Mathematical Sciences

MA 500. BASIC REAL ANALYSIS

This course covers basic set theory, topology of Rn, continuous functions, uniform convergence, compactness, infinite series, theory of differentiation and integration. Other topics covered may include the inverse and implicit function theorems and Riemann-Stieltjes integration. Students may not count both MA 3831 and MA 500 toward their undergraduate degree requirements.

MA 501. ENGINEERING MATHEMATICS

This course develops mathematical techniques
used in the engineering disciplines. Preliminary
concepts will be reviewed as necessary, including
vector spaces, matrices and eigen values. The principal
topics covered will include vector calculus,
Fourier transforms, fast Fourier transforms and
Laplace transformations. Applications of these
techniques for the solution of boundary value and
initial value problems will be given. The problems
treated and solved in this course are typical of
those seen in applications and include problems
of heat conduction, mechanical vibrations and
wave propagation. (Prerequisite: A knowledge of
ordinary differential equations, linear algebra and
multivariable calculus is assumed.)

MA 502. LINEAR ALGEBRA

This course provides an introduction to the theory and methods of applicable linear algebra. The goal is to bring out the fundamental concepts and techniques that underlie and unify the many ways in which linear algebra is used in applications.
The course is suitable for students in mathematics and other disciplines who wish to obtain deeper insights into this very important subject than are normally offered in undergraduate courses. It is also intended to provide a foundation for further
study in subjects such as numerical linear algebra and functional analysis.

MA 503. LEBESGUE MEASURE AND INTEGRATION

This course begins with a review of topics normally
covered in undergraduate analysis courses:
open, closed and compact sets; liminf and limsup;
continuity and uniform convergence. Next the
course covers Lebesgue measure in Rn including
the Cantor set, the concept of a sigma-algebra, the
construction of a nonmeasurable set, measurable
functions, semicontinuity, Egorov’s and Lusin’s
theorems, and convergence in measure. Next
we cover Lebesgue integration, integral convergence
theorems (monotone and dominated),
Tchebyshev’s inequality and Tonelli’s and Fubini’s
theorems. Finally Lp spaces are introduced with
emphasis on L2 as a Hilbert space. Other related
topics will be covered at the instructor’s discretion. (Prerequisite: Basic knowledge of undergraduate
analysis is assumed.)

MA 504. FUNCTIONAL ANALYSIS

This course will give a comprehensive presentation of fundamental concepts and theorems in Banach and Hilbert spaces. Whenever possible, the theory will be illustrated by examples in Lebesgue spaces. Topics include: The Hahn-Banach theorems, the Uniform Boundedness principle (Banach-Steinhaus Theorem), the Open Mapping and Closed Graph theorems, and weak topologies and convergence. Additional topics will be covered at the instructor’s discretion. (Prerequisite: MA 503 or equivalent.) 

MA 505. COMPLEX ANALYSIS

This course will provide a rigorous and thorough treatment
of the theory of functions of one
complex variable. The topics to be covered include
complex numbers, complex differentiation, the
Cauchy-Riemann equations, analytic functions,
Cauchy’s theorem, complex integration, the
Cauchy integral formula, Liouville’s theorem,
the Gauss mean value theorem, the maximum
modulus theorem, Rouche's theorem, the Poisson
integral formula, Taylor-Laurent expansions,
singularity theory, conformal mapping with applications,
analytic continuation, Schwarz’s reflection
principle and elliptic functions. (Prerequisite:
knowledge of undergraduate analysis.)

MA 508. MATHEMATICAL MODELING

This course introduces mathematical model
building
using dimensional analysis, perturbation
theory and variational principles. Models
are selected from the natural and social sciences
according to the interests of the instructor and
students. Examples are: planetary orbits, spring-mass
systems, fluid flow, isomers in organic chemistry,
biological competition, biochemical kinetics
and physiological flow. Computer simulation of
these models will also be considered. (Prerequisite:
knowledge of ordinary differential equations and
of analysis at the level of MA 501 is assumed.)

