# Mathematical Sciences

## Undergraduate Courses

### MA 1020. CALCULUS I WITH PRELIMINARY TOPICS

Cat. I (14-week course)

This course includes the topics of MA 1021 and also presents selected topics

from algebra, trigonometry, and analytic geometry.

This course, which extends for 14 weeks and offers 1/3 unit of credit, is

designed for students whose precalculus mathematics is not adequate for MA

1021.

Although the course will make use of computers, no programming experience

is assumed.

Students may not receive credit for both MA 1020 and MA 1021.

### MA 1021. CALCULUS I

Cat. I

This course provides an introduction to differentiation and its applications.

Topics covered include: functions and their graphs, limits, continuity,

differentiation, linear approximation, chain rule, min/max problems, and

applications of derivatives.

Recommended background: Algebra, trigonometry and analytic geometry.

Although the course will make use of computers, no programming experience

is assumed.

Students may not receive credit for both MA 1021 and MA 1020.

### MA 1022. CALCULUS II

Cat. I

This course provides an introduction to integration and its applications.

Topics covered include: inverse trigonometric functions, Riemann sums,

fundamental theorem of calculus, basic techniques of integration, volumes of

revolution, arc length, exponential and logarithmic functions, and applications.

Recommended background: MA 1021. Although the course will make use of

computers, no programming experience is assumed.

### MA 1023. CALCULUS III

Cat. I

This course provides an introduction to series, parametric curves and vector

algebra.

Topics covered include: numerical methods, indeterminate forms, improper

integrals, sequences, Taylor's theorem with remainder, convergence of series and

power series, polar coordinates, parametric curves and vector algebra.

Recommended background: MA 1022. Although the course will make use of

computers, no programming experience is assumed.

### MA 1024. CALCULUS IV

Cat. I

This course provides an introduction to multivariable calculus.

Topics covered include: vector functions, partial derivatives and gradient,

multivariable optimization, double and triple integrals, polar coordinates, other

coordinate systems and applications.

Recommended background: MA 1023. Although the course will make use of

computers, no programming experience is assumed.

### MA 1033. THEORETICAL CALCULUS III

This course will cover the same material as MA 1023 Calculus III but from a different perspective. A more rigorous study of sequences and series will be undertaken: starting from the least upper bound property in R, the fundamental theorems for convergent series will be proved. Convergence criteria for series will be rigorously justified and L'Hospital's rule will be introduced and proved. Homework problems will include a blend of computational exercises as usually assigned in MA 1023 Calculus III and problems with a stronger theoretical flavor.

Recommended background: Differential and integral calculus (MA1021 and MA 1022, or equivalent).

Note: Students can receive credit for this class and MA1023 Calculus III.

### MA 1034. THEORETICAL CALCULUS IV

Cat. I

This course will cover the same material as MA1024 Calculus IV from a more mathematically rigorous perspective. The course gives a rigorous introduction of differentiation and integration for functions of one variable. After introducing vector functions, differentiation and integration will be extended to functions of several variables.

Recommended background: Theoretical Calculus III (MA1033, or equivalent).

Note: Students can receive credit for this class and MA1024 Calculus IV.

### MA 1120. CALCULUS II (SEMESTER VERSION)

Cat.I

The topics for integral calculus (MA 1022) are covered in this course: the

concept of the definite integral, the Fundamental Theorem of Calculus,

integration techniques, and applications of integration. Applications include: area, volume, arc length, center of mass, work, force, and exponential growth

and decay. Logarithmic and exponential functions are studied in depth.

Arithmetic and geometric sequences and series will also be covered. Key historical events in the development of integral calculus are examined.

Technology will be used as appropriate to support the material being studied.

This course extends for 14 weeks and offers 1/3 unit of credit. It is designed

for students who would benefit from additional contact hours and who need to

strengthen their mathematical background. Although the course will make use

of computers, no programming experience is assumed.

Students may not receive credit for both MA 1120 and MA 1022 or MA 1102.

### MA 1801. DENKSPORT

Problem solving is a fundamental mathematical skill. In this course students will be exposed to problems coming from a wide range of mathematical disciplines; and will work together in a collaborative environment to explore potential solutions. Discussion problems may be inspired by the research of faculty leading the discussion, by past mathematical competitions (such as the Putnam Competition) or elsewhere. This course meets once per week, with an emphasis on discussion and exploration of problems. There will be no exam and no assigned homework. Grading is by participation only. This course may be taken multiple times; content will vary depending on the speakers. Grading for this course will be on a Pass/NR basis.

