Graduate Calendar Archives: 2001 / 2002

Ottawa-Carleton Institute of Mathematics and Statistics

The Institute

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, as well as 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 p.91.

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 p.209.

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

• N.U. Ahmed, Nonlinear Functional Analysis, Control Theory (UO)
• Mayer Alvo, Nonparametric Statistics, Sequential Analysis (UO)
• Y. Billig, Algebra (C)
• Amitava Bose, Stochastic Modelling, Probability Theory (C)
• W.D. Burgess, Algebra, Non-Commutative Rings (UO)
• Charles Castonguay, Demography (UO)
• M.P. Closs, Native American Mathematics (UO)
• Miklós Csörgó, Probability and Statistics (C)
• A.R. Dabrowski, Invariance Principles, Weakly Dependent Variables (UO)
• Daniel Daigle, Algebraic Geometry, Commutative Algebra (UO)
• D.A. Dawson, Stochastic Processes and Probability Theory (C)
• Benoit Dionne, Ordinary Differential Equations, Bifurcation Theory (UO)
• J.D. Dixon, Group Theory, Algebra Computation (C)
• Vlastimil Dlab, Finite Dimensional Algebras, Representation Theory (C)
• P. Farrell, Sampling, Discrete Data, Applied Statistics (C)
• Che-Kao Fong, Operator Theory (C)
• Zhicheng Gao, Graph Theory (C)
• C.W.L. Garner, Foundations of Geometry (C)
• Thierry Giordano, Operator Algebras, Ergodic Theory (UO)
• D.E. Handelman, K-theory, Operator Algebras, Ring Theory (UO)
• Roger Herz-Fischler, History and Sociology of Mathematics (C)
• B.G. Ivanoff, Probability, Point Processes, Martingales (UO)
• W. Jaworski, Probability (C)
• Barry Jessup, Rational Homotopy (UO)
• Daniel Krewski, Applied Statistics in Medicine (C)
• E.O. Kreyszig, Partial Differential Equations, Numerical Analysis (C)
• L.E. May, Numerical Analysis (C)
• D.R. McDonald, Applied Probability (UO)
• Sam Melkonian, Non-linear Differential Equations (C)
• S.E. Mills, Applied Statistics, Statistical Methods, Inference (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)
• Erhard Neher, Jordan Algebras (UO)
• L.D. Nel, Nonnormable Analysis and Calculus (C)
• D. Panario, Finite Fields, Combinatronics, Analysis of Algorithms (C)
• J.N. Pandey, Generalized Functions, Partial Differential Equations (C)
• J.C. Poland, Group Theory (C)
• I.S. Pressman, Optimization, Algebra (C)
• Michel Racine, Jordan Algebras (UO)
• Mizanur Rahman, Special Functions (C)
• J.N.K. Rao, Sample Surveys Theory and Methods (C)
• Luis Ribes, Group Theory (C)
• R.B. Richter, Graph Theory, Combinatorics (C)
• Ivan Rival, Combinatorics, Algorithms(UO)
• Wulf Rossmann, Lie Groups (UO)
• Damien Roy, Number Theory (UO)
• A.K. Md. E. Saleh, Order Statistics, Mathematical Statistics (C)
• Iona Schiopu-Kratina, Probability Theory, Stochastic Processes (UO)
• P.J. Scott, Logic, Category Theory (UO)
• Barbara Szyszkowicz, Statistics (C)
• Rem� Vaillancourt, Partial Differential Equations, Numerical Methods (UO)
• K. S. Williams, Number Theory (C)
• 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 (3 year) degree with at least high honours standing may be admitted to a qualifying-year program.

