• E.O. Kreyszig, Partial differential equations, numerical analysis (C)
• V. LeBlanc, Differential equations, bifurcation theory, dynamical systems (UO)
• J. Levy, Group representations (UO)
• I.A. Manji, Homological methods in commutative algebra and algebraic geometry, cryptography (C)
• D.R. McDonald, Applied probability (UO)
• Sam Melkonian, Non-linear differential equations (C)
• Paul Mezo, Algebra and number theory (C)
• S.E. Mills, Applied statistics, statistical methods, inference, data mining (C)
• A.B. Mingarelli, Ordinary differential equations, difference equations (C)
• M. Mojirsheibani, Resampling, classification and pattern recognition (C)
• B.C. Mortimer, Group theory, coding theory (C)
• Lucia Moura, Combinatorial algorithms and optimization, combinatorics, (UO)
• Erhard Neher, Jordan algebras and groups, lie algebras (UO)
• Matthias Neufang, Analysis (C)
• Monica Nevens, Representation theory of p-adic Lie groups (UO)
• Nathan Ng, Analytic number theory (UO)
• Arian Novruzi, Partial differential equations, shape optimization, numerical Analysis (UO)
• Mohamedou Ould Haye, Statistics (C)
• D. Panario, Finite fields, combinatorics, analysis of algorithms (C)
• J.N. Pandey, Generalized functions, partial differential equations (C)
• Paul-Eugène Parent, Algebraic topology, homotopy theory (UO)
• Chul Gyu Park, Statistics (C)
• Vladimir Pestov, Topological transformation groups, geometry of large dimensions (UO)
• Michel Racine, Jordan algebras, al gebra, polynomial identities (UO)
• Mizanur Rahman, Special functions (C)
• J.N.K. Rao, Sample surveys theory and methods (C)
• P. Révész, Probability (CU)
• Luis Ribes, Group theory (C)
• R.B. Richter, Graph theory, combinatorics (C)
• Wulf Rossmann, Representations of semisimple lie groups (UO)
• Damien Roy, Transcendental number theory (UO)
• A.K. Md. E. Saleh, Order statistics, mathematical statistics (C)
• Mateja Sajna, Graph theory (UO)
• David Sankoff, Mathematical genomics, (UO)
• P. Sawyer, Spherical functions (UO)
• P.J. Scott, Logic, Category theory (UO)
• A. Sebbar, Number theory, quantum groups (UO)
• P. Selinger, Logic, category theory (UO)
• A. Singh, Statistics (C)
• Sanjoy Sinha, Biostatistics, longitudinal data analysis, robust inference, time series analysis (C)
• Benjamin Steinberg, Algebra (C)
• Natalia Stepanova, Statistics (C)
• Brett Stevens, Combinatorics (C)
• I. Stojmenovic, Discrete mathematics, combinatorial algorithms, multiple-value logic, theoretical computer science (UO)
• Barbara Szyszkowicz, Probability (C)
• François Theberge, Applied probability (UO)
• Rémi Vaillancourt, Scientific computation (UO)
• G. Walsh, Number theory, diophantine equations (UO)
• Qiang (Steven) Wang, Discrete mathematics and algebra (C)
• K. S. Williams, Number theory (C)
• M. Zarepour, Resampling and nonparametric Bayesian inference, time series analysis (UO)
• Y. Zhao, Applied probability (C)

Master of Science

Admission Requirements

The normal requirement for admission to the master's program is an Honours bachelor's degree in mathematics, or the equivalent, with at least high honours standing. Applicants holding a general (three-year) degree with at least high honours standing may be admitted to a qualifying-year program.

Subsequent admission to the regular master's program depends on performance during the qualifying-year program and will be decided no later than one year after admission to the qualifying-year program. Details are outlined in the General Regulations section of this Calendar. Students with outstanding academic performance and research promise while in the M.Sc. program may be permitted to transfer to the Ph.D. program without completing the M.Sc. program.

Special consideration may be given, for acceptance in the high-technology concentration, to graduates in computer science or engineering with a strong mathematical background and work experience in the high-technology sector.