MA 510. NUMERICAL METHODS

This course provides an introduction to a broad range of modern numerical techniques that are widely used in computational mathematics, science, and engineering. It is suitable for both mathematics majors and students from other departments. It covers introductory-level material for subjects treated in greater depth in MA 512 and MA 514 and also topics not addressed in either of those courses. Subject areas include numerical methods for systems of linear numerical methods for systems of linear and nonlinear equations, interpolation and approximation, differentiation and integration, and differential equations. Specific topics include basic direct and iterative methods for linear systems; classical rootfinding methods; Newton’s method and related methods for non-linear systems; fixed-point iteration; polynomial, piecewise polynomial, and spline interpolation methods: least-squares approximation; orthogonal functions and approximation; basic techniques for numerical differentiation; numerical integration, including adaptive quadrature; and methods for initial-value problems for ordinary differential equations. Additional topics may be included at the instructor’s discretion as time permits. Both theory and practice are examined. Error estimates, rates of convergence, and the consequences of finite precision arithmetic are also discussed. Topics from linear algebra and elementary functional analysis will be introduced as needed. These may include norms and inner products, orthogonality and orthogonalization, operators and projections, and the concept of a function space. (Prerequisite: knowledge of undergraduate linear algebra and differential equations is assumed, as is familiarity with MATLAB or a higher-level programming language.)

MA 511. APPLIED STATISTICS FOR ENGINEERS AND SCIENTISTS

This course is an introduction to statistics for
graduate students in engineering and the sciences.
Topics covered include basic data analysis, issues in the design of studies, an introduction to probability,
point and interval estimation and hypothesis
testing for means and proportions from one
and two samples, simple and multiple regression,
analysis of one and two-way tables, one-way analysis
of variance. As time permits, additional topics,
such as distribution-free methods and the design
and analysis of factorial studies will be considered.
(Prerequisites: Integral and differential calculus.)

MA 512. NUMERICAL DIFFERENTIAL EQUATIONS

This course begins where MA 510 ends in the
study of the theory and practice of the numerical
solution of differential equations. Central topics
include a review of initial value problems,
including Euler’s method, Runge-Kutta methods,
multi-step methods, implicit methods and predictor-
corrector methods; the solution of two-point
boundary value problems by shooting methods
and by the discretization of the original problem
to form systems of nonlinear equations; numerical
stability; existence and uniqueness of solutions;
and an introduction to the solution of partial
differential equations by finite differences. Other
topics might include finite element or boundary
element methods, Galerkin methods, collocation,
or variational methods. (Prerequisites: graduate or
undergraduate numerical analysis. Knowledge of a
higher-level programming language is assumed.)

MA 514. NUMERICAL LINEAR ALGEBRA

This course provides students with the skills
necessary to develop, analyze and implement
computational methods in linear algebra. The
central topics include vector and matrix algebra,
vector and matrix norms, the singular value
decomposition, the LU and QR decompositions,
Householder transformations and Givens
rotations, and iterative methods for solving linear
systems including Jacobi, Gauss-Seidel, SOR
and conjugate gradient methods; and eigenvalue
problems. Applications to such problem areas as
least squares and optimization will be discussed.
Other topics might include: special linear systems,
such as symmetric, positive definite, banded or
sparse systems; preconditioning; the Cholesky decomposition;
sparse tableau and other least-square
methods; or algorithms for parallel architectures.
(Prerequisite: basic knowledge of linear algebra or
equivalent background. Knowledge of a higher level
programming language is assumed.)

MA 520. FOURIER TRANSFORMS AND DISTRIBUTIONS

The course will cover L1, L2, L∞ and basic facts
from Hilbert space theory (Hilbert basis, projection
theorems, Riesz theory). The first part of
the course will introduce Fourier series: the L2
theory, the C∞ theory: rate of convergence, Fourier
series of real analytic functions, application to
the trapezoidal rule, Fourier transforms in L1,
Fourier integrals of Gaussians, the Schwartz class
S, Fourier transforms and derivatives, translations,
convolution, Fourier transforms in L2, and
characteristic functions of probability distribution
functions. The second part of the course will cover
tempered distributions and applications to partial
differential equations. Other related topics will be
covered at the instructor’s discretion. (Prerequisite:
MA 503.)

MA 521. PARTIAL DIFFERENTIAL EQUATIONS

This course considers a variety of material in partial
differential equations (PDE). Topics covered
will be chosen from the following: classical linear
elliptic, parabolic and hyperbolic equations and
systems, characteristics, fundamental/ Green’s solutions,
potential theory, the Fredholm alternative,
maximum principles, Cauchy problems, Dirichlet/
Neumann/Robin problems, weak solutions and
variational methods, viscosity solutions, nonlinear
equations and systems, wave propagation, free
and moving boundary problems, homogenization.
Other topics may also be covered. (Prerequisites:
MA 503 or equivalent.)

MA 522. HILBERT SPACES AND APPLICATIONS TO PDE

The course covers Hilbert space theory with special
emphasis on applications to linear ODs and
PDEs. Topics include spectral theory for linear operators
in n-dimensional and infinite dimensional
Hilbert spaces, spectral theory for symmetric compact
operatos, linear and bilinear forms, Riesz and
Lax-Milgram theorems, weak derivatives, Sobolev
spaces H1, H2, Rellich compactness theorem, weak
and classical solutions for Dirichlet and Neumann
problems in one variable and in Rn, Dirichlet
variational principle, eigenvalues and eigenvectors.
Other related topics will be covered at the
instructor's discretion. (Prerequisite: MA 503.)

