# Master of Science in Mathematics

## Message from the Director

Because of its beauty, precision, and usefulness, mathematics has always attracted not only the most profound and theoretical minds, but also pragmatic thinkers who are eager to apply its insights to the problems of the world around us.

The master's degree program in mathematics is designed for students who have a strong undergraduate background in mathematics or a related field. Our program caters to students in many different situations, including, but not limited to, middle- and secondary-school teachers, those seeking to teach in two-year colleges, business professionals whose work is quantitative in nature, students desiring solid preparation for entrance into a doctoral program, and those who are just attracted by the beauty of mathematics.

Full-time Fairfield University faculty members teach in the master's program, bringing a wealth of expertise to the classroom. The breadth of their specialties, together with their commitment to excellence in teaching and making a difference in individual students’ lives, enriches the program and the options available to students. This benefit translates into an ability to allow our students to design individualized programs of study, in consultation with a faculty advisor, related to their background, interest, and personal goals.

The curriculum features a common core of six credits and six credits of proof-intensive coursework, supplemented by a series of electives that make specialization possible. Because our program caters to working adults, classes mostly meet one evening a week during the fall and spring semesters and are available in the summer, as well.

As director of the graduate program in mathematics, I invite you to peruse the course descriptions and faculty credentials that follow and join us in a more focused study within the field I so enjoy.

**Stephen F. Sawin, Ph.D.**

*Director of the MS in Mathematics Program*

The Master of Science program in mathematics welcomes students of ability and with a strong undergraduate background in mathematics or a related field, such as computer science, engineering, physics, finance, economics, or certain social sciences.

## Requirements

To earn a Master of Science degree in Mathematics, students complete the following in consultation with a faculty advisor:

Code | Title | Credits |
---|---|---|

Core Courses | ||

MA 0435 | Linear Algebra ^{1} | 3 |

MA 0471 | Real Analysis ^{1} | 3 |

Select two of the following Proof-Intensive courses: | 6 | |

Abstract Algebra | ||

Statistics Theory | ||

Complex Analysis | ||

Advanced Abstract Algebra | ||

Number Theory | ||

Geometry | ||

Topology | ||

Elective Courses | ||

Select six elective courses from the following: ^{2} | 18 | |

For Teachers and Prospective Teachers | ||

Applied Statistical Methods | ||

Applied Statistics II | ||

Probability Theory | ||

Foundations and Set Theory | ||

Number Theory | ||

Use of Technology in the Classroom | ||

Geometry | ||

Topology | ||

For Business-Oriented Professionals | ||

Introduction to Applied Mathematics | ||

Applied Statistical Methods | ||

Applied Statistics II | ||

Probability Theory | ||

Statistics Theory | ||

Dynamical Systems | ||

Partial Differential Equations | ||

Classical Financial Mathematics | ||

Numerical Analysis | ||

Mathematics of Financial Derivatives | ||

For Those Interested in Pure Mathematics | ||

Abstract Algebra | ||

Complex Analysis | ||

Foundations and Set Theory | ||

Dynamical Systems | ||

Advanced Abstract Algebra | ||

Number Theory | ||

Geometry | ||

Topology | ||

Numerical Analysis | ||

Capstone | ||

MA 0590 | Capstone Project (Pass/Fail) ^{3} | 0 |

Total Credits | 30 |

^{1} | One or both of these courses may be substituted with another proof-intensive course with permission of Program Director. |

^{2} | These examples illustrate three possible areas in which students might specialize within the MS program. In each case, students complete all required courses, in addition to electives such as those listed above. These are suggestions only; a student needs not restrict himself or herself to those courses in a specific category. |

^{3} | Each student should complete, generally in his/her final semesters, a capstone consisting of a project or an oral or written exposition of mathematics, in consultation with a faculty advisor. Capstones are generally associated with a course the student is taking, though it may be associated with an independent study. The faculty advisor may or may not be the instructor of the associated course, and each student, with the help of his/her advisor, should develop a proposal in advance for his/her capstone. |

**MA 0401 Introduction to Applied Mathematics**3 Credits

This course provides an introduction to essential techniques in the study of ordinary differential equations, including separation of variables, characteristic equations for linear equations, variation of parameters and Laplace transforms. The course also includes an introduction to fundamentals of applied linear algebra, including solutions of systems of linear equations, vector spaces, matrices, determinants, eigenvalues and eigenvectors.