Program Requirements

The two options for the M.Sc. program are:

• 2.5 credits and a thesis
• 4.0 credits

The courses must be chosen from those at the graduate level except that a student may take up to 1.0 credit of undergraduate courses at the 4000-level to satisfy these requirements. Not all these courses may be taken in the same field of mathematics; at least 1.0 credit must be in another field. All master's students ar e required to participate actively in a seminar or project under the guidance of their adviser. A maximum of 1.0 credit taken outside of the School of Mathematics and Statistics at Carleton University or the Department of Mathematics and Statistics at the University of Ottawa may be allowed for credit.

Students who plan to specialize in probability or statistics are strongly advised that during their master's program they include, where possible, the courses STAT 5600, STAT 5501 in mathematical statistics, STAT 4502, STAT 5505 in applied statistics, and STAT 4501, STAT 5701 in probability, together with 1.0 credit further in the School of Mathematics and Statistics. In addition, a graduate course in another field, such as biology, biostatistics, economics, computer science, systems analysis, and stochastic modeling, is highly recommended.

High-Technology Concentration in the M.Sc.

An M.Sc. with a high-technology concentration is available. This concentration is intended for mathematics graduates interested in employment in the high technology area; it is also intended for science or engineering graduates currently employed in the high-technology area who require a greater understanding of mathematics for their work. The course requirement for the high-technology designation on a student's transcript is completion of a minimum of five courses for the thesis option and six courses for the non-thesis option, selected from the list of high-technology courses authorized by the Director of the Institute. Each student will be assigned an adviser who will be responsible for approving course selection.

Doctor of Philosophy

Admission Requirements

The normal requirement for admission to the Ph.D. program is a master's degree in mathematics, or the equivalent, with at least high honours standing. Details are outlined in the General Regulations section of this Calendar.

Program Requirements

Course requirements, which are determined at the time of admission, include a minimum of 3.0 credits and a suitable thesis. Not all of these courses may be taken in the same field of mathematics; at least 1.0 credit must be in another field.

All candidates must take comprehensive examinations, and must satisfy a language requirement. The language requirement is determined by the candidate's advisory committee and normally requires the ability to read mathematical literature in a language considered useful for his/her research or career, and other than the candidate's principal language of study.

Students specializing in mathematics or probability undertake a comprehensive examination in the following areas:

• The candidate's general area of specialization at the Ph.D. level
• Examinations on two topics chosen from algebra, analysis, probability, topology, and statistics. (This choice excludes the student's specialty.)

Students specializing in statistics must write an examination in the following areas:

• Mathematical statistics which includes multivariate analysis
• An examination in probability, and
• An examination in either (i) applied statistics, or (ii) analysis

In all cases, the examination must be completed successfully within twenty months of initial registration in the Ph.D. program in the case of full-time students, and within thirty-eight months of initial registration in the case of part-time students.

All Ph.D. candidates are also required to undertake a final oral examination on the subject of their thesis.

Selection of Courses

The following undergraduate courses may, with the approval of the School of Mathematics and Statistics, be selected by master's candidates in partial fulfillment of their degree requirements:

Mathematics and Statistics
MATH 4001 Vector Calculus
MATH 4105 Rings and Modules
MATH 4107 Commutative Algebra
MATH 4207 Foundations of Geometry
MATH 4208 Introduction to Differentiable Manifolds
MATH 4405 Analytical Dynamics
MATH 4406 Hydrodynamics and Elasticity
MATH 4407 Tensor Analysis and Relativity Theory
STAT 4501 Probability Theory
STAT 4502 Sampling: Theory and Methods
STAT 4503 Applied Multivariate Analysis
STAT 4506 Non-Parametric Methods
STAT 4508 Stochastic Models
STAT 4509 Stochastic Optimization
MATH 4702 Integral Transforms
MATH 4703 Qualitative Theory of Ordinary Differential Equations
MATH 4802 Introduction to Mathematical Logic
MATH 4803 Topics in Applied Logic
MATH 4804 Design and Analysis of Algorithms
MATH 4806 Numerical Analysis
MATH 4808 Graph Theory and Algorithms

Graduate Courses

Not all of the following courses are offered in a given year. For an up-to-date statement of course offerings for 2006-2007 and to determine the term of offering, consult the Registration Instructions and Class Schedule booklet, published in the summer and available online at carleton.ca/cu/programs/sched_dates/

University of Ottawa course numbers (in parentheses) follow the Carleton course number and credit information.