MA 524. CONVEX ANALYSIS AND OPTIMIZATION

This course covers topics in functional analysis
that are critical to the study of convex optimization
problems. The first part of the course will
include the minimization theory for quadratic and
convex functionals on convex sets and cones, the
Legendre-Fenchel duality, variational inequalities
and complementarity systems. The second part
will include optimal stopping time problems in
deterministic control, value functions and Hamilton-
Jacobi inequalities and linear and quadratic
programming, duality and Kuhn-Tucker multipliers.
Other related topics will be covered at the
instructor’s discretion. (Prerequisite: MA 503.)

MA 528. MEASURE THEORETIC PROBABILITY THEORY

This course is designed to give graduate students interested in financial mathematics and stochastic analysis the necessary background in measuretheoretic probability and provide a theoretical foundation for Ph.D. students with research interests in analysis and mathematical statistics. Besides classical topics such as the axiomatic foundations of probability, conditional probabilities and independence, random variables and their distributions, and limit theorems, this course focuses on concepts crucial for the understanding of stochastic processes and quantitative finance: conditional expectations, filtrations and martingales as well as change of measure techniques and the Radon-Nikodym theorem. A wide range of illustrative examples from a topic chosen by the instructor’s discretion (e.g financial mathematics, signal processing, actuarial mathematics) will be presented. (Prerequisite:MA500 Basic Real Analysis or equivalent.)

MA 529. STOCHASTIC PROCESSES

This course is designed to introduce students to continuous-time stochastic processes. Stochastic processes play a central role in a wide range of applications from signal processing to finance and also offer an alternative novel viewpoint to several areas of mathematical analysis, such as partial differential equations and potential theory.
The main topics for this course are martingales, maximal inequalities and applications, optimal stopping and martingale convergence theorems, the strong Markov property, stochastic integration, Ito's formula and applications, martingale representation theorems, Girsanov's theorem and applications, and an introduction to stochastic differential equations, the Feynman-Kac formula, and connections to partial differential equations.
Optional topics (at the instructor's discretion) include Markov processes and Poisson-and jump-processes. (Prerequisite: MA 528. Measure- Theoretic Probability Theory, which can be taken concurrently (or, with special permission by the instructor, MA 540)).

MA 530. DISCRETE MATHEMATICS

This course provides the student of mathematics
or computer science with an overview of discrete
structures and their applications, as well as the basic
methods and proof techniques in combinatorics.
Topics covered include sets, relations, posets,
enumeration, graphs, digraphs, monoids, groups,
discrete probability theory and propositional calculus.
(Prerequisites: college math at least through
calculus. Experience with recursive programming
is helpful, but not required.)

MA 533. DISCRETE MATHEMATICS II

This course is designed to provide an in-depth
study of some topics in combinatorial mathematics
and discrete optimization. Topics may vary
from year to year. Topics covered include, as time
permits, partially ordered sets, lattices, matroids,
matching theory, Ramsey theory, discrete programming
problems, computational complexity of
algorithms, branch and bound methods.

MA 535. ALGEBRA

Fundamentals of group theory: homomorphisms
and the isomorphism theorems, finite groups,
structure of finitely generated Abelian groups.
Structure of rings: homomorphisms, ideals, factor
rings and the isomorphism theorems, integral
domains, factorization. Field theory: extension
fields, finite fields, theory of equations. Selected
topics from: Galois theory, Sylow theory, Jordan-
Hölder theory, Polya theory, group presentations,
basic representation theory and group characters,
modules. Applications chosen from mathematical
physics, Gröbner bases, symmetry, cryptography,
error-correcting codes, number theory.

MA 540. PROBABILITY AND MATHEMATICAL STATISTICS I

Intended for advanced undergraduates and
beginning graduate students in the mathematical
sciences, and for others intending to pursue the
mathematical study of probability and statistics.
Topics covered include axiomatic foundations, the
calculus of probability, conditional probability and
independence, Bayes’ Theorem, random variables, discrete and continuous distributions, joint,
marginal and conditional distributions, covariance
and correlation, expectation, generating
functions, exponential families, transformations
of random variables, types of convergence, laws of
large numbers the Central Limit Theorem, Taylor
series expansion, the delta method. (Prerequisite:
knowledge of basic probability at the level of
MA 2631 and of advanced calculus at the level of
MA 3831/3832 is assumed.)