Their subsequent admission to the regular master's program depends on their 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 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 400-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 are 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 70.560, 70.551 in mathematical statistics; 70.452, 70.555 in applied statistics, and 70.451, 70.571 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 modelling, 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 are; 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 six courses selected from the list of high-technology courses authorized by the Director of the Institute. Each student will be assigned an advisor 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

The 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 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:

70.401 Vector Calculus

70.415 Rings and Modules

70.417 Commutative Algebra

70.427 Foundations of Geometry

70.428 Introduction to Differentiable Mani- folds

70.445 Analytical Dynamics

70.446 Hydrodynamics and Elasticity

70.447 Tensor Analysis and Relativity Theory

70.451 Probability Theory

70.452 Sampling: Theory and Methods

70.453 Applied Multivariate Analysis

70.456 Non-Parametric Methods

70.458 Stochastic Models

70.459 Stochastic Optimization

70.472 Integral Transforms

70.473 Qualitative Theory of Ordinary Differential Equations

70.482 Introduction to Mathematical Logic

70.483 Topics in Applied Logic

70.484 Design and Analysis of Algorithms

70.486 Numerical Analysis

70.488 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 2001-2002, please consult the Registration Instructions and Class Schedule booklet published in the summer.

F,W,S indicates term of offering. Courses offered in the fall and winter are followed by T. The number following the letter indicates the credit weight of the course: 1 denotes 0.5 credit, 2 denotes 1.0 credit, etc.

Mathematics 70.501W1 (MAT5120)

Abstract Measure Theory

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

Prerequisite: Mathematics 70.407.

Mathematics 70.503F1 (MAT5122)

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.

Mathematics 70.504F1 (MAT5129)

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: Mathematics 70.302 and 70.403.

Mathematics 70.505F1 (MAT5127)

Complex Analysis

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

Mathematics 70.506F1 (MAT5316)

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: Mathematics 70.403.

Mathematics 70.507F1 (MAT5125)

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 Daniell-Stone theory. Also offered , with different requirements, as Mathematics 70.407 for which additional credit is precluded.

Prerequisites: Mathematics 70.301 and 70.302 (MAT3125) or permission of the Department.

Mathematics 70.508W1 (MAT5126)

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 Mathematics 70.403 for which additional credit is precluded.

Prerequisite: Mathematics 70.507 (MAT5125) or permission of the Department.

Mathematics 70.509F1 (MAT5121)

Introduction to Hilbert Space

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

Prerequisites: Mathematics 70.301, 70.302, and 70.403.

Mathematics 70.512F1 (MAT5148)

Group Representations and Applications

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

Mathematics 70.513F1 (MAT5146)

Rings and Modules

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

Mathematics 70.514F1 (MAT5143)

Lie Algebras

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

Prerequisites: Mathematics 70.517 (MAT5141) and 70.519 (MAT5142) or permission of the Department.

Mathematics 70.516W1 (MAT5145)

Group Theory

Fundamental principles as applied to abelian, nilpotent, solvable, free, and finite groups; representations. Also offered, with different requirements, as Mathematics 70.416, for which additional credit is precluded.

Prerequisite: Mathematics 70.310 or permission of the Department.

Mathematics 70.517F1 (MAT5141)

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 Department.

Mathematics 70.518W1 (MAT5147)

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 Mathematics 70.418 for which additional credit is precluded.

Prerequisite: Mathematics 70.310 or permission of the Department.

Mathematics 70.519W1 (MAT5142)

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: Mathematics 70.517 (MAT5141) and permission of the Department.

Mathematics 70.521W1 (MAT5150)

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 Department.

Mathematics 70.522F1 (MAT5168)

Homology Theory

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

Prerequisite: Mathematics 70.425.

Mathematics 70.525F1 (MAT5151)

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 Mathematics 70.425 for which additional credit is precluded.

Prerequisite: Mathematics 70.301 or permission of the Department.

Mathematics 70.526W1 (MAT5152)

Topology II

Covering spaces, homology via the Eilenberg-Steenrod Axioms, applications, construction of a homology functor. Also offered, with different requirements, as Mathematics 70.426 for which additional credit is precluded.

Prerequisites: Mathematics 70.310 (MAT3143) and 70.525 (MAT5151) or permission of the Department.

Mathematics 70.527F1 (MAT5169)

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: Mathematics 70.310 (may be taken concurrently) or permission of the Department.

Mathematics 70.528F1 (MAT5155)

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: Mathematics 70.301 or permission of the Department.