Recommended background: Curiosity about Mathematics

### MA 1971. BRIDGE TO HIGHER MATHEMATICS

Cat. I

The principal aim of this course is to introduce and enhance mathematical thinking. The course is intended not only for beginning mathematics, statistics or actuarial students, but also for students seeking to further their mathematical interests and those simply curious about logic and reason. Students in the course will be expected to explain, justify, defend, disprove, conjecture and verify mathematical ideas, both verbally and in writing. One expected by-product of this training is that students will develop concrete proof-writing skills which will improve their prospects for success in more advanced mathematics courses. When appropriate, course discussion will touch on current events in the mathematical sciences, including recently solved problems and open challenges facing today's scientists.

Recommended background: at least two courses in Mathematical Sciences at WPI, or equivalent.

### MA 2051. ORDINARY DIFFERENTIAL EQUATIONS

Cat. I

This course develops techniques for solving ordinary differential equations.

Topics covered include: introduction to modeling using first-order differential

equations, solution methods for linear higher-order equations, qualitative

behavior of nonlinear first-order equations, oscillatory phenomena including

spring-mass system and RLC-circuits and Laplace transform. Additional topics

may be chosen from power series method, methods for solving systems of

equations and numerical methods for solving ordinary differential equations.

Recommended background: MA 1024.

### MA 2071. MATRICES AND LINEAR ALGEBRA I

Cat. I

This course provides an introduction to the theory and techniques of matrix algebra and linear algebra. Topics covered include: operations on matrices, systems of linear equations, linear transformations, determinants, eigenvalues and eigenvectors, least squares, vector spaces, inner products, introduction to numerical techniques, and applications of linear algebra. Credit may not be earned for this course and MA 2072.

Recommended background: None, although basic knowledge of equations for planes and lines in space would be helpful.

### MA 2072. ACCELERATED MATRICES AND LINEAR ALGEBRA I

Cat. I

This course provides an accelerated introduction to the theory and techniques of matrix algebra and linear algebra, aimed at Mathematical Sciences majors and others interested in advanced concepts of linear algebra. Topics covered include: matrix algebra, systems of linear equations, linear transformations, determinants, eigenvalues and eigenvectors, the method of least squares, vector spaces, inner products, non-square matrices and singular value decompositions. Students will be exposed to computational and numerical techniques, and to applications of linear algebra, particularly to Data Science. Credit may not be earned for this course and MA 2071.

Recommended background: Basic knowledge of matrix algebra

### MA 2073. MATRICES AND LINEAR ALGEBRA II

Cat. I

This course provides a deeper understanding of topics introduced in MA 2071, and continues the development of linear algebra. Topics covered include: abstract vector spaces, linear transformations, matrix representations of a linear transformation, determinants, characteristic and minimal polynomials, diagonalization, eigenvalues and eigenvectors, the matrix exponential, inner product spaces. This course is designed primarily for Mathematical Science majors and those interested in the deeper mathematical issues underlying linear algebra. Recommended background: MA 2071 or MA 2072.

### MA 2201. DISCRETE MATHEMATICS

Cat. I

This course serves as an introduction to some of the more important concepts, techniques, and structures of discrete mathematics providing a bridge between computer science and mathematics. Topics include sets, functions and relations, propositional and predicate calculus, mathematical induction, properties of integers, counting techniques and graph theory. Students will be expected to develop simple proofs for problems drawn primarily from computer science and applied mathematics.

Recommended background: None

### MA 2210. MATHEMATICAL METHODS IN DECISION MAKING

Cat. I

This course introduces students to the principles of decision theory as applied to

the planning, design and management of complex projects. It will be useful to

students in all areas of engineering, actuarial mathematics as well as those in

such interdisciplinary areas as environmental studies. It emphasizes quantitative,

analytic approaches to decision making using the tools of applied mathematics,

operations research, probability and computations. Topics covered include: the

systems approach, mathematical modeling, optimization and decision analyses.

Case studies from various areas of engineering or actuarial mathematics are used

to illustrate applications of the materials covered in this course.

Recommended background: MA 1024.

Suggested background: Familiarity

with vectors and matrices. Although the course makes use of computers, no

programming experience is assumed.

Students who have received credit for

CE 2010 may not receive credit for MA 2210.