MA 541. PROBABILITY AND MATHEMATICAL STATISTICS II

This course is designed to provide background in
principles of statistics. Topics covered include estimation
criteria: method of moments, maximum
likelihood, least squares, Bayes, point and interval
estimation, Fisher’s information, Cramer-Rao
lower bound, sufficiency, unbiasedness, and
completeness, Rao-Blackwell Theorem, efficiency,
consistency, interval estimation pivotal quantities,
Neyman-Person Lemma, uniformly most powerful
tests, unbiased, invariant and similar tests,
likelihood ratio tests, convex loss functions, risk
functions, admissibility and minimaxity, Bayes
decision rules. (Prerequisite: knowledge of the
material in MA 540 is assumed.)

MA 542. REGRESSION ANALYSIS

Regression analysis is a statistical tool that utilizes
the relation between a response variable and one
or more predictor variables for the purposes of
description, prediction and/or control. Successfu l
use of regression analysis requires an appreciation
of both the theory and the practical problems that
often arise when the technique is employed with
real-world data. Topics covered include the theory
and application of the general linear regression
model, model fitting, estimation and prediction,
hypothesis testing, the analysis of variance and
related distribution theory, model diagnostics and
remedial measures, model building and validation,
and generalizations such as logistic response
models and Poisson regression. Additional topics
may be covered as time permits. Application of
theory to real-world problems will be emphasized
using statistical computer packages. (Prerequisite:
knowledge of probability and statistics at the level
of MA 511 and of matrix algebra is assumed.)

MA 543. STATISTICAL METHODS FOR DATA SCIENCE

This course surveys the statistical methods most useful in data science applications. Topics covered include predictive modeling methods, including multiple linear regression, and time series; data dimension reduction; discrimination and classification methods, clustering methods; and committee methods. Students will implement these methods using statistical software.
Prerequisites: Statistics at the level of MA 2611 and MA2612 and linear algebra at the level of MA 2071.

MA 546. DESIGN AND ANALYSIS OF EXPERIMENTS

Controlled experiments—studies in which treatments
are assigned to observational units—are the
gold standard of scientific investigation. The goal
of the statistical design and analysis of experiments
is to (1) identify the factors which most affect a
given process or phenomenon; (2) identify the
ways in which these factors affect the process or
phenomenon, both individually and in combination;
(3) accomplish goals 1 and 2 with minimum
cost and maximum efficiency while maintaining
the validity of the results. Topics covered in this
course include the design, implementation and
analysis of completely randomized complete
block, nested, split plot, Latin square and repeated
measures designs. Emphasis will be on the application
of the theory to real data using statistical computer packages. (Prerequisite: knowledge of
basic probability and statistics at the level of
MA 511 is assumed.)

MA 547. DESIGN AND ANALYSIS OF OBSERVATIONAL AND SAMPLING STUDIES

Like controlled experiments, observational studies
seek to establish cause-effect relationships,
but unlike controlled experiments, they lack the
ability to assign treatments to observational units.
Sampling studies, such as sample surveys, seek to
characterize aspects of populations by obtaining
and analyzing samples from those populations.
Topics from observational studies include:
prospective and retrospective studies; overt and
hidden bias; adjustments by stratification and
matching. Topics from sampling studies include:
simple random sampling and associated estimates for means, totals, and proportions; estimates for
subpopulations; unequal probability sampling;
ratio and regression estimation; stratified, cluster,
systematic, multistage, double sampling designs,
and, time permitting, topics such as model-based
sampling, spatial and adaptive sampling.
(Prerequisite: knowledge of basic probability and
statistics, at the level of MA 511 is assumed.)

MA 548. QUALITY CONTROL

This course provides the student with the basic
statistical tools needed to evaluate the quality of
products and processes. Topics covered include the
philosophy and implementation of continuous
quality improvement methods, Shewhart control
charts for variables and attributes, EWMA and
Cusum control charts, process capability analysis,
factorial and fractional factorial experiments for
process design and improvement, and response
surface methods for process optimization. Additional
topics will be covered as time permits.
Special emphasis will be placed on realistic applications
of the theory using statistical computer
packages. (Prerequisite: knowledge of basic
probability and statistic, at the level of MA 511 is
assumed.)