**MA 0417 Applied Statistical Methods**3 Credits

This course introduces students to the techniques in applied statistical methods as used in the physical sciences, social sciences and business. Topics include probability (reliability, discrete and continuous distributions); descriptive and exploratory statistics using analytic and graphical tools; basic statistical testing (sampling techniques, theory of estimation and standard hypothesis testing); regression analysis (normal linear model, multivariate regression, and model building as time permits); correlation techniques; analysis of variance and factorial designs if time permits; proportion tests, chi-squared analysis and other discrete data techniques as time permits. Included is the use of computer software, such as R, SPSS and Minitab.

**MA 0418 Applied Statistics II**3 Credits

**Prerequisite: **MA 0417.

This course is a continuation of MA 0417, Applied Statistics, and covers additional statistical concepts used in the physical sciences, social sciences, business and health studies. Topics include, but are not limited to, confidence intervals, regression analysis (multiple regression, logistic regression and regression with categorical predictors), analysis of variance (two-way, factorial design, repeated measures and mixed models), analysis of categorical variables (measures of association, chi-squared tests, odds ratio, relative risk, McNemar's test) and non-parametric tests. One statistical package such as R, SPSS and Minitab, will be used throughout the course. Students should have a laptop.

**MA 0435 Linear Algebra**3 Credits

**Prerequisite: **MA 0401.

This graduate-level treatment of linear algebra includes general vector spaces; basis and dimension; linear transformations; linear operators and the relationship to matrices; inner product spaces and orthonormalization, least squares approximations, Hilbert spaces; diagonalization and other canonical forms for matrices; eigenvalues, eigenvectors, and applications to ordinary differential equations; and Hermitian, unitary, and positive definite matrices. The course also incorporates a discussion of the historical development of linear algebra, the relationship of linear algebra to analysis, and a coordinated introduction to a symbolic algebra program such as Maple or Mathematica.

**MA 0436 Abstract Algebra**3 Credits

This graduate level treatment of abstract algebra with a focus on ring theory includes the integers, the division algorithm divisibility criteria, primes and unique factorization; equivalence relations and congruence classes, modular arithmetic; rings, basic properties of rings, ideals, ring homeomorphisms; ring of polynomials, divisibility algorithm, irreducible elements and unique factorization properties, roots and irreducibility; quotients rings, prime and maximal ideals; Euclidian domains, principal ideals domains, factorization domains, field of quotients of an integral domain; introduction to group theory. This is a proof-intensive course.

**MA 0451 Probability Theory**3 Credits

**Prerequisite: **MA 0417.

This graduate-level treatment of the theory of probability includes a brief review of probability spaces and finite counting techniques, random variables and distribution functions, density, mass functions, and expectation. The course also examines the standard random variables; multivariate distributions; functions and sums of random variables; limit theorems - weak and strong law of large numbers and the central limit theorem. The course also discusses the historical development of probability.

**MA 0452 Statistics Theory**3 Credits

**Prerequisite: **MA 0451.

This graduate-level treatment of the theory of mathematical statistics includes theory of estimators, maximum likelihood techniques; theory of estimation; hypothesis testing theory - decision analysis; and Bayesian methods. The course also discusses the historical development of statistics.

**MA 0471 Real Analysis**3 Credits

This graduate-level treatment of real analysis includes the completeness of the real numbers; the topology of Euclidean n-space and its generalizations to metric and topological spaces; convergence and continuous functions; sequences of functions; general differentiability; the theory of integration and the Lebesgue integral; infinite series and uniform convergence; and a discussion of the historical development of real analysis.

**MA 0472 Complex Analysis**3 Credits

This graduate-level treatment of complex analysis includes the complex number field and its properties; complex analytic functions and their differences with real functions; the complex integral; Cauchy's Theorem and consequences; and a discussion of the historical development of complex analysis. This is a proof-intensive course.