Industrial Engineering majors cannot receive credit for both MA 2210 and BUS 2080.

### MA 2211. THEORY OF INTEREST I

An introduction to actuarial mathematics is provided for those who may be interested in the actuarial profession. Topics usually included are: measurement of interest, including accumulated and present value factors; annuities certain; amortization schedules and sinking funds; and bonds.

Recommended background: Single variable calculus (MA 1021 and MA 1022 or equivalent) and the ability to work with appropriate computer software.

Students may not receive credit for both MA 2211 and MA 3211

### MA 2212. THEORY OF INTEREST II

This course covers topics in fixed income securities. Topics are chosen to cover the mechanics and pricing of modern-day fixed income products and can include: yield curve theories; forward rates; interest rate swaps; credit-default swaps; bonds with credit risk and options; bond duration and convexity; bond portfolio construction; asset- backed securities, including collateralized debt obligations and mortgage-backed securities with prepayment risk; asset-liability hedging; applications of binomial interest rate trees.

Recommended background: An introduction to theory of interest (MA 2211 or equivalent) and the ability to work with appropriate computer software.

### MA 2251. VECTOR AND TENSOR CALCULUS

Cat. I

This course provides an introduction to tensor and vector calculus, an essential

tool for applied mathematicians, scientists, and engineers.

Topics covered include: scalar and vector functions and fields, tensors, basic

differential operations for vectors and tensors, line and surface integrals, change of variable theorem in integration, integral theorems of vector and tensor

calculus. The theory will be illustrated by applications to areas such as

electrostatics, theory of heat, electromagnetics, elasticity and fluid mechanics.

Recommended background: MA 1024.

### MA 2273. COMBINATORICS

Cat. II

This course introduces the concepts and techniques of combinatorics, a part of

mathematics with applications in computer science and in the social, biological,

and physical sciences. Emphasis will be given to problem solving. Topics will be

selected from: basic counting methods, inclusion-exclusion principle, generating

functions, recurrence relations, systems of distinct representatives, combinatorial

designs, combinatorial algorithms and applications of combinatorics.

This course is designed primarily for Mathematical Sciences majors and those

interested in the deeper mathematical issues underlying combinatorics.

Undergraduate credit may not be earned both for this course and for MA 3273.

Recommended background: MA 2071.

This course will be offered in 2015-16, and in alternating years thereafter.

### MA 2431. MATHEMATICAL MODELING WITH ORDINARY DIFFERENTIAL EQUATIONS

Cat. I

This course focuses on the principles of building mathematical models from a physical, chemical or biological system and interpreting the results. Students will learn how to construct a mathematical model and will be able to interpret solutions of this model in terms of the context of the application. Mathematical topics focus on solving systems of ordinary differential equations, and may include the use of stability theory and phase-plane analysis. Applications will be chosen from electrical and mechanical oscillations, control theory, ecological or epidemiological models and reaction kinetics. This course is designed primarily for students interested in the deeper mathematical issues underlying mathematical modeling. Students may be required to use programming languages such as Matlab or Maple to further investigate different models. Recommended background: multivariable calculus (MA 1024 or equivalent), ordinary differential equations (MA 2051 or equivalent), and linear algebra (MA 2071 or equivalent).

### MA 2610. APPLIED STATISTICS FOR THE LIFE SCIENCES

Cat. I

This course is designed to introduce the student to statistical methods and

concepts commonly used in the life sciences. Emphasis will be on the practical

aspects of statistical design and analysis with examples drawn exclusively from

the life sciences, and students will collect and analyze data. Topics covered

include analytic and graphical and numerical summary measures, probability

models for sampling distributions, the central limit theorem, and one and two

sample point and interval estimation, parametric and non-parametric hypothesis

testing, principles of experimental design, comparisons of paired samples and

categorical data analysis.

Undergraduate credit may not be earned for both this

course and for MA 2611.

Recommended background: MA 1022.

### MA 2611. APPLIED STATISTICS I

Cat. I

This course is designed to introduce the student to data analytic and applied

statistical methods commonly used in industrial and scientific applications as

well as in course and project work at WPI. Emphasis will be on the practical

aspects of statistics with students analyzing real data sets on an interactive

computer package.

Topics covered include analytic and graphical representation of data,

exploratory data analysis, basic issues in the design and conduct of experimental

and observational studies, the central limit theorem, one and two sample point

and interval estimation and tests of hypotheses.

Recommended background: MA 1022.