MA 549. ANALYSIS OF LIFETIME DATA

Lifetime data occurs frequently in engineering,
where it is known as reliability or failure time
data, and in the biomedical sciences, where it is
known as survival data. This course covers the
basic methods for analyzing such data. Topics
include: probability models for lifetime data, censoring,
graphical methods of model selection and
analysis, parametric and distribution-free inference,
parametric and distribution-free regression
methods. As time permits, additional topics such
as frailty models and accelerated life models will
be considered. Special emphasis will be placed on
realistic applications of the theory using statistical
computer packages. (Prerequisite: knowledge of
basic probability and statistics at the level of
MA 511 is assumed.)

MA 550. TIME SERIES ANALYSIS

Time series are collections of observations made
sequentially in time. Examples of this type of data
abound in many fields ranging from finance to engineering. Special techniques are called for in
order to analyze and model these data. This course
introduces the student to time and frequency
domain techniques, including topics such as
autocorrelation, spectral analysis, and ARMA
and ARIMA models, Box-Jenkins methodology,
fitting, forecasting, and seasonal adjustments.
Time permitting, additional topics will be chosen
from: Kalman filter, smoothing techniques,
Holt-Winters procedures, FARIMA and GARCH
models, and joint time-frequency methods such
as wavelets. The emphasis will be in application
to real data situations using statistical computer
packages. (Prerequisite: knowledge of MA 511 is
assumed. Knowledge of MA 541 is also assumed,
but may be taken concurrently.)

MA 552. DISTRIBUTION-FREE AND ROBUST STATISTICAL METHODS

Distribution-free statistical methods relax the
usual distributional modeling assumptions of
classical statistical methods. Robust methods are
statistical procedures that are relatively insensitive
to departures from typical assumptions, while
retaining the expected behavior when assumptions
are satisfied. Topics covered include, time
permitting, order statistics and ranks; classical
distribution-free tests such as the sign, Wilcoxon
signed rank, and Wilcoxon rank sum tests, and associated
point estimators and confidence intervals;
tests pertaining to one and two-way layouts; the
Kolmogorov-Smirnov test; permutation methods;
bootstrap and Monte Carlo methods; M, L, and
R estimators, regression, kernel density estimation
and other smoothing methods. Comparisons will
be made to standard parametric methods. (Prerequisite:
knowledge of MA 541 is assumed, but may
be taken concurrently.)

MA 554. APPLIED MULTIVARIATE ANALYSIS

This course is an introduction to statistical methods
for analyzing multivariate data. Topics covered
are multivariate sampling distributions, tests and
estimation of multivariate normal parameters,
multivariate ANOVA, regression, discriminant
analysis, cluster analysis, factor analysis and principal
components. Additional topics will be covered
as time permits. Students will be required to analyze
real data using one of the standard packages
available. (Prerequisite: knowledge of MA 541 is
assumed, but may be taken concurrently. Knowledge
of matrix algebra is assumed.)

MA 556. APPLIED BAYESIAN STATISTICS

Bayesian statistics makes use of an inferential
process that models data summarizing the results
in terms of probability distributions for the model
parameters. A key feature is that in the Bayesian
approach, past information can be updated with
new data in an elegant way in order to aid in
decision making. Topics included in the courses:
statistical decision theory, the Bayesian inferential
framework (model specification, model fitting and
model checking); computational methods for posterior
simulation integration; regression models, hierarchical models, and ANOVA; time permitting,
additional topics will include generalized
linear models, multivariate models, missing data
problems, and time series analysis. (Prerequisites:
knowledge of MA 541 is assumed.)

MA 559. STATISTICS GRADUATE SEMINAR

This seminar introduces students to issues and
trends in modern statistics. In the seminar, students
and faculty will read and discuss survey and
research papers, make and attend presentations,
and participate in brainstorming sessions toward
the solution of advanced statistical problems.

MA 560. GRADUATE SEMINAR

Designed to introduce graduate students to study
of original papers and afford them opportunity to
give account of their work by talks in the seminar.

MA 562. PROFESSIONAL MASTER'S SEMINAR

This seminar will introduce professional master’s
students to topics related to general writing,
presentation, group communication and interviewing
skills, and will provide the foundations
to successful cooperation within interdisciplinary
team environments. All full-time students will be
required to take both components A and B of the
seminar during their professional master’s studies.

MA 571. FINANCIAL MATHEMATICS I

This course provides an introduction to many of
the central concepts in mathematical finance. The
focus of the course is on arbitrage-based pricing
of derivative securities. Topics include stochastic
calculus, securities markets, arbitrage-based pricing
of options and their uses for hedging and risk
management, forward and futures contracts, European
options, American options, exotic options,
binomial stock price models, the Black-Scholes-
Merton partial differential equation, risk-neutral
option pricing, the fundamental theorems of
asset pricing, sensitivity measures (“Greeks”), and
Merton’s credit risk model. (Prerequisite: MA 540,
which can be taken concurrently.)

