Department of Mathematics and Statistics

In the Spring of 2010,  the department will offer the core courses Math-502 (Deterministic Math Models), Math-503 (Mathematical Statistics, two sections), and Math-504 (Numerical Methods). The following elective courses will be offered: Math-603 (Signal Processing), Math-656 (Data Mining), Math-657 (Categorical Data Analysis),  Math-701 (Statistical Consulting Practicum),  and Math-703 (Internship, only for students who entered before fall 2009).

In addition, many non-mathematical elective courses at Georgetown University  are available to graduate students. Details may be found  here.

The  Biostatistics and Epidemiology graduate program is offering BIST-512 (Statistical Modeling I), BIST-513 (Statistical Modeling II), BIST-531 (Pattern Recognition), and BIST-541 (Principles of Epidemiology).

BIST-512, BIST-513, and BIST-531 may be taken as math/stats elective courses. BIST-512 is similar to MATH-651 (Regression Models), a course that is usually offered in the fall. BIST-531 is very similar to MATH-656 (Data Mining). Students in this program are encouraged to take MATH-656 instead of BIST-531. BIST-541 may be taken as a non-math elective course.

BIST-502 (Applied Biostatistics) and BIST-505 (Epidemiology and Public Health) are service course for graduate students in the Medical Center and do not count towards the MS degree in Mathematics and Statistics. 

Students who wish to take a course in the BIST program should always contact the BIST instructor first.

Spring 2010 Schedule at a Glance

Course	Title	Instructor	Time	Place
Math-502	Deterministic Math Models	Shaw	M 6:15-8:45pm	TBA
Math-503.1	Mathematical Statistics	Arab	W 6:15-8:45pm	TBA
Math-503.2	Mathematical Statistics	Arab	T 6:15-8:45pm	TBA
Math-504	Numerical Methods	Luo	R 6:15-8:45pm	STM 343
Math-603	Signal Processing	Benke	R 6:15-8:45pm	Comp. Lab
Math-656	Data Mining	Wilson	T 6:15-8:45pm	STM 343
Math-657	Categorical Data Analysis	Tadesse	W 6:15-8:45pm	TBA
Math-701	Statistics Consulting Practicum	TBA	TBA (3 hours per week)	STM 3rd floor
Math-702	Applied Math Clinic	TBA	TBA (3 hours per week)	STM 3rd floor
Math-703	Internship	Staff	various	Various
BIST-512	Statistical Modeling I	Luta	MW 9:00-10:15am	NRB W402
BIST-511	Statistical Modeling II	Bebu	TR 9:00-10:15am	NRB W402
BIST-531	Pattern Recognition	Liu	MW 4:00-5:15pm	NRB W402
BIST-541	Principles of Epidemiology	Loffredo	TR 4:00-5:15pm	Lombardi SS131

 
In the Fall of 2009,  the department offered the core courses Math-501 (Probability Theory with Applications) in two sections and Math-502 (Deterministic Math Models), as well as the elective courses Math-410 (Applied Combinatorics), Math-510 (Mathematical and Statistical Computing),  Math-604 (Stochastic processes),  Math-651 (Regression/General Linear Models), Math-701 (Statistical Consulting Practicum), Math-702 (Applied Mathematics Clinic), and Math-703 (Internship).

During the Summer of 2009, the department offered Math-658 (Survey Sampling). The course was taught for eight weeks, June 1 - July 24, Mo We 6:00pm-8:00pm. The instructor was Dr. Tommy Wright. Math-703 (Internship) was also offered during the summer. 

Math-501: Probability Theory and Applications. Fall. This is an advanced introduction to probability theory. The notions of probability measure, random variable, independence, and conditional probability will be presented, along with the frequentist interpretation of probability embodied in the Law of Large Numbers. Basic properties of discrete and continuous random variables will be developed, including means, variances, c.d.f.s, functions of random variables, generating functions, and Chebyshev’s inequality. Joint distributions will be considered, along with covariances, random vectors, and changes of variable. The course will also include an introduction to stochastic processes of Gaussian and Poisson type, and a discussion of different notions of convergence, culminating in the Central Limit Theorem. Applications will be given in population studies, information theory, and Bayesian inference.