### MA 2621. PROBABILITY FOR APPLICATIONS

Cat. I

This course is designed to introduce the student to probability.

Topics to be covered are: basic probability theory including Bayes theorem;

discrete and continuous random variables; special distributions including the

Bernoulli, Binomial, Geometric, Poisson, Uniform, Normal, Exponential, Chisquare,

Gamma, Weibull, and Beta distributions; multivariate distributions;

conditional and marginal distributions; independence; expectation; transformations

of univariate random variables.

Recommended background: MA 1024.

### MA 2631. PROBABILITY

Cat. I

The purpose of this course is twofold:

- To introduce the student to probability. Topics to be covered will be chosen

from: axiomatic development of probability; independence; Bayes theorem;

discrete and continuous random variables; expectation; special distributions

including the binomial and normal; moment generating functions; multivariate

distributions; conditional and marginal distributions; independence

of random variables; transformations of random variables; limit theorems.

- To introduce fundamental ideas and methods of mathematics using the

study of probability as the vehicle. These ideas and methods may include

systematic theorem-proof development starting with basic axioms;

mathematical induction; set theory; applications of univariate and

multivariate calculus.

This course is designed primarily for Mathematical Sciences majors and those

interested in the deeper mathematical issues underlying probability theory.

Recommended background: MA 1024.

Undergraduate credit may not be earned both for this course and for MA 2621.

### MA 3212. ACTUARIAL MATHEMATICS I

A study of actuarial mathematics with emphasis on the theory and application of contingency mathematics in various areas of insurance. Topics usually included are: survival functions and life tables; life insurance; property insurance; annuities; net premiums; and premium reserves.

Recommended background: An introduction to the theory of interest, and familiarity with basic probability (MA 2211 and either MA 2621 or MA 2631, or equivalent).

### MA 3213. ACTUARIAL MATHEMATICS II

A continuation of the study of actuarial mathematics with emphasis on calculations in various areas of insurance, based on multiple insureds, multiple decrements, and multiple state models. Topics usually included are: survival functions; life insurance; property insurance; common shock; Poisson processes and their application to insurance settings; gross premiums; and reserves.

Recommended background: An introduction to actuarial mathematics (MA 3212 or equivalent)

### MA 3231. LINEAR PROGRAMMING

Cat. I The mathematical subject of linear programming deals with those problems in optimal resource allocation which can be modeled by a linear profit (or cost) function together with feasibility constraints expressible as linear inequalities. Such problems arise regularly in many industries, ranging from manufacturing to transportation, from the design of livestock diets to the construction of investment portfolios. This course considers the formulation of such real-world optimization problems as linear programming problems, the most important algorithms for their solution, and techniques for their analysis. The core material includes problem formulation, the primal and dual simplex algorithms, and duality theory. Further topics may include: sensitivity analysis; applications such as matrix games or network flow models; bounded variable linear programs; interior point methods. Recommended background: Matrices and Linear Algebra (MA 2071, or equivalent).

### MA 3233. DISCRETE OPTIMIZATION

Cat. II

Discrete optimization is a lively field of applied mathematics in which techniques from combinatorics, linear programming, and the theory of algorithms are used to solve optimization problems over discrete structures, such as networks or graphs. The course will emphasize algorithmic solutions to general problems, their complexity, and their application to real-world problems drawn from such areas as VLSI design, telecommunications, airline crew scheduling, and product distribution. Topics will be selected from: Network flow, optimal matching, integrality of polyhedra, matroids, and NP-completeness.

Recommended background: At least one course in graph theory, combinatorics or optimization (e.g., MA 2271, MA 2273 or MA 3231).

### MA 3257. NUMERICAL METHODS FOR LINEAR AND NONLINEAR SYSTEMS

Cat. I

This course provides an introduction to modern computational methods for

linear and nonlinear equations and systems and their applications.

Topics covered include: solution of nonlinear scalar equations, direct and

iterative algorithms for the solution of systems of linear equations, solution of

nonlinear systems, the eigenvalue problem for matrices. Error analysis will be

emphasized throughout.

Recommended background: MA 2071. An ability to write computer programs

in a scientific language is assumed.

### MA 3457. NUMERICAL METHODS FOR CALCULUS AND DIFFERENTIAL EQUATIONS

Cat. I

This course provides an introduction to modern computational methods for

differential and integral calculus and differential equations.