MA 572. FINANCIAL MATHEMATICS II

The course is devoted to the mathematics of fixed
income securities and to the financial instruments
and methods used to manage interest rate risk.
The first topics covered are the term-structure of
interest rates, bonds, futures, interest rate swaps
and their uses as investment or hedging tools and
in asset-liability management. The second part of
the course is devoted to dynamic term-structure
models, including risk-neutral interest rate trees,
the Heath-Jarrow-Morton model, Libor market
models, and forward measures. Applications of
these models are also covered, including the pricing
of non-linear interest rate derivatives such as
caps, floors, collars, swaptions and the dynamic
hedging of interest rate risk. The course concludes
with the coverage of mortgage-backed and asset backed
securities. (Prerequisite: MA 571.)

MA 573. COMPUTATIONAL METHODS OF FINANCIAL MATHEMATICS

Most realistic quantitative finance models are too
complex to allow explicit analytic solutions and
are solved by numerical computational methods.
The first part of the course covers the application
of finite difference methods to the partial differential
equations and interest rate models arising
in finance. Topics included are explicit, implicit
and Crank-Nicholson finite difference schemes
for fixed and free boundary value problems, their
convergence and stability. The second part of the
course covers Monte Carlo simulation methods,
including random number generation, variance reduction
techniques and the use of low discrepancy
sequences. (Prerequisites: MA 571 and programming
skills at the level of MA 579, which can be
taken concurrently.)

MA 574. PORTFOLIO VALUATION AND RISK MANAGEMENT

Balancing financial risks vs returns by the use of
asset diversification is one of the fundamental
tasks of quantitative financial management. This
course is devoted to the use of mathematical
optimization and statistics to allocate assets, to
construct and manage portfolios and to measure
and manage the resulting risks. The fist part of the
course covers Markowitz’s mean-variance optimization
and efficient frontiers, Sharpe’s single index
and capital asset pricing models, arbitrage pricing
theory, structural and statistical multi-factor models,
risk allocation and risk budgeting. The second
part of the course is devoted to the intertwining
of optimization and statistical methodologies in
modern portfolio management, including resampled
efficiency, robust and Bayesian statistical
methods, the Black-Litterman model and robust
portfolio optimization.

MA 575. MARKET AND CREDIT RISK MODELS AND MANAGEMENT

The objective of the course is to familiarize
students with the most important quantitative
models and methods used to measure and
manage financial risk, with special emphasis on
market and credit risk. The course starts with the
introduction of metrics of risk such as volatility,
value-at-risk and expected shortfall and with
the fundamental quantitative techniques used in
financial risk evaluation and management. The
next section is devoted to market risk including
volatility modeling, time series, non-normal heavy
tailed phenomena and multivariate notions of
co-dependence such as copulas, correlations and
tail-dependence. The final section concentrates
on credit risk including structural and dynamic
models and default contagion and applies the
mathematical tools to the valuation of default
contingent claims including credit default swaps,
structured credit portfolios and collateralized debt
obligations. (Prerequisite: knowledge of MA 540
assumed but can be taken concurrently.)

MA 584. STATISTICAL METHODS IN GENETICS AND BIOINFORMATICS

This course provides students with knowledge and
understanding of the applications of statistics in
modern genetics and bioinformatics. The course
generally covers population genetics, genetic
epidemiology, and statistical models in bioinformatics.
Specific topics include meiosis modeling,
stochastic models for recombination, linkage and
association studies (parametric vs. nonparametric
models, family-based vs. population-based
models) for mapping genes of qualitative and
quantitative traits, gene expression data analysis,
DNA and protein sequence analysis, and molecular
evolution. Statistical approaches include log-likelihood
ratio tests, score tests, generalized linear
models, EM algorithm, Markov chain Monte
Carlo, hidden Markov model, and classification
and regression trees. Students may not receive
credit for both MA 584 and MA 4603. (Prerequisite:
knowledge of probability and statistics at the
undergraduate level.)

MME 518. GEOMETRICAL CONCEPTS

This course focuses primarily on the foundations
and applications of Euclidean and non-Euclidean
geometries. The rich and diverse nature of the
subject also implies the need to explore other
topics, for example, chaos and fractals. The course
incorporates collaborative learning and the investigation
of ideas through group projects. Possible
topics include geometrical software and computer
graphics, tiling and tessellations, two- and three dimensional
geometry, inversive geometry, graphical
representations of functions, model construction,
fundamental relationship between algebra
and geometry, applications of geometry, geometry
transformations and projective geometry, and
convexity.