The course will use the book  by Neil Weiss, A Course in Probability, Pearson Addison-Wesley, First Edition, 2006. Available at Amazon.com for about $80 (new) or $60 (used). Some previous exposure to elementary probability theory and random variables is desirable.

Math-502: Deterministic Mathematical Models. Fall and Spring. This is a course on differential and difference equations with an emphasis on derivation and analysis of models of physical phenomena. The course will begin with a brief review of matrix techniques including eigenvalues. As a preliminary to deriving models, a treatment of dimensional analysis will be given, leading to the Buckingham pi-theorem. The course will then introduce first order systems of ordinary differential and difference equations with constant coefficients. Analytical techniques such as linearization, scaling, perturbation, characteristics and variational methods will be studied. Applications of this material will include age-structured populations and single-species non-linear models. The course will include an introduction to partial differential equations, with applications to conservation laws, shock formation and traffic congestion.

Textbook: David A Logan, Applied Mathematics. Either the second or third edition is suitable. Either book may be purchased new or used from any source such as Amazon.com, eBay, etc. For the second edition go to amazon.com, select Books and enter 'David A Logan, Applied Mathematics, Second Edition' in the search field to find this information. Price: $86.53. Used & new available from $40.00. For the third edition enter David A Logan, Applied Mathematics in the search field and find a price of $110, with other new and used from about $80

Math-503: Mathematical Statistics. Spring. This is a first course in the mathematical theory of statistical inference. The emphasis is on classical methods, with appropriate attention also to Bayesian methods. Topics include principles of data reduction (sufficiency, completeness), point estimation (method of moments, maximum likelihood, Bayes and minimax estimators), criteria for point estimation (mean squared error, bias, consistency, efficiency), some asymptotic properties of point estimators, interval estimation, tests of hypotheses (likelihood ratio, most powerful and uniformly most powerful tests, Bayesian tests), criteria for hypothesis tests (error probabilities and power, power functions), asymptotics of some large sample tests, nonparametric methods (e.g. sign test, rank sum test, Wilcoxon, and Mann-Whitney tests), elements of the analysis of variance and linear regression (if time permits).

The course will use Casella/Berger, Statistical Inference, 2nd edition, Chapters 6-12. Additional References: Hogg/McKean/Craig, Introduction to Mathematical Statistics, 6th edition. Bain/Engelhardt, Introduction to Probability and Mathematical Statistics, 2nd edition.

Prerequisites: Calculus of one and several variables, some linear algebra (matrix algebra), Math-501 (Probability Theory).

Math-504: Numerical Methods. Fall and Spring.

This course introduces numerical algorithms for solving nonlinear equations; systems of linear equations; approximation and interpolation; numerical integration; fast Fourier transform and least squares methods, and ordinary differential equations. We will also discuss issues associated with computer
arithmetic. Error analysis and convergence of numerical schemes will be
examined as well.

Prerequisites: multivariable calculus, linear algebra and some exposure
to differential equations. Some knowledge in one computer programming language is required, such as Matlab.

The main text for the course will be Numerical Analysis by Tim Sauer.

Math-510: Mathematical and Statistical Computing. Fall. The objective of this course is to provide students with programming background sufficient for graduate level study in mathematics and statistics. The statistical packages most widely used by practicing statisticians are R and SAS. The first portion of the course (about 8 weeks) will be structured around statistical methods and examples will be worked out using both computing environments. Topics to be covered include data management, descriptive statistics, graphical displays, hypothesis testing, correlation, regression models, and multivariate analysis methods. The emphasis of this part of the course is therefore both on learning R and SAS through hands-on experience with real data, as well as acquiring skills to perform data analysis, interpret the output, and draw conclusions. The last part of the course will cover the essentials of the programming language Matlab which is used widely for scientific computation. This includes basic structure, commands, M-files, do-loops and special commands. The course will proceed by moving through topics in numerical linear algebra and, possibly, selected topics in analysis as time permits. This portion of the course is thus object-oriented, which serves the twin purposes of reprising linear algebra and learning Matlab.