**MA 0495 Special Topics (Shell)**3 Credits

Mathematical topics not currently among the department's offerings can be offered one-time or to allow a professor the opportunity to "test drive"' a course for the first time.

**MA 0510 Foundations and Set Theory**3 Credits

The foundations of modern mathematics lie in set theory and logic. This course provides a graduate-level treatment of these areas in the foundation of theoretical mathematics. It is also a good preparation for proof-intensive courses for those without a solid undergraduate foundation in theoretical mathematics.

**MA 0531 Dynamical Systems**3 Credits

**Prerequisite: **MA 0401.

This course provides an introduction to the study of dynamical systems from the point of view of both continuous time and discrete time systems. Topics include fixed point and stability analysis for linear and nonlinear flows in one and two dimensions, phase plane analysis, bifurcations and limit cycles, one-dimensional maps, chaos and Lyapunov exponents.

**MA 0532 Partial Differential Equations**3 Credits

**Prerequisite: **MA 0401.

This graduate-level treatment of partial differential equations includes boundary value problems; Fourier series, and Fourier transforms.

**MA 0535 Advanced Abstract Algebra**3 Credits

**Prerequisite: **MA 0436.

A collection of topics in advanced abstract algebra, this course includes group theory, field extensions and Galois. This is a proof-intensive course.

**MA 0537 Number Theory**3 Credits

This graduate-level survey of the problems and techniques of number theory includes elementary number theory and introductions to analytic and algebraic number theory. This is a proof-intensive course.

**MA 0550 Classical Financial Mathematics**3 Credits

This course covers the basic mathematics of classical financial investments. It will include the basic formulas for compound interest and effective yields, infinite series and exponential functions, annuities and perpetuities, amortization and sinking funds, time value of money, and bond and stock discounts.

**MA 0565 Use of Technology in the Classroom**3 Credits

Designed for teachers, this course surveys various computer software mathematics packages suitable for use in the classroom, such as Maple, Mathematica, MATLAB, SKETCHPAD, and ISETL. The course includes a description of the programs and discusses how they can be integrated into a classroom setting.

**MA 0577 Numerical Analysis**3 Credits

This course provides a graduate-level treatment of numerical analysis and the numerical solution of mathematical problems and includes an introduction to computer implementation of numerical algorithms.

**MA 0578 Mathematics of Financial Derivatives**3 Credits

**Prerequisite: **MA 0550.

This course covers the theory of financial derivatives, including an explanation of option pricing theory and investments, the idea of financial derivatives, stochastic differential equations, and the Black-Scholes model.

**MA 0583 Geometry**3 Credits

This course offers a graduate-level treatment of Euclidean and non-Euclidean geometry and is highly recommended for teachers. This is a proof-intensive course.

**MA 0585 Topology**3 Credits

**Prerequisite: **MA 0471.

This course provides an introductory, graduate-level treatment of point-set and algebraic topology and topological methods. This is a proof-intensive course.

**MA 0590 Capstone Project**0 Credits

This is an independent project or presentation planned by the student with the help of a faculty mentor and produced by the student through original work. The project is typically based on the content of a course and is worked on in conjunction with that course, but students can also learn the necessary material in a three-credit independent study with their mentor.

**MA 0599 Independent Study**3 Credits

The Master's Degree Program in Mathematics affords each student the opportunity to do an independent study course with a professor/mentor. This can either be an existing course in the program or a course on an advanced topic in mathematics. In the latter case the syllabus and requirements are developed by the student and the faculty mentor.

Professors in the program are full-time faculty of the College of Arts and Sciences, with highly regarded expertise in a wide range of areas of mathematics and a deep commitment to teaching and making a difference in individual students’ lives.

## Professor

Bernhardt

Coleman

Dennin

Fine

Mulvey, *Chair*

Sawin, *Director*

Weiss

## Associate Professor

Baginski

Demers

McSweeney

Rafalski

Staecker

Striuli