MME 522. APPLICATIONS OF CALCULUS

There are three major goals for this course: to
establish the underlying principles of calculus,
to reinforce students’ calculus skills through
investigation of applications involving those skills,
and to give students the opportunity to develop
projects and laboratory assignments for use by
first-year calculus students. The course will focus
heavily on the use of technology to solve problems
involving applications of calculus concepts. In addition,
MME students will be expected to master
the mathematical rigor of these calculus concepts
so that they will be better prepared to develop
their own projects and laboratory assignments. For
example, if an MME student chose to develop a
lab on convergence of sequence, he/she would be
expected to understand the rigorous definition of
convergence and how to apply it to gain sufficient
and/or necessary conditions for convergence. The
process of developing these first-year calculus
assignments will enable the MME students to
increase their own mathematical understanding of
concepts while learning to handle mathematical
and computer issues which will be encountered by
their own calculus students. Their understanding
of the concepts and applications of calculus will be
further reinforced through computer laboratory
assignments and group projects. Applications
might include exponential decay of drugs in the
body, optimal crank shaft design, population
growth, or development of cruise control systems.

MME 523. ANALYSIS WITH APPLICATIONS

This course introduces students to mathematical
analysis and its use in modeling. It will emphasize
topics of calculus (including multidimensional)
in a rigorous way. These topics will be motivated
by their usefulness for understanding concepts of
the calculus and for facilitating the solutions of
engineering and science problems. Projects involving
applications and appropriate use of technology
will be an essential part of the course. Topics
covered may include dynamical systems and differential
equations; growth and decay; equilibrium;
probabilistic dynamics; optimal decisions and
reward; applying, building and validating models;
functions on n-vectors; properties of functions;
parametric equations; series; applications such as
pendulum problems; electromagnetism; vibrations;
electronics; transportation; gravitational
fields; and heat loss.

MME 524. PROBABILITY, STATISTICS AND DATA ANALYSIS I

This course introduces students to probability, the
mathematical description of random phenomena,
and to statistics, the science of data. Students in
this course will acquire the following knowledge
and skills:
• Probability models- mathematical models used
to describe and predict random phenomena.
Students will learn several basic probability
models and their uses, and will obtain experience
in modeling random phenomena.

• Data analysis- the art/science of finding patterns
in data and using those patterns to explain the
process which produced the data. Students
will be able to explore and draw conclusions
about data using computational and graphical
methods. The iterative nature of statistical
exploration will be emphasized.

• Statistical inference and modeling- the use of
data sampled from a process and the probability
model of that process to draw conclusions
about the process. Students will attain
proficiency in selecting, fitting and criticizing
models, and in drawing inference from data.
• Design of experiments and sampling studies
– the proper way to design experiments and
sampling studies so that statistically valid inferences
can be drawn. Special attention will be
given to the role of experiments and sampling
studies in scientific investigation. Through lab
and project work, students will obtain practical
skills in designing and analyzing studies and
experiments. Course topics will be motivated
whenever possible by applications and
reinforced by experimental and computer lab
experiences. One in-depth project per semester
involving design, data collection, and statistical
or probabilistic analysis will serve to integrate
and consolidate student skills and understanding.
Students will be expected to learn and use a
statistical computer package such as MINITAB.

MME 525. PROBABILITY, STATISTICS AND DATA ANALYSIS II

This course introduces students to probability, the
mathematical description of random phenomena,
and to statistics, the science of data. Students in
this course will acquire the following knowledge
and skills:
• Probability models- mathematical models used
to describe and predict random phenomena.
Students will learn several basic probability
models and their uses, and will obtain experience
in modeling random phenomena.

• Data analysis- the art/science of finding patterns
in data and using those patterns to explain the
process which produced the data. Students
will be able to explore and draw conclusions
about data using computational and graphical
methods. The iterative nature of statistical
exploration will be emphasized.

• Statistical inference and modeling- the use of
data sampled from a process and the probability
model of that process to draw conclusions
about the process. Students will attain
proficiency in selecting, fitting and criticizing
models, and in drawing inference from data.
• Design of experiments and sampling studies
– the proper way to design experiments and
sampling studies so that statistically valid inferences
can be drawn. Special attention will be
given to the role of experiments and sampling
studies in scientific investigation. Through lab
and project work, students will obtain practical
skills in designing and analyzing studies and
experiments. Course topics will be motivated
whenever possible by applications and
reinforced by experimental and computer lab
experiences. One in-depth project per semester
involving design, data collection, and statistical
or probabilistic analysis will serve to integrate
and consolidate student skills and understanding.
Students will be expected to learn and use a
statistical computer package such as MINITAB.