Topics covered include: interpolation and polynomial approximation,

approximation theory, numerical differentiation and integration, numerical

solutions of ordinary differential equations. Error analysis will be emphasized

throughout.

Recommended background: MA 2051. An ability to write computer programs

in a scientific language is assumed.

Undergraduate credit may not be earned for

both this course and for MA 3255/CS 4031.

### MA 3471. ADVANCED ORDINARY DIFFERENTIAL EQUATIONS

Cat. II

The first part of the course will cover existence and uniqueness of solutions,

continuous dependence of solutions on parameters and initial conditions,

maximal interval of existence of solutions, Gronwall's inequality, linear systems

and the variation of constants formula, Floquet theory, stability of linear and

perturbed linear systems. The second part of the course will cover material

selected by the instructor. Possible topics include: Introduction to dynamical

systems, stability by Lyapunov's direct method, study of periodic solutions,

singular perturbation theory and nonlinear oscillation theory.

Recommended background: MA 2431 and MA 3832.

This course will be offered in 2015-16, and in alternating years thereafter.

### MA 3475. CALCULUS OF VARIATIONS

Cat. II

This course covers the calculus of variations and select topics from optimal control

theory. The purpose of the course is to expose students to mathematical concepts

and techniques needed to handle various problems of design encountered in

many fields, e. g. electrical engineering, structural mechanics and manufacturing.

Topics covered will include: derivation of the necessary conditions of a

minimum for simple variational problems and problems with constraints,

variational principles of mechanics and physics, direct methods of minimization

of functions, Pontryagin's maximum principle in the theory of optimal control

and elements of dynamic programming.

Recommended background: MA 2051.

This course will be offered in 2016-17, and in alternating years thereafter.

### MA 3627. INTRODUCTION TO THE DESIGN AND ANALYSIS OF EXPERIMENTS

Cat. II

This course will teach students how to design experiments in order to collect meaningful data for analysis and decision making. This course continues the exploration of statistics for scientific and industrial applications begun in MA 2611 and MA 2612. The course offers comprehensive coverage of the key elements of experimental design used by applied researchers to solve problems in the field, such as random assignment, replication, blocking, and confounding. Topics covered include the design and analysis of general factorial experiments; two-level factorial and fractional factorial experiments; principles of design; completely randomized designs and one-way analysis of variance (ANOVA); complete block designs and two-way analysis of variance; complete factorial experiments; fixed, random, and mixed models; split-plot designs; nested designs.

Recommended background: Applied Statistics (MA 2611 and MA2612, or equivalent).

### MA 3631. MATHEMATICAL STATISTICS

Cat. I

This course introduces students to the mathematical principles of statistics.

Topics will be chosen from: Sampling distributions, limit theorems, point and

interval estimation, sufficiency, completeness, efficiency, consistency; the Rao-

Blackwell theorem and the Cramer-Rao bound; minimum variance unbiased

estimators and maximum likelihood estimators; tests of hypotheses including

the Neyman-Pearson lemma, uniformly most powerful and likelihood radio tests.

Recommended background: MA 2631.

### MA 3823. GROUP THEORY

This course provides an introduction to one of the major areas of modern

algebra. Topics covered include: groups, subgroups, permutation groups, normal

subgroups, factor groups, homomorphisms, isomorphisms and the fundamental

homomorphism theorem.

Recommended background: MA 2073.

### MA 3825. RINGS AND FIELDS

Cat. II

This course provides an introduction to one of the major areas of modern

algebra. Topics covered include: rings, integral domains, ideals, quotient rings,

ring homomorphisms, polynomial rings, polynomial factorization, extension

fields and properties of finite fields.

Recommended background: MA 2073.

Undergraduate credit may not be earned both for this course and for MA 3821.

This course will be offered in 2015-16, and in alternating years thereafter.

### MA 3831. PRINCIPLES OF REAL ANALYSIS I

Cat. I

Principles of Real Analysis is a two-part course giving a rigorous presentation of the important concepts of classical real analysis. Topics covered in the sequence include: basic set theory, elementary topology of Euclidean spaces, metric spaces, compactness, limits and continuity, differentiation, Riemann-Stieltjes integration, infinite series, sequences of functions, and topics in multivariate calculus. Recommended background: at least one course focused on proof-based mathematics (e.g., MA 1971 Bridge to Higher Mathematics, MA1033 Theoretical Calculus III).