MATH 5001 [0.5 credit] (MAT 5120)

Abstract Measure Theory

Abstract measure and integral, L-spaces, complex measures, product measures, differentiation theory, Fourier transforms.

Prerequisite: MATH 4007.

MATH 5003 [0.5 credit] (MAT 5122)

Banach Algebras

Commutative Banach algebras; the space of maximal ideals; representation of Banach algebras as function algebras and as operator algebras; the spectrum of an element. Special types of Banach algebras: for example, regular algebras with involution, applications.

MATH 5004 [0.5 credit] (MAT 5129)

Integral Equations

A survey of the main results in the theory of non-singular linear integral equations; Volterra and Fredholm equations of first and second kind in the L2 case, with special results for the continuous case; Hermitian kernels; eigen-function expansions; compact operators.

Prerequisites: MATH 3002 and MATH 4003.

MATH 5005 [0.5 credit] (MAT 5127)

Complex Analysis

Complex differentiation and integration, harmonic functions, maximum modulus principle, Runge's theorem, conformal mapping, entire and meromorphic functions, analytic continuation.

MATH 5006 [0.5 credit] (MAT 5316)

Topological Vector Spaces

Construction of new topological vector spaces out of given ones; local convexity and the Hahn-Banach theorem; compactness and the Krein-Milman theorem; conjugate spaces, polar sets.

Prerequisite: MATH 4003.

MATH 5007 [0.5 credit] (MAT 5125)

Real Analysis I (Measure Theory and Integration)

General measure and integral, Lebesgue measure and integration on R, Fubini's theorem, Lebesgue-Radon-Nikodym theorem, absolute continuity and differentiation, LP-spaces. Selected topics such as Da niell-Stone theory. Also offered, with different requirements, as MATH 4007 for which additional credit is precluded. Prerequisites: MATH 3001 and MATH 3002 (MAT 3125) or permission of the School.

MATH 5008 [0.5 credit] (MAT 5126)

Real Analysis II (Functional Analysis)

Banach and Hilbert spaces, bounded linear operators, dual spaces. Topics selected from: weak-topologies, Alaoglu's theorem, compact operators, differential calculus in Banach spaces, Riesz representation theorems. Also offered, with different requirements, as MATH 4003 for which additional credit is precluded.

Prerequisite: MATH 5007 (MAT 5125) or permission of the School.

MATH 5009 [0.5 credit] (MAT 5121)

Introduction to Hilbert Space

Geometry of Hilbert Space, spectral theory of linear operators in Hilbert Space.

Prerequisites: MATH 3001, MATH 3002, and MATH 4003.

MATH 5102 [0.5 credit] (MAT 5148)

Group Representations and Applications

An introduction to group representations and character theory, with selected applications.

MATH 5103 [0.5 credit] (MAT 5146)

Rings and Modules

Generalizations of the Wedderburn-Artin theorem and applications, homological algebra.

MATH 5104 [0.5 credit] (MAT 5143)

Lie Algebras

Basic concepts: ideals, homomorphisms, nilpotent, solvable, semi-simple. Representations, universal enveloping algebra. Semi-simple Lie algebras: structure theory, classification, and representation theory.

Prerequisites: MATH 5107 (MAT 5141) and MATH 5109 (MAT 5142) or permission of the School.

MATH 5106 [0.5 credit] (MAT 5145)

Group Theory

Fundamental principles as applied to abelian, nilpotent, solvable, free, and finite groups; representations. Also offered, with different requirements, as MATH 4106, for which additional credit is precluded.

Prerequisite: MATH 3100 or perm ission of the School.

MATH 5107 [0.5 credit] (MAT 5141)

Algebra I

Groups, Sylow subgroups, finitely generated abelian groups. Rings, field of fractions, principal ideal domains, modules. Polynomial algebra, Euclidean algorithm, unique factorization.

Prerequisite: permission of the School.

MATH 5108 [0.5 credit] (MAT 5147)

Homological Algebra and Category Theory

Axioms of set theory, categories, functors, natural transformations; free, projective, injective and flat modules; tensor products and homology functors, derived functors; dimension theory. Also offered, with different requirements, as MATH 4108 for which additional credit is precluded.