MME 526. LINEAR MODELS I

This two-course sequence imparts computational
skills, particularly those involving matrices, to
deepen understanding of mathematical structure
and methods of proof; it also includes discussion
on a variety of applications of the material developed,
including linear optimization. Topics in this
sequence may include systems of linear equations,
vector spaces, linear independence, bases, linear
transformations, determinants, eigenvalues and
eigenvectors, systems of linear inequalities, linear
programming problems, basic solutions, duality
and game theory. Applications may include economic
models, computer graphics, least squares
approximation, systems of differential equations,
graphs and networks, and Markov processes.

MME 527. LINEAR MODELS II

This two-course sequence imparts computational
skills, particularly those involving matrices, to
deepen understanding of mathematical structure
and methods of proof; it also includes discussion
on a variety of applications of the material developed,
including linear optimization. Topics in this
sequence may include systems of linear equations,
vector spaces, linear independence, bases, linear
transformations, determinants, eigenvalues and
eigenvectors, systems of linear inequalities, linear
programming problems, basic solutions, duality
and game theory. Applications may include economic
models, computer graphics, least squares
approximation, systems of differential equations,
graphs and networks, and Markov processes.

MME 528. MATHEMATICAL MODELING AND PROBLEM SOLVING

This course introduces students to the process of
developing mathematical models as a means for
solving real problems. The course will encompass
several different modeling situations that utilize a
variety of mathematical topics. The mathematical
fundamentals of these topics will be discussed, but
with continued reference to their use in finding
the solutions to problems. Problems to be covered
include balance in small group behavior, traffic
flow, air pollution flow, group decision making,
transportation, assignment, project planning
and the critical path method, genetics, inventory
control and queueing.

MME 529. NUMBERS, POLYNOMIALS AND ALGEBRAIC STRUCTURES

This course enables secondary mathematics
teachers to see how commonly taught topics
such as number systems and polynomials fit into
the broader context of algebra. The course will
begin with treatment of arithmetic, working
through Euclid’s algorithm and its applications,
the fundamental theorem of arithmetic and
its applications, multiplicative functions, the
Chinese remainder theorem and the arithmetic
of Z/n. This information will be carried over
to polynomials in one variable over the rational
and real numbers, culminating in the construction
of root fields for polynomials via quotients
of polynomial rings. Arithmetic in the Gaussian
integers and the integers in various other quadratic
fields (especially the field of cube roots of unity)
will be explored through applications such as the
generation of Pythagorean triples and solutions to
other Diophantine equations (like finding integer-sided
triangles with a 60 degree angle). The course
will then explore cyclotomy, and the arithmetic in
rings of cyclotomic integers. This will culminate
in Gauss’s construction of the regular 5-gon and
17-gon and the impossibility of constructing a
9-gon or trisecting a 60-degree angle. Finally,
solutions of cubics and quartics by radicals will be
studied. All topics will be based on the analysis of
explicit calculations with (generalized) numbers.
The proposed curriculum covers topics that are
part of the folklore for high school mathematics
(the impossibility of certain ruler and compass
constructions), but that many teachers know only
as facts. There are also many applications of the
ideas that will allow the teachers to use results and
ideas from abstract algebra to construct for their
students problems that have manageable solutions.

MME 531. DISCRETE MATHEMATICS

This course deals with concepts and methods
which emphasize the discrete nature in many
problems and structures. The rapid growth of this
branch of mathematics has been inspired by its
wide range of applicability to diverse fields such as
computer science, management, and biology. The
essential ingredients of the course are:
Combinatorics - The Art of Counting.
Topics include basic counting principles and
methods such as recurrence relations, generating
functions, the inclusion-exclusion principle
and the pigeonhole principle. Applications may
include block designs, latin squares, finite projective
planes, coding theory, optimization and
algorithmic analysis.
Graph Theory. This includes direct graphs and
networks. Among the parameters to be examined
are traversibility, connectivity, planarity, duality
and colorability.

MME 532. DIFFERENTIAL EQUATIONS

This course would have concepts and techniques for both Ordinary and Partial
Differential Equations. Topics from ordinary differential equations include
existence and uniqueness for first order, single variable problems as well as
separation of variables and linear methods for first order problems. Second order,
linear equations would be solved for both the homogeneous and nonhomogeneous
cases. The phenomena of beats and resonance would be analyzed. The Laplace Transform would be introduced for appropriate second order nonhomogeneous problems. Partial Differential Equations would focus on boundary value problems arising from the Heat and Wave equations in one variable. Fourier Series expansions would be used to satisfy initial conditions and the concepts of orthogonality and convergence addressed.