# Mathematics

**Christopher K. King, PhD**

Professor and Chair

567 Lake Hall

617.373.2450

617.373.5658 (fax)

Solomon M. Jekel, Associate Professor and Head Advisor, s.jekel@northeastern.edu

Mathematics is of ever-increasing importance to our society and everyday life. It has long been the language of science and technology and provides a rich source of methods for analyzing and solving problems encountered in the physical world. Today, mathematics is essential in virtually all fields of human endeavor, including business, the arts, and the social sciences.

The Bachelor of Arts degree requires at least eleven mathematics courses and two physics courses, in addition to the study of a foreign language; this program is appropriate for students who wish a broader liberal arts education. The Bachelor of Science degree requires at least fourteen mathematics courses and two physics courses but no foreign language study; it is more specialized, and it is recommended for those strongly interested in mathematics and science. The department also offers a minor degree in mathematics.

The major programs provide flexibility with elective courses. Students may take advantage of a range of interdisciplinary programs and may join a major in mathematics with one in such fields as computer science, physics, and biology.

Exceptional students are accepted into the Honors Program and have the option to enroll in honors sections of several of their mathematics courses. All math majors may benefit from co-op opportunities in the scientific and business communities in Boston and elsewhere.

Many of the mathematics courses that we offer use computers for visualization, modeling, and numerical approximation.

Students planning to teach secondary-school mathematics must major in mathematics and take a specific minor in education, which includes course work and student teaching.

Mathematical training may lead to opportunities in applied research (natural sciences, engineering, economics, management, computer science) as well as in mathematical research, teaching, or industry.

## Academic Progression Standards

Students who begin as freshman mathematics majors must, after the fourth semester, satisfy the following:

- Have completed at least six of their required mathematics and physics courses
- Have a grade average of C or better in the following courses:
Course List Code Title Hours MATH 1365 Introduction to Mathematical Reasoning 4 MATH 1341 Calculus 1 for Science and Engineering 4 MATH 1342 Calculus 2 for Science and Engineering 4 MATH 2321 Calculus 3 for Science and Engineering 4 MATH 2331 Linear Algebra 4 - Have a grade-point average (GPA) of at least 2.000 in all required mathematics and physics courses

Students who transfer to the major must, after two semesters in the major, satisfy the following:

- Have completed at least four of their required mathematics and physics courses
- Have a GPA of at least 2.000 in all required mathematics and physics courses
- Have grades of C or better in the following courses (if already taken):
Course List Code Title Hours MATH 1365 Introduction to Mathematical Reasoning 4 MATH 1341 Calculus 1 for Science and Engineering 4 MATH 1342 Calculus 2 for Science and Engineering 4 MATH 2321 Calculus 3 for Science and Engineering 4 MATH 2331 Linear Algebra 4

Students who fail to achieve the above conditions will be placed on departmental probation. Students who remain two consecutive semesters on departmental probation will be dismissed from the major.

## Bachelor of Arts (BA)

## Bachelor of Science (BS)

- Mathematics
- Biology and Mathematics
- Computer Science and Mathematics
- Economics and Mathematics
- Graphic and Information Design and Mathematics
- Mathematics and Business Administration
- Mathematics and Physics

## Dual Degrees

## Minor

## Mathematics Courses

**MATH 0100. Algebra Review. 4 Hours.**

Designed for arts and sciences, criminal justice, and other majors who need to build their algebraic skills in order to succeed in the next math or math-related courses required by their major. Most students are directed to this course as a result of placement tests. Concepts include solving first- and second-degree equations, understanding slopes and graphs of lines, solving simultaneous equations in several variables, solving rational equations, and graphing inequalities. Requires the analysis and solution of word problems. (Does not count toward graduation credit.).

**MATH 1000. Mathematics at Northeastern. 1 Hour.**

Designed for freshman math majors to introduce them to one another, their major, their college, and the University. Students are introduced to our advising system, register for next semester’s courses, and learn more about co-op. Also helps students develop the academic and interpersonal skills necessary to succeed as a university student. *Prereq. Mathematics majors only.*

**MATH 1110. College Algebra. 4 Hours.**

Covers laws of exponents, roots, graphing of equations and inequalities, special curves (that is, conic sections), functions and operations on functions, complex numbers, matrices, and vectors. If time permits, also explores elementary discrete probability and least squares curve fitting. *Prereq. Not open to students in the College of Computer and Information Science.*

**MATH 1120. Precalculus. 4 Hours.**

Focuses on linear, polynomial, exponential, logarithmic, and trigonometric functions. Emphasis is placed on understanding, manipulating, and graphing these basic functions, their inverses and compositions, and using them to model real-world situations (that is, exponential growth and decay, periodic phenomena). Equations involving these functions are solved using appropriate techniques. Special consideration is given to choosing reasonable functions to fit numerical data. *Prereq. Not open to students in the College of Computer and Information Science.*

**MATH 1130. College Math for Business and Economics. 4 Hours.**

Introduces students to some of the important mathematical concepts and tools (such