There are no prerequisites besides calculus, linear algebra, and some knowledge of elementary statistics. Students will be required to purchase the student edition of Matlab 7 which comes with a user's manual and is available for $99.

Textbooks: L. Delwiche, S. Slaughter, The Little SAS Book, 3rd  edition 2003.  SAS Publishing (ISBN: 1590473337). Available at Amazon.com for about $40.
W. M. Venables, D. M. Smith, An Introduction to R. 2nd edition 2009. Network Theory Ltd. ISBN: 0954612086. Available in the Georgetown bookstore, at Amazon.com for about $14, and as a free pdf file at http://cran.r-project.org/doc/manuals/R-intro.pdf
T. Driscoll, Learning Matlab. 2009. SIAM. ISBN 0898716837. Available at siam.org or at Amazon.org for $28.
 

Math-520: Nonlinear Differential Equations. Spring. Not offered 2010. Nonlinear  differential equations arising  from various branches of sciences,  economics and finance and have been  playing  an essential role in applied mathematics. Solutions to nonlinear equations always exhibit very interesting and rich phenomena and behavior such as strange attractors, chaos, shock waves and solitary waves. This course emphasizes the study of qualitative dynamical behavior of solutions to some nonlinear equations.  The topics include stability, bifurcations (changes in the nature of solutions as parameters are varied, for example, the creation of periodic orbits from an equilibrium point), chaos for ordinary differential equations, and some nonlinear wave phenomena such as shock waves  and solitary waves for partial differential equations.  Most of the material in this course will be presented by means of examples.

The prerequisite for this course is a first course in ordinary differential equations, for example,  Math-502 or Math-201.

Math-601: Partial Differential Equations. Fall. Not offered 2009. As with any area of applied mathematics, the field of partial differential equations is interesting not only because of its application, but also because it has taken on a mathematical life of its own. The purpose of this course is to give students with minimal background some basic knowledge about partial differential equations. Additionally, this course will also expose students to frontier research. This course will cover the classification of second-order equations and the methods and techniques for finding fundamental solutions for Laplace, Heat, and Wave Equations. The methods include the Fourier transform method, geometric method involving the action along the geodesics, using the eigenfunction expansion, and path integrals method.

Prerequisites: Math-502 (Deterministic Mathematical Models). The textbook will be Applied Partial Differential Equations with Fourier Series and BoundaryValue Problem by Richard Haberman, Pearson Education Inc., 4e, 2003.

Math-602: Optimization. Spring. Not offered 2010. This course presents the most important numerical algorithms for solving unconstrained and constrained optimization problems. We will cover the basic theoretical results in optimization theory, and we will describe the structure of several optimization algorithms including: gradient methods, Newton and quasi-Newton methods, conjugate gradient methods, gradient projection methods, penalty methods, interior-point methods, etc.

Several representative applications of optimization methods will be described in detail, and the students will be required to solve numerically three such applications.

Prerequisite: Math-504 (Numerical Methods) and knowledge of Matlab or Mathematica.

Textbook: tba

Math-603: Signal Processing. Spring.  This course concerns the analysis and implementation of computational algorithms for the processing of digital signals. Students will learn the mathematics governing the theory and the algorithms involved, and will learn the details of the algorithms through writing programs which implement the algorithms. The topics covered include discrete-time signals and systems, the z-transform, frequency analysis of signals and systems, sampling and reconstruction, the fast Fourier transform, digital filters, linear prediction and optimum linear filters. If time permits some aspects of multi-rate signal processing, power spectrum estimation, and wavelet analyisis will be covered. Applications to speech processing, image processing and data compression will also be discussed.

Pre-requisites: Background in multivariable calculus and linear algebra, and some knowledge of real analysis,  complex analysis and computer programming.

