| **MATHEMATICS AND STATISTICS: COURSES**
All courses listed will not necessarily be offered in any given year.
*MATHEMATICS*
**62-510. Functions of a Real Variable I**
Lebesgue measure, abstract measure, integration, monotone and dominated convergence theorems, Radon-Nikodym theorem, Hahn decomposition theorem, Fubini's theorem, Lp spaces.
**62-511. Functions of a Real Variable II **
Metric spaces, topological spaces, compactness, Stone-Weierstrass and Ascoli theorems, Baire category theorem, classical Banach spaces.
**62-512. Functional Analysis I **
Normed linear spaces and examples, Hahn-Banach theorem, open mapping theorem, principle of uniform boundedness, weak and weak* topologies on Banach spaces, Hilbert spaces and bounded linear operators on Hilbert spaces.
**62-513. Functional Analysis II **
Banach algebras and spectral theory, operator theory, C*-algebras and their representations, elementary von Neumann algebra theory.
**62-520. Abstract Algebra **
Elements of group theory are explored including such topics as: the Sylow Theorems, classification of groups of low order, Jordan-Holder Theorem, solvable groups, nilpotent groups, groups in terms of generators and relation, representations of groups, basic operations on representations, orthogonality relations
**62-521. Ring Theory and Modules**
This course is designed to introduce students to the structure theory of general rings and their modules. It will provide an appropriate foundation for more advanced graduate material in algebra at the doctoral level and will be an excellent preparation for doctoral comprehensive examinations. Topics covered will include: semisimple rings, Wedderburn-Artin Theorem, modules over a principal ideal domain, projective, injective and flat modules, introduction to homology theory.
**62-523. Lie Algebras **
Engel's Theorem, Lie's Theorem, criterion for semi simplicity, root space decomposition, universal enveloping algebra, PBW basis, representation theory, finite dimensional modules, Harish-Chandra's Theorem.
**62-525. Matrix Algebra and Analysis **
Aspects of measure theory and probability, convergence theorems for integrations and expectations, moments and inequalities, construction of Lebesgue-Stieltjes measure, Riemann-Stieltjes integral, comparison of Riemann and Lebesgue integrals, introduction to complex variable, contour integration, characteristics functions, elementary theorems on linear and matrix algebra, generalized and conditional inverses, distributions of quadratic forms. This course is designed for graduate students in Statistics.
**62-530. General Topology **
Elementary concepts of topology, product and quotient spaces, continuity and homeomorphisms, nets and filters, separation and countability, compactness, connectedness.
**62-551. Advanced Linear Programming**
By presenting results and their proofs, the student will acquire a solid understanding of the theory, algorithms and applications of linear programming. This course is a prerequisite for more advanced courses on integer programming, combinatorial optimization and networks flows. Topics emphasized include: formulations of linear programming problems, convex sets and convex functions, separation theorem, Farkas' lemma, duality theory, economic interpretation of duality, optimality conditions, primal and dual simplex algorithms, cycling, sensitivity analysis, interior-point methods and central path, primal-dual methods, convergence results.
**62-552. Nonlinear Programming**
This course will provide an introduction to the field of nonlinear programming. By presenting results and their proofs, the student will acquire a solid understanding of the theory behind most algorithms for solving nonlinear optimization problems. He/she will also acquire the knowledge and skills needed to conduct research in this area. Topics covered will include: unconstrained optimization, necessary and sufficient conditions for optimality, convex sets and convex functions, steepest descent method, Newton's method, conjugate gradient methods, quasi-Newton's methods, separation theorem and Farkas' lemma, Karush-Kuhn-Tucker conditions, constraint qualification conditions, duality theory, Barrier methods, and quadratic programming.
**62-553. Integer Programming**
This course will provide the student with a rigorous introduction to the field of integer programming. Topics covered will include: modelling with integer variables, elements of computational complexity theory, elements of polyhedral theory, total unimodularity, branch and bound methods, cutting plane methods, implicit enumeration, Bender decomposition, dynamic programming, lagrangian relaxation, knapsack problems, set covering/packing/partitioning problems, heuristic methods.
**62-554 Combinatorial Optimization**
This course will provide a rigorous introduction to combinatorial optimization. The student will develop a solid understanding of the theory, algorithms and applications of these problems and their connections to integer programming, linear programming and complexity theory. Topics will include: formulation of combinatorial optimization problems, polytopes and polyhedra, elements of computational complexity theory, shortest paths, bipartite and non-bipartite matchings, max-flow min-cut theorem, multi-commodity flow problems, clique and coloring problems, perfect graphs, traveling salesman problem, spanning trees, matroids.
**62-561. Partial Differential Equations **
First-order equations, classification of second-order equations, canonical forms and general solutions of second-order equations, diffusion equations, Laplace equations, the maximum principle and uniqueness for the Dirichlet problem, wave equations, Riemann's method for linear hyperbolic equation, Green's functions and transform methods.