Prerequisite: MATH 3100 or permission of the School.

MATH 5109 [0.5 credit] (MAT 5142)

Algebra II

Field theory, algebraic and transcendental extensions, finite fields, Galois groups. Modules over principal ideal domains, decomposition of a linear transformation, Jordan normal form.

Prerequisites: MATH 5107 (MAT 5141) and permission of the School.

MATH 5201 [0.5 credit] (MAT 5150)

Topics in Geometry

Various axiom systems of geometry. Detailed examinations of at least one modern approach to foundations, with emphasis upon the connections with group theory.

Prerequisite: permission of the School.

MATH 5202 [0.5 credit] (MAT 5168)

Homology Theory

The Eilenberg-Steenrod axioms and their consequences, singular homology theory, applications to topology and algebra.

Prerequisite: MATH 4205.

MATH 5205 [0.5 credit] (MAT 5151)

Topology I

Topological spaces, product and identification topologies, countability and separation axioms, compactness, connectedness, homotopy, fundamental group, net and filter convergence. Also offered, with different requirements, as MATH 4205 for which additional credit is precluded.

Prerequisite: MATH 3001 or permission of the School.

MATH 5206 [0.5 credit] (MAT 5152)

Topology II

Covering spaces, homology via the Eilenberg-Steenrod Axioms, applications, construction of a homology functor. Also offered, with different requirements, as MATH 4206 for which additional credit is precluded.

Prerequisites: MATH 3100 (MAT 3143) and MATH 5205 (MAT 5151) or permission of the School.

MATH 5207 [0.5 credit] (MAT 5169)

Foundations of Geometry

A study of at least one modern axiom system of Euclidean and non-Euclidean geometry, embedding of hyperbolic and Euclidean geometries in the projective plane, groups of motions, models of non-Euclidean geometry.

Prerequisite: MATH 3100 (may be taken concurrently) or permission of the School.

MATH 5208 [0.5 credit] (MAT 5155)

Differentiable Manifolds

A study of differentiable manifolds from the point of view of either differential topology or differential geometry. Topics such as smooth mappings, transversality, intersection theory, vector fields on manifolds, Gaussian curvature, Riemannian manifolds, differential forms, tensors, and connections are included.

Prerequisite: MATH 3001 or permission of the School.

MATH 5300 [0.5 credit] (MAT 5160)

Mathematical Cryptography

Analysis of cryptographic methods used in authentication and data protection, with particular attention to the underlying mathematics, e.g. Algebraic Geometry, Number Theory, and Finite Fields. Advanced topics on Public-Key Cryptography: RSA and integer factorization, Diffie-Hellman, discrete logarithms, elliptic curves. Topics in current research.

Prerequisite: undergraduate honours algebra, including group theory and finite fields.

MATH 5301 [0.5 credit] (MAT 5161)

Mathematical Logic

A basic graduate course in mathematical logic. Propositional and predicate logic, proof th eory, Gentzen's Cut-Elimination, completeness, compactness, Henkin models, model theory, arithmetic and undecidability. Special topics (time permitting) depending on interests of instructor and audience.

Prerequisites: Honours undergraduate algebra, analysis and topology or permission of the instructor.

MATH 5305 [0.5 credit] (MAT 5163)

Analytic Number Theory

Dirichlet series, characters, Zeta-functions, prime number theorem, Dirichlet's theorem on primes in arithmetic progressions, binary quadratic forms. Also offered at the undergraduate level, with different requirements, as MATH 4305, for which additional credit is precluded.

Prerequisite: MATH 3007 or permission of the School.

MATH 5306 [0.5 credit] (MAT 5164)

Algebraic Number Theory

Algebraic number fields, bases, algebraic integers, integral bases, arithmetic in algebraic number fields, ideal theory, class number. Also offered, with different requirements, as MATH 4306 for which additional credit is precluded.

Prerequisite: MATH 3100 or permission of the School.

MATH 5403 (MAT 5187)

Topics in Applied Mathematics

MATH 5405 [0.5 credit] (MAT 5131)

Ordinary Differential Equations

Linear systems, fundamental solution. Nonlinear systems, existence and uniqueness, flow. Equilibria, periodic solutions, stability. Invariant manifolds and hyperbolic theory. One or two specialized topics taken from, but not limited to: perturbation and asymptotic methods, normal forms and bifurcations, global dynamics. Prerequisite: MATH 3008 or permission of the School.