The main text for the course will be Digital Signal Processing, 4/E by Proakis and Manolakis.

Math-604: Stochastic Processes. Fall 2009. This course is an introduction to stochastic processes without the use of measure theory.  We will start by considering discrete time stochastic processes by covering Markov chains and martingales.  We will then consider continuous time stochastic processes by covering Poisson processes, general Markov processes and Brownian motion.  The course will emphasize problem solving in the sense that we will introduce and study various theorems that allow us to solve problems of interest.  Computer simulation of stochastic processes through MATLAB will be part of the course.

Pre-requisites: Math-501, background in ordinary differential equations.

Textbook: Rick Durrett, Essentials of Stochastic Processes.

Math-605: Introduction to Financial Mathematics. Spring. Not offered in 2010. This is a course on mathematical finance, emphasizing mathematical models and techniques for pricing financial derivative instruments. Financial markets of stocks, bonds, futures and options; present value analysis; Brownian motion and Ito's formula; option pricing with random walk models, partial differential equations (Black-Scholes), and binomial trees.

Pre-requisites: Calculus of one and several variables, some linear algebra (matrix algebra).

Textbook: The Mathematics  of Financial Derivatives by Paul Wilmott, Sam Howison, Jeff Dewynne, Cambridge University Press, 1995 ISBN 0-521-49789-2 (paperback edition).

Math-610: Combinatorics and Combinatorial Optimization. Not offered in 2009/2010. The course will cover the following topics, most of which are in the textbook.  For the others, there will be some supplementary notes.

• Network flow problems
• Optimal matching algorithms
• Programming, polytopes, and geometry
• String-search and bio-informatics
• The "traveling salesman" problem
• Neural networks and genetic algorithms

The textbook for this class is Combinatorial Optimization,  by Cook, Cunningham, Pulleyblank, and Schrijver, Wiley-Interscience, 1997 (1st ed.).

Math-651: Regression Methods and Generalized Linear Models. Fall. Simple and multiple regression, inference and prediction, model building and diagnostics, analysis of variance (ANOVA), analysis of covariance (ANCOVA), generalized linear models, and other extensions as time permits (e.g. mixed models, nonlinear regression). This course covers both theoretical and applied aspects of regression analysis.  Examples and illustrations will use SAS and/or R.

Pre-requisite: MATH 503 (Mathematical Statistics) or equivalent.

Textbook: Kutner, Nachtsheim, Li, Applied Linear Statistical Models. (5/E).

Math-652: Data Analysis and Multivariate Methods. Spring. Not offered in 2010. This course is an introduction to analysis of multivariate data. Topics include matrix algebra and random vectors, the multivariate normal distribution, Hotelling's T², multivariate linear regression models, principal components, factor analysis, and discriminant analysis.

Prerequisite: Math-501 (Probability Theory) or equivalent and Math-503 (Mathematical Statistics).

Textbook: Applied Multivariate Statistical Analysis (6th Edition) (Hardcover) by Richard A. Johnson and Dean W. Wichern. Prentice-Hall, ISBN 978-0131877153.

Math-654: Computer-intensive Statistical Methods. Not offered in  2009/2010.  Stochastic simulation methods are playing increasingly important roles in science, business, and engineering. The course introduces the basic theory and some applications of stochastic simulation techniques and then focuses on modern methods of statistical inference for problems that are difficult to treat with conventional parametric or asymptotic methods. Topics: Stochastic simulation as an experiment. Random number generation. Transformation methods. Elements of Monte Carlo simulation: rejection, Metropolis algorithm. Simulating random processes. Simulation of and random sampling from data. Empirical cumulative distribution function. Nonparametric tests: Permutation, rank, and randomization tests. Expectation-maximization for missing and censored data. Jack-knife procedures. Bootstrap methods for bias reduction, variance estimation, and confidence intervals. Parametric and nonparametric estimation of probability density functions. The course uses the software package R.

Prerequisites: Math-503 (Mathematical Statistics).