### MA 4213. LOSS MODELS I - RISK THEORY

This course covers topics in loss models and risk theory as it is applied, under specified assumptions, to insurance. Topics covered include: economics of insurance, short term individual risk models, single period and extended period collective loss models, and applications.

Recommended background: An introduction to probability (MA 2631 or equivalent).

### MA 4214. LOSS MODELS II - SURVIVAL MODELS

Survival models are statistical models of times to occurrence of some event. They are widely used in areas such as the life sciences and actuarial science (where they model such events as time to death, or to the development or recurrence of a disease), and engineering (where they model the reliability or useful life of products or processes). This course introduces the nature and properties of survival models, and considers techniques for estimation and testing of such models using realistic data. Topics covered will be chosen from: parametric and nonparametric survival models, censoring and truncation, nonparametric estimation (including confidence intervals and hypothesis testing) using right-,

left-, and otherwise censored or truncated data.

Recommended background: An introduction to mathematical statistics (MA 3631 or equivalent).

### MA 4216. ACTUARIAL SEMINAR

This pass/fail graduation requirement will be offered every term, under the supervision of the actuarial professors. In order to receive a passing grade, students will need to complete some or all of the following: attend speaker talks, attend company visits to campus, take part and help out with Math Department activities, take part and help out with Actuarial Club activities, prepare for actuarial exams, or complete other activities as approved by the instructor(s).

Recommended background: Interest in being an actuarial mathematics major.

### MA 4222. TOP NUMERICAL ALGORITHMS OF THE CENTURY

This course will highlight top algorithms that have tremendous impact on the development and practice of modern science and engineering. Class discussions will focus on introducing students to the mathematical theory behind the algorithms, and their applications. In particular, the course will address issues of computational efficiency, implementation, and error analysis. Algorithms to be considered may include the Fast Multipole Method, Metropolis Algorithm for the Monte Carlo Method, Fast Fourier Transform, Kalman filters and Singular Value Decomposition. Students will be expected to apply these algorithms to real-world problems. For example, we will look at image processing and audio compression (Fast Fourier Transform), recommendation systems (Singular Value Decomposition), and the tracking and prediction of an object’s position (Kalman Filters). In addition to studying these algorithms, students will learn about high performance computing and will have access to a machine with parallel and gpu capabilities to run code for applications with large data sets.

Recommended background: MA2071 (Linear Algebra), MA2621 or MA2631 (Probability), MA3257 (Numerical Methods for Calculus and Differential Equations), MA3457 (Numerical Methods for Linear and NonLinear Systems), at least one course in Computer Science. The ability to write computer programs in a scientific language is assumed.

### MA 4235. MATHEMATICAL OPTIMIZATION

Cat. II

This course explores theoretical conditions for the existence of solutions and

effective computational procedures to find these solutions for optimization

problems involving nonlinear functions.

Topics covered include: classical optimization techniques, Lagrange multipliers

and Kuhn-Tucker theory, duality in nonlinear programming, and algorithms for

constrained and unconstrained problems.

Recommended background: Vector calculus at the level of MA 2251.

This course will be offered in 2015-16, and in alternating years thereafter.

### MA 4237. PROBABILISTIC METHODS IN OPERATIONS RESEARCH

Cat. II

This course develops probabilistic methods useful to planners and decision

makers in such areas as strategic planning, service facilities design, and failure of

complex systems.

Topics covered include: decisions theory, inventory theory, queuing theory,

reliability theory, and simulation.

Recommended background: Probability theory at the level of MA 2621

or MA 2631.

This course will be offered in 2015-16, and in alternating years thereafter.

### MA 4291. APPLIED COMPLEX VARIABLES

Cat. I

This course provides an introduction to the ideas and techniques of complex

analysis that are frequently used by scientists and engineers. The presentation

will follow a middle ground between rigor and intuition.

Topics covered include: complex numbers, analytic functions, Taylor and Laurent

expansions, Cauchy integral theorem, residue theory, and conformal mappings.

Recommended background: MA 1024 and MA 2051.

### MA 4411. NUMERICAL ANALYSIS OF DIFFERENTIAL EQUATIONS

Cat. II

This course is concerned with the development and analysis of numerical

methods for differential equations.

Topics covered include: well-posedness of initial value problems, analysis of

Euler's method, local and global truncation error, Runge-Kutta methods, higher

order equations and systems of equations, convergence and stability analysis of one-step methods, multistep methods, methods for stiff differential equations

and absolute stability, introduction to methods for partial differential equations.