MATH 5406 [0.5 credit] (MAT 5133)

Partial Differential Equations

First-order equations, characteristics method, classification of second-order equations, separation of variables, Green's functions. Lp and Sobolev spaces, distributions, variational formulation and weak solutions, Lax-Milgram theorem, Galerkin approximation. Parabolic PDEs. Wave equations, hyperbolic systems, nonlinear PDEs, reactiondiffusion equations, infinite-dimensional dynamical systems, regularity.

Prerequisite: MATH 3002 or permission of the School.

MATH 5407 [0.5 credit] (MAT 5134)

Topics in Partial Differential Equations

Theory of distributions, initial-value problems based on two-dimensional wave equations, Laplace transform, Fourier integral transform, diffusion problems, Helmholtz equation with application to boundary and initial-value problems in cylindrical and spherical coordinates. Also offered, with different requirements, as MATH 4701 for which additional credit is precluded.

Prerequisite: MATH 5406 or permission of the School.

STAT 5500 [0.5 credit] (MAT 5177)

Multivariate Normal Theory

Multivariate normal distribution properties, characterization, estimation of means, and covariance matrix. Regression approach to distribution theory of statistics; multivariate tests; correlations; classification of observations; Wilks' criteria.

Prerequisite: MATH 3500.

STAT 5501 [0.5 credit] (MAT 5191)

Mathematical Statistics II

Confidence intervals and pivotals; Bayesian intervals; optimal tests and Neyman-Pearson theory; likelihood ratio and score tests; significance tests; goodness-of-fit-tests; large sample theory and applications to maximum likelihood and robust estimation. Also offered, with different requirements, as MATH 4507 for which additional credit is precluded.

Prerequisite: MATH 4500 or STAT 5600 or permission of the School.

STAT 5502 [0.5 credit] (MAT 5192)

Sampling Theory and Methods

Unequal probability sampling with and without replacement; unified theory for standard errors; prediction approach; ratio and regression estimation; stratification and optimal designs; multistage cluster sampling; double sampling; domains of study; post-stratification; nonresponse; measurement errors; related topics.

Prerequisite: MATH 4502 or permission of the School.

STAT 5503 [0.5 credit] (MAT 5193)

Linear Models

Theory of non full rank linear models; estimable functions, best linear unbiased estimators, hypotheses testing, confidence regions; multi-way classifications; analysis of covariance; variance component models; maximum likelihood estimation, Minque, Anova methods; miscellaneous topics.

Prerequisite: MATH 4500 or STAT 5600 or permission of the School.

STAT 5504 [0.5 credit] (MAT 5194)

Stochastic Processes and Time Series Analysis

Stationary stochastic processes, inference for stochastic processes, applications to time series and spatial series analysis.

Prerequisite: MATH 4501 or permission of the School.

STAT 5505 [0.5 credit] (MAT 5195)

Design of Experiments

Overview of linear model theory; orthogonality; randomized block and split plot designs; latin square designs; randomization theory; incomplete block designs; factorial experiments: confoundin g and fractional replication; response surface methodology. Miscellaneous topics.

Prerequisite: STAT 3505 or STAT 4500 or STAT 5600 or permission of the School.

STAT 5506 [0.5 credit] (MAT 5175)

Robust Statistical Inference

Nonparametric tests for location, scale, and regression parameters; derivation of rank tests; distribution theory of linear rank statistics and their efficiency. Robust estimation of location, scale and regression parameters; Huber's M-estimators, Rank-methods, L-estimators. Influence function. Adaptive procedures. Also offered, with different requirements, as MATH 4506 for which additional credit is precluded.

Prerequisite: MATH 4500 or STAT 5600 or permission of the School.

STAT 5507 [0.5 credit] (MAT 5176)

Advanced Statistical Inference

Pure significance test; uniformly most powerful unbiased and invariant tests; asymptotic comparison of tests; confidence intervals; large-sample theory of likelihood ratio and chi-square tests; likelihood inference; Bayesian inference; fiducial and structural methods; resampling methods.

Prerequisite: MATH 4507 or STAT 5501 or permission of the School.