Mathematics 70.531F1 (MAT5161)

Mathematical Logic

A basic graduate course in mathematical logic. Propositional and predicate logic, proof theory, 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 alegebra, analysis and topology or permission of the instructor.

Mathematics 70.535F1 (MAT5163)

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 Mathematics 70.435, for which additional credit is precluded.

Prerequisite: Mathematics 70.307 or permission of the Department.

Mathematics 70.536W1 (MAT5164)

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 Mathematics 70.436 for which additional credit is precluded.

Prerequisite: Mathematics 70.310 or permission of the Department.

Mathematics 70.543 (MAT5187)

Topics in Applied Mathematics

Mathematics 70.545F1 (MAT5131)

Ordinary Differential Equations

Existence and uniqueness theorems, boundary value problems, qualitative theory.

Prerequisite: Mathematics 70.308 or permission of the Department.

Mathematics 70.546F1 (MAT5133)

Introduction to Partial Differential Equations

First order linear, quasi-linear, and nonlinear equations; second order equations in two or more variables; systems of equations; the wave equation; Laplace and Poisson equations; Dirichlet and Neumann problems; Green's functions. Also offered ,with different requirements, as Mathematics 70.470 for which additional credit is precluded.

Prerequisites: Mathematics 70.302, or 70.307 and 70.308, or permission of the Department.

Mathematics 70.547W1 (MAT5134)

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 Mathematics 70.471 for which additional credit is precluded.

Prerequisite: Mathematics 70.546 or permission of the Department.

Mathematics 70.550F1 (MAT5177)

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: Mathematics 70.350.

Mathematics 70.551W1 (MAT5191)

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 Mathematics 70.457 for which additional credit is precluded.

Prerequisite: Mathematics 70.450 or 70.560 or permission of the Department.

Mathematics 70.552W1 (MAT5192)

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: Mathematics 70.452 or permission of the Department.

Mathematics 70.553F1 (MAT5193)

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: Mathematics 70.450 or 70.560 or permission of the Department.

Mathematics 70.554F1 (MAT5194)

Stochastic Processes and Time Series Analysis

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

Prerequisite: Mathematics 70.451 or permission of the Department.

Mathematics 70.555W1 (MAT5195)

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: confounding and fractional replication; response surface methodology. Miscellaneous topics.

Prerequisite: Mathematics 70.355 or 70.450 or 70.560 or permission of the Department.

Mathematics 70.556W1 (MAT5175)

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 Mathematics 70.456 for which additional credit is precluded.

Prerequisite: Mathematics 70.450 or 70.560 or permission of the Department.

Mathematics 70.557W1 (MAT5176)

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: Mathematics 70.457 or 70.551 or permission of the Department.

Mathematics 70.558F1 (MAT5172)

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 modelling, population models, etc.

Prerequisites: Mathematics 70.356 or 70.451, or permission of the Department.

Mathematics 70.559F1 (MAT5196)

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 Mathematics 70.453, for which additional credit is precluded.

Prerequisite: Mathematics 70.450 or 70.560 or permission of the Department.

Mathematics 70.560F1(MAT5190)

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 Mathematics 70.450 for which additional credit is precluded.

Prerequisite: Mathematics 70.350 or permission of the Department.

Mathematics 70.561F1 (MAT5197)

Stochastic Optimization

Topics chosen from stochastic dynamic programing, Markov decision processes, search theory, optimal stopping. Also offered at the undergraduate level, with different requirements, as Mathematics 70.459, for which additional credit is precluded.

Prerequisite: Mathematics 70.356 or permission of the Department.

Mathematics 70.562F1 (MAT5317)

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; applications biological

Prerequisites: Mathematics 70.450 or 70.560, 70.457 or 70.551, or permission of the Department.

Mathematics 70.563W1 (MAT5318)

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: Mathematics 70.450 or 70.560, 70.457 or 70.551 or permission of the Department.

Mathematics 70.564F1 (MAT5173)

Stochastic Analysis

Brownian motion, continuous martingales, and stochastic integration.