Recommended background: MA 2071 and MA 3457/CS 4033. An ability to

write computer programs in a scientific language is assumed.

This course will be offered in 2016-17, and in alternating years thereafter.

### MA 4451. BOUNDARY VALUE PROBLEMS

Cat. I

Science and engineering majors often encounter partial differential equations in

the study of heat flow, vibrations, electric circuits and similar areas. Solution

techniques for these types of problems will be emphasized in this course.

Topics covered include: derivation of partial differential equations as models of

prototype problems in the areas mentioned above, Fourier Series, solution of

linear partial differential equations by separation of variables, Fourier integrals

and a study of Bessel functions.

Recommended background: MA 1024 or and MA 2051.

### MA 4473. PARTIAL DIFFERENTIAL EQUATIONS

Cat. II

The first part of the course will cover the following topics: classification of

partial differential equations, solving single first order equations by the method

of characteristics, solutions of Laplace's and Poisson's equations including the

construction of Green's function, solutions of the heat equation including the

construction of the fundamental solution, maximum principles for elliptic and

parabolic equations. For the second part of the course, the instructor may

choose to expand on any one of the above topics.

Recommended background: MA 2251 and MA 3832.

This course will be offered in 2016-17, and in alternating years thereafter.

### MA 4603. STATISTICAL METHODS IN GENETICS AND BIOINFORMATICS

Cat. II

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.

Recommended background: MA 2612, MA 2631 (or MA 2621), and one or

more biology courses.

This course will be offered in 2015-16, and in alternating years thereafter.

### MA 4631. PROBABILITY AND MATHEMATICAL STATISTICS I

Cat. I (14 week course)

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, this course begins by covering the material of MA 3631 at a more advanced level. Additional topics covered are: one-to-one and many-to-one transformations of random variables; sampling distributions; order statistics, limit theorems.

Recommended background: MA 2631 or MA 3631, MA 3831, MA 3832.

### MA 4632. PROBABILITY AND MATHEMATICAL STATISTICS II

Cat. I (14 week course)

This course is designed to provide background in

principles of statistics.

Topics covered include: point and interval estimation; sufficiency, completeness, efficiency, consistency; the Rao-Blackwell Theorem and the Cramer-Rao bound; minimum variance unbiased estimators, maximum likelihood estimators and Bayes estimators; tests of hypothesis including uniformly most powerful, likelihood ratio, minimax and bayesian tests.

Recommended background: MA 3631 or MA 4631.

### MA 4635. DATA ANALYTICS AND STATISTICAL LEARNING

The focus of this class will be on statistical learning – the intersection of applied statistics and modeling techniques used to analyze and to make predictions and inferences from complex real-world data. Topics covered include: regression; classification/clustering; sampling methods (bootstrap and cross validation); and decision tree learning.

Recommended background: Linear Algebra (MA2071 or equivalent), Applied Statistics II (MA2612 or equivalent), Probability (MA2631 or MA2621 or equivalent). The ability to write computer programs in a scientific language is assumed.

### MA 4892. TOPICS IN ACTUARIAL MATHEMATICS

Topics covered in this course would vary from one offering to the next. The purpose of this course will be to introduce actuarial topics that typically arise in the professional actuarial organization’s curriculum beyond the point where aspiring actuaries are still in college. Topics might include ratemaking, estimation of unpaid claims, equity linked insurance products, simulation, or stochastic modeling of insurance products.

Recommended background: Could vary by the specific topics being covered, but would typically include an introduction to the theory of interest and an introduction to actuarial mathematics (MA 2211 and MA 3212 or equivalent)

### MA 4895. DIFFERENTIAL GEOMETRY

Cat. II

The course gives an introduction to differential geometry with a focus on Riemannian geometry. Starting with the geometry of curves and surfaces in the three-dimensional Euclidean space and Riemannian metrics in 2 and higher dimensions, the course introduces the first fundamental form, tangent bundles, vector fields, distance functions and geodesics, followed by covariant derivatives and second fundamental form. The proof of Gauss’s Theorema Egregium is highlighted. Additional topics are by instructor’s discretion. Students may not receive credit for both MA 489X and MA 4895.

Recommended background: Advanced Linear Algebra and Real Analysis (e.g., MA 2073 Theoretical Linear Algebra and MA 3831 Principles of Real Analysis, or equivalent)

## Graduate Courses

### 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 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.