STAT 5508 [0.5 credit] (MAT 5172)

Topics in Stochastic Processes

Course contents will vary, but will include topics drawn from Markov processes. Brownian motion, stochastic differential equations, martingales, Markov random fields, random measures, and infinite particle systems, advanced topics in modeling, population models, etc.

Prerequisites: STAT 3506 or STAT 4501, or permission of the School.

STAT 5509 [0.5 credit] (MAT 5196)

Multivariate Analysis

Multivariate methods of data analysis, including principal components, cluster analysis, factor analysis, canonical correlation, MANOVA, profile analysis, discriminant analysis, path analysis. Also offered at the undergraduate level, with different requirements, as MATH 4503, f or which additional credit is precluded.

Prerequisite: MATH 4500 or STAT 5600 or permission of the School.

STAT 5600 [0.5 credit] (MAT 5190)

Mathematical Statistics I

Statistical decision theory; likelihood functions; sufficiency; factorization theorem; exponential families; UMVU estimators; Fisher's information; Cramer-Rao lower bound; maximum likelihood, moment estimation; invariant and robust point estimation; asymptotic properties; Bayesian point estimation. Also offered, with different requirements, as MATH 4500 for which additional credit is precluded.

Prerequisite: MATH 3500 or permission of the School.

STAT 5601 [0.5 credit] (MAT 5197)

Stochastic Optimization

Topics chosen from stochastic dynamic programming, Markov decision processes, search theory, optimal stopping. Also offered at the undergraduate level, with different requirements, as MATH 4509, for which additional credit is precluded.

Prerequisite: STAT 3506 or permission of the School.

STAT 5602 [0.5 credit] (MAT 5317)

Analysis of Categorical Data

Analysis of one-way and two-way tables of nominal data; multi-dimensional contingency tables, log-linear models; tests of symmetry, marginal homogeneity in square tables; incomplete tables; tables with ordered categories; fixed margins, logistic models with binary response; measures of association and agreement.

Prerequisites: MATH 4500 or STAT 5600, MATH 4507 or STAT 5501, or permission of the School.

STAT 5603 [0.5 credit] (MAT 5318)

Reliability and Survival Analysis

Types of censored data; nonparametric estimation of survival function; graphical procedures for model identification; parametric models and maximum likelihood estimation; exponential and Weibull regression models; nonparametric hazard function models and associate statistical inference; rank tests with censored data applications.

Prerequisites: MATH 4500 or STAT 5600, MATH 4507 or STAT 5501 or permission of the School.

STAT 5604 [0.5 credit] (MAT 5173)

Stochastic Analysis

Brownian motion, continuous martingales, and stochastic integration.

Prerequisites: MATH 4501 or STAT 5708 or permission of the School.

MATH 5605 [0.5 credit] (MAT 5165)

Theory of Automata

Algebraic structure of sequential machines, de-composition of machines; finite automata, formal languages; complexity. Also offered, with different requirements, as MATH 4805/COMP 4805 for which additional credit is precluded.

Prerequisite: MATH 2100 or permission of the School.

MATH 5607 [0.5 credit] (MAT 5324)

Game Theory

Two-person zero-sum games; infinite games; multi-stage games; differential games; utility theory; two-person general-sum games; bargaining problem; n-person games; games with a continuum of players. Also offered, with different requirements, as MATH 4807 for which additional credit is precluded.

Prerequisite: MATH 3001 or permission of the School.

MATH 5609 [0.5 credit] (MAT 5301)

Topics in Combinatorial Mathematics

Prerequisite: permission of the School.

STAT 5701 [0.5 credit] (MAT 5198)

Stochastic Models

Markov systems, stochastic networks, queuing networks, spatial processes, approximation methods in stochastic processes and queuing theory. Applications to the modeling and analysis of computer-communications systems and other distributed networks. Also offered, with different requirements, as MATH 4508 for which additional credit is precluded.

Prerequisite: STAT 3506 or permission of the School.

STAT 5702 [0.5 credit] (MAT 5182)

Modern Applied and Computational Statistics

Resampling and computer intensive methods: bootstrap, jackknife with applications to bias estimation, variance estimation, confidence intervals, and regre ssion analysis. Smoothing methods in curve estimation; statistical classification and pattern recognition: error counting methods, optimal classifiers, bootstrap estimates of the bias of the misclassification error.