Math-656: Data Exploration and Data Mining. Spring.  Huge volumes of data are constantly being generated by businesses, in science, in telecommunications, and elsewhere, doubling the amount of information available in the world roughly every nine months. This course presents an introduction to computer-based methods for discovering patterns in  large data sets. It focuses on statistical aspects and computational algorithms for numerical and categorical data. After a brief review of graphical exploration methods, the course discusses linear and nonlinear methods of feature extraction and dimension reduction, data tours, clustering methods, and multivariate visualization techniques.  Statistical techniques for classification and prediction such as rule-based methods, decision trees, artificial neural  networks, and Bayes networks are also covered in the course. The course will also discuss methods for assessing the quality of results (cross validation, bootstrap) and for combining results (bagging, boosting). The course will use software such as perl, R, and WEKA.

Prerequisites: Linear algebra (matrix methods), some previous experience with elementary statistics and probability, familiarity with programming. 

Textbooks: TBA. perl, R and WEKA are available as free open source software and are also installed in the classroom and in the departmental computer lab.

Math 657: Categorical Data Analysis. Spring. This course deals with statistical models for the analysis of categorical data. Topics to be covered include inference for contingency tables, generalized linear models with emphasis on logistic regression and loglinear models, and models for clustered/repeated measures. The goal of the course is not to memorize formulae, but to understand and apply statistical concepts and techniques to real data.  Most examples will be illustrated using SAS. Students are free to use the software of their preference.

Pre-requisite: MATH 503 or equivalent
Co-requisite: MATH 651 or BIST512 or equivalent

Textbook: Agresti, A. (2002) Categorical Data Analysis, 2nd Ed. Wiley. Optional Textbook: Stokes M.E., Davis C.S. and Koch G.G. (2000) Categorical Data Analysis using the SAS System, 2nd Ed. Wiley.

Math-658: Survey Sampling. Summer 2009. This course covers design and analysis of sample surveys.  Sample designs include simple random sampling, systematic, stratified, cluster, double, and multistage sampling.  Analytical methods include sample size determination, ratio and regression estimation, imputation for missing data, and nonsampling error to adjustment.

Pre-requisite: Math-501 or equivalent.

Math-701: Statistical Consulting Practicum. Fall and Spring. Students will conduct statistical consulting and data analysis activities, guided by departmental faculty. Tasks will include the design of research protocols, the formulation of statistical models, recommendations regarding appropriate methodologies, data analysis, and interpretation of results. An important component will be communications. Students will be expected to co-author written reports and to give oral presentations to clients.

Math-701 may be taken as a two-credit pass/fail course by students who entered the program before fall 2009. All other students can only register for three-credit and cannot take this course pass/fail.

Prerequisites: Familiarity with statistical software (R and/or SAS) and background in regression analysis are highly recommended. Students are encouraged to take Math-651 concurrently.

Textbook: We will be using statistical reference texts and online sources as needed.

Time and Room: To be determined (weekday afternoons in St. Mary's). Students will be required to be present for a total of three hours per week.  

Math-702: Applied Mathematics Clinic.  Not offered in spring 2010. Students will be introduced to topics of current interest in mathematics and its applications in science, finance etc. Typical topics will be mathematical models in biology and bioinformatics, computational methods in finance, or new methods in signal processing. Students will work in teams and will give regular presentations.

Math-702 may be taken as a two-credit pass/fail course by students who entered the program before fall 2009. All other students can only register for three-credit and cannot take this course pass/fail.

Prerequisites: Math-501 (Probability Theory), Math-502 (Deterministic Models) and Math-504 (Numerical Methods) or equivalent. Some familiarity with computational software such as R, Mathematica or Matlab.

Textbook: Reference texts and online resources.

Time and Room: To be determined (weekday late afternoon or evening in St. Mary's).

Math-703: Internship. Fall, Spring, Summer. Industrial internship for students in the MS Degree Program in Mathematics and Statistics. Math-703 may only be taken by students who entered the program before the fall semester 2009. All other students cannot take this course.  Please consult the student handbook for details, or contact the program.

Instructor: Math department faculty