**62-568. Numerical Analysis I**
General error analysis, direct solution of linear algebraic equations, iterative solution of linear equations, algebraic eigenvalue problems, numerical solution of a system of nonlinear equations, error analysis.
**62-569. Numerical Analysis II**
Interpolation and approximation, numerical integration and differentiation, finite differences. Numerical solution of ordinary and partial differential equations using finite differences.
**62-598. Special Topics **
**62-795. Seminar**
Presentations from graduate students, faculty and visiting researchers on various research topics of mathematics and statistics. All graduate students are expected to attend each and every seminar; however students must attend no fewer than 75 percent of all seminars. Students must register in this course in each term of full-time registration in the M.Sc. programs. This course will be graded on a PASS/FAIL basis.
**62-796. Major Paper **
**62-797. Thesis (M.Sc.)**
*STATISTICS*
**65-540. Theory of Probability **
Basic probability model, random variables and their distributions, expectation, convergence of random variables and their distributions, independence and conditional dependence. Zero-one laws, characteristic functions, generating functions, Law of large numbers, Central Limit Theorem.
**65-541. Stochastic Processes**
Discrete and continuous time Markov processes, renewal theory, branching processes, Brownian motion.
**65-542. Advanced Mathematical Statistics **
A review of probability theory, transformations and expectations, common families of distributions, inequalities and identities, properties of a random sample, data reduction and best estimation strategies, asymptotic approximation.
**65-543. Statistical Inference **
Measure of performance, pure significance test and formal hypothesis testing, interval estimation, asymptotic evaluations, analysis of variance and regression, analysis of categorical data.
**65-544. Multivariate Analysis**
This course is aimed at giving theoretical and methodological background on inference procedure for the analysis of multivariate continuous data mainly under the assumption of normaility.
**65-546. Statistical Data Analysis **
This course takes a computer-oriented approach to equip students with the experience of data analysis, beginning with designing of experiment to presentation of results. Depending on the background of the students, different topics will be emphasized.
**65-548. Non-parametric Statistics **
Nonparametric tests including Wilcoxon, Mann-Whitney, Smirnov, Fisher's exact test, Cox and Stuart test for trend, runs test. Estimation. Theory and applications.
**65-549. Discrete Multivariate Analysis**
This course is aimed at giving theoretical and methodological background for the analysis of discrete data mainly in the form of contingency tables. Other discrete models as part of the generalized linear models may be covered.
**65-550. Generalized Linear Models **
This course is aimed at giving theoretical and methodological background for the analysis of discrete or continuous data using the generalized linear models and other semi-parametric models where full distributional assumptions cannot be justified.
**65-552. Experimental Design **
Factorial designs with and without interactions, randomized block, Latin square, balanced incomplete block, nested design, confounding factorial and other designs. Fixed, random and mixed models.
**65-554. Theory of Sampling and Surveys**
Sampling methods including simple random, stratified, cluster, PPS and multistage, ratio and regression estimates. Theory and applications.
**65-555. Regression Analysis**
Simple and multiple linear regression, inference on regression parameters, residual analysis, stepwise regression, polynomial regression, diagnostics and remedial measures for multicollinearity and influential observations, weighted least squares, logistic regression, nonlinear regression.
**65-557. Large Sample Theory**
This course will present the basic large sample theory with a minimum coating of abstraction and at a level with the usual program in statistics and applied statistics. The main objective is to present the essentials of large sample theory of statistics with a view toward its application to a variety of problems that generally crop up in other areas. Topics to be covered will include: mathematical background, stochastic convergence, weak convergence and central limit theorems, asymptotic behaviour of estimators and test statistic, multivariate extensions, bootstrapping.
**65-558 Sequential Analysis**
This course will equip graduate students in Statistics and Biological Sciences with a firm knowledge of the increasingly important sequential analysis methodology. Both theoretical and practical aspects of the sequential analysis applied to medical clinical trials and to biological studies will be covered in this course. Methodologies for designing and analyzing sequential clinical trials using both fully and group sequential methods: permuted block design, sequential biased coin design, Wald's SPRT procedures, O'Brien-Fleming and Pocock group sequential procedures, alpha- and beta-spending function approach, Whitehead's triangular tests, and post-trail estimation methods. Software such as SAS and Splus will be used for analyzing real and simulated trials.
**65-559. Topics in Statistics**
Topics offered may include queueing theory, statistical quality control, statistical consulting, survival analysis, time series analysis, decision theory, and Bayesian analysis.
**65-795. Seminar**
Presentations from graduate students, faculty and visiting researchers on various research topics of mathematics and statistics. All graduate students are expected to attend each and every seminar; however students must attend no fewer than 75 percent of all seminars. Students must register in this course in each term of full-time registration in the M.Sc. / Ph.D programs. This course will be graded on a PASS/FAIL basis.
**65-796. Major Paper**
**65-797. Thesis (M.Sc.)**
**65-798. Dissertation (Ph.D.)** |