Prerequisite: permission of the instructor.

STAT 5703 [0.5 credit] (MAT 5181)

Data Mining

Visualization and knowledge discovery in massive datasets; unsupervised learning: clustering algorithms; dimension reduction; supervised learning: pattern recognition, smoothing techniques, classification. Computer software will be used.

Prerequisite: permission of the instructor.

STAT 5704 [0.5 credit] (MAT 5174)

Network Performance

Advanced techniques in performance evaluation of large complex networks. Topics may include classical queueing theory and simulation analysis; models of packet networks; loss and delay systems; blocking probabilities.

Prerequisite: some familiarity with probability and stochastic processes and queueing, or permission of the instructor.

STAT 5708 [0.5 credit] (MAT 5170)

Probability Theory I

Probability spaces, random variables, expected values as integrals, joint distributions, independence and product measures, cumulative distribution functions and extensions of probability measures, Borel-Cantelli lemmas, convergence concepts, independent identically distributed sequences of random variables.

Prerequisites: MATH 3001, MATH 3002, and MATH 3500, or permission of the School.

STAT 5709 [0.5 credit] (MAT 5171)

Probability Theory II

Laws of large numbers, characteristic functions, central limit theorem, conditional probabilities and expectations, basic properties and convergence theorems for martingales, introduction to Brownian motion.

Prerequisite: STAT 5708 (MAT 5170) or permission of the School.

MATH 5801 [0.5 credit] (MAT 5303)

Linear Optimization

Linear programming problems; simplex method, upper bounded variables, free variables; duality; postoptimality analysis; linear programs having special structures; integer programming problems; unimodularity; knapsack problem.

Prerequisite: course in linear algebra and permission of the School.

MATH 5802 [0.5 credit] (MAT 5325)

Introduction to Information and Systems Science

Introduction to the process of applying computers in problem solving. Emphasis on the design and analysis of efficient computer algorithms for large, complex problems. Applications: data manipulation, databases, computer networks, queuing systems, optimization. (Also listed as SYSC 5802, COMP 5802 and ISYS 5802.)

MATH 5803 [0.5 credit] (MAT 5304)

Nonlinear Optimization

Methods for unconstrained and constrained optimization problems; Kuhn-Tucker conditions; penalty functions; duality; quadratic programming; geometric programming; separable programming; integer nonlinear programming; pseudo-Boolean programming; dynamic programming.

Prerequisite: permission of the School.

MATH 5804 [0.5 credit] (MAT 5307)

Topics in Operations Research

MATH 5805 [0.5 credit] (MAT 5308)

Topics in Algorithm Design

MATH 5806 [0.5 credit] (MAT 5180)

Numerical Analysis

Error analysis for fixed and floating point arithmetic; systems of linear equations; eigen-value problems; sparse matrices; interpolation and approximation, including Fourier approximation; numerical solution of ordinary and partial differential equations.

Prerequisite: permission of the School.

MATH/COMP 5807 [0.5 credit] (MAT 5167)

Formal Language and Syntax Analysis

Computability, unsolvable and NP-hard problems. Formal languages, classes of language automata. Principles of compiler design, syntax analysis, parsing (top-down, bottom-up), ambiguity, operator precedence, automatic construction of efficient parsers, LR, LR(O), LR(k), SLR, LL(k). Syntax directed translation.

Prerequisites: MATH 5605 or MATH 4805 or COMP 3002, or permission of the School.

MATH 5808 [0.5 credit] (MAT 5305)

Combinatorial Optimization I

Network flow theory and related material. Topics will include shortest paths, minimum spanning trees, maximum flows, minimum cost flows. Optimal matching in bipartite graphs.

Prerequisite: permission of the School.

MATH 5809 [0.5 credit] (MAT 5306)

Combinatorial Optimization II

Topics include optimal matching in non-bipartite graphs, Euler tours and the Chinese Postman problem. Other extensions of network flows: dynamic flows, multicommodity flows, and flows with gains, Bottleneck problems. Matroid optimization. Enumerative and heuristic algorithms for the Traveling Salesman and other "hard" problems.

Prerequisite: MATH 5808.