Prerequisites: Mathematics 70.451 or 70.578 or permission of the Department

Mathematics 70.565F1 (MAT5165)

Theory of Automata

Algebraic structure of sequential machines, de-composition of machines; finite automata, formal languages; complexity. Also offered ,with different requirements, as Mathematics 70.485/Computer Science 95.485 for which additional credit is precluded.

Prerequisite: Mathematics 70.210 or permission of the Department.

Mathematics 70.567F1 (MAT5324)

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 Mathematics 70.487 for which additional credit is precluded.

Prerequisite: Mathematics 70.301 or permission of the Department.

Mathematics 70.569F1 (MAT5301)

Topics in Combinatorial Mathematics

Prerequisite: Permission of the Department.

Mathematics 70.571W1 (MAT5198)

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 Mathematics 70.458 for which additional credit is precluded.

Prerequisite: Mathematics 70.356 or permission of the Department

Mathematics 70.578F1 (MAT5170)

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: Mathematics 70.301, 70.302, and 70.350, or permission of the Department.

Mathematics 70.579W1 (MAT5171)

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: Mathematics 70.578 (MAT5170) or permission of the Department.

Mathematics 70.581F1 (MAT5303)

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 Department.

Mathematics 70.582F1 (MAT5325)

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 Engineering 94.582, Computer Science 95.582, Information and Systems Science 93.582)

Mathematics 70.583W1 (MAT5304)

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 Department.

Mathematics 70.584F1, W1, S1 (MAT5307)

Topics in Operations Research

Mathematics 70.585F1, W1, S1 (MAT5308)

Topics in Algorithm Design

Mathematics 70.586F1 (MAT5180)

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 Department.

Mathematics 70/95.587F1 (MAT5167)

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: Mathematics 70.565 or 70.485 or Computer Science 95.302, or permission of the Department.

Mathematics 70.588W1 (MAT5305)

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 Department.

Mathematics 70.589W1 (MAT5306)

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 Travelling Salesman and other "hard" problems.

Prerequisite: Mathematics 70.588.

Mathematics 70.590F1, W1, S1 (MAT5990)

Seminar

Mathematics 70.591F1, W1, S1 (MAT5991)

Directed Studies

Mathematics 70.592 (MAT5992)

Seminar in Biostatistics

Students work in teams on the analysis of experimental data orexperimental plans. The participation of experimenters in these teams isencouraged. Student teams present their results in the seminar, and preparea brief written report on their work.

Mathematics 70.593F1, W1, S1

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.

Mathematics 70.594F1, W1, S1

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.

Mathematics 70/94/95.595F4, W4, S4

M.C.S. Thesis

Mathematics 70.596 (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 as well as upon oral and written presentation of results.

Prerequisite: Permission of the Institute.

Mathematics 70/93/94/95.598 F3, W3, S3

M.Sc. Thesis in Information and Systems Science

Mathematics 70.599F3, W3, S3

M.Sc. Thesis

Mathematics 70.602W1 (MAT5309)

Harmonic Analysis on Groups

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

Mathematics 70.608F1, W1, S1 (MAT5326)

Topics in Analysis

Mathematics 70.609F1, W1, S1 (MAT5329)

Topics in Analysis

Mathematics 70.611F1, W1, S1 (MAT5327)

Topics in Algebra

Mathematics 70.612F1, W1, S1 (MAT5330)

Topics in Algebra

Mathematics 70.613F1, W1, S1 (MAT5331)

Topics in Algebra

Mathematics 70.614W1 (MAT5158)

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: Mathematics 70.507 and 50.517 or permission of the Department.

Mathematics 70.621F1, W1, S1 (MAT5312)

Topics in Topology

Mathematics 70.657F1, W1, S1 (MAT5313)

Topics in Probability and Statistics

Mathematics 70.658F1, W1, S1 (MAT5314)

Topics in Probability and Statistics

Mathematics 70.686F1, W1, S1 (MAT5361)

Topics in Mathematical Logic

Mathematics 70.687F1 (MAT5162)

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.

Mathematics 70.690F1, W1, S1 (MAT6990)

Seminar

Mathematics 70.691F1, W1, S1 (MAT6991)

Directed Studies

Mathematics 70.699F, W, S

Ph.D. Thesis