MATH 5818 [0.5 credit] (MAT 5166)

Graph Theory

Paths and cycles, trees, connectivity, Euler tours and Hamilton cycles, edge colouring, independent sets and cliques, vertex colouring, planar graphs, directed graphs. Selected topics from one or more of the following areas: algebraic graph theory, topolog ical graph theory, random graphs.

Prerequisite: MATH 3805 or permission of the School.

MATH 5819 [0.5 credit]

Combinatorial Enumeration

Ordinary and exponential generating functions, product formulas, permutations, rooted trees, cycle index, WZ method. Lagrange inversions, singularity analysis of generating functions and asymptotics. Selected topics from one or more of the following areas: random graphs, random combinatorial structures, hypergeometric functions.

Prerequisite: MATH 3805 or permission of the School.

MATH 5900 [0.5 credit] (MAT 5990)

Seminar

MATH 5901 [0.5 credit] (MAT 5991)

Directed Studies

STAT 5902 [0.5 credit] (MAT 5992)

Seminar in Biostatistics

Students work in teams on the analysis of experimental data or experimental plans. The participation of experimenters in these teams is encouraged. Student teams present their results in the seminar, and prepare a brief written report on their work.

MATH 5903 [0.5 credit]

Project

Intended for students registered in Information and Systems Science and M.C.S. programs. Students pursuing the non-thesis option will conduct a study, analysis, and/or design project. Results will be given in the form of a typewritten report and oral presentation.

STAT 5904 [0.5 credit]

Statistical Internship

This project-oriented course allows students to undertake statistical research and data analysis projects as a cooperative project with governmental or industrial sponsors. Practical data analysis and consulting skills will be emphasized. The grade will be based upon oral and written presentation of results.

Prerequisite: permission of the Institute.

MATH/SYSC/COMP 5905 [2.0 credits]

M.C.S. Thesis

MATH 5906 (MAT 5993)

Research Internship

This course affords students the opportunity to undertake research in mathematics as a cooperative project with governmental or industrial sponsors. The grade will be based upon the mathematical content and upon oral and written presentation of results.

Prerequisite: permission of the Institute.

MATH/ISYS/SYSC/COMP 5908 [1.5 credits]

M.Sc. Thesis in Information and Systems Science

MATH 5909 [1.5 credits]

M.Sc. Thesis

MATH 6002 [0.5 credit] (MAT 5309)

Harmonic Analysis on Groups

Transformation groups; Haar measure; unitary representations of locally compact groups; completeness and compact groups; character theory; decomposition.

MATH 6008 [0.5 credit] (MAT 5326)

Topics in Analysis

MATH 6009 [0.5 credit] (MAT 5329)

Topics in Analysis

MATH 6101 [0.5 credit] (MAT 5327)

Topics in Algebra

MATH 6102 [0.5 credit] (MAT 5330)

Topics in Algebra

MATH 6103 [0.5 credit] (MAT 5331)

Topics in Algebra

MATH 6104 [0.5 credit] (MAT 5158)

Lie Groups

Matrix groups: one-parameter groups, exponential map, Campbell-Hausdorff formula, Lie algebra of a matrix group, integration on matrix groups. Abstract Lie groups.

Prerequisites: MATH 5007 and PADM 5107 or permission of the School.

MATH 6201 [0.5 credit] (MAT 5312)

Topics in Topology

MATH 6507 [0.5 credit] (MAT 5313)

Topics in Probability and Statistics

MATH 6508 [0.5 credit] (MAT 5314)

Topics in Probability and Statistics

MATH 6806 [0.5 credit] (MAT 5361)

Topics in Mathematical Logic

MATH 6807 [0.5 credit] (MAT 5162)

Mathematical Foundations of Computer Science

Foundations of functional languages, lambda calculi (typed, polymorphically typed, untyped), Curry-Howard Isomorphism, proofs-as-programs, normalization and rewriting theory, operational semantics, type assignment, introduction to denotational semantics of programs, fixed-point programming.

Prerequisites: honours undergraduate algebra and either topology or analysis, permission of the instructor or some acquaintance with logic.

MATH 6900 [0.5 credit] (MAT 6990)

Seminar

MATH 6901 [0.5 credit] (MAT 6991)

Directed Studies

MATH 6909

Ph.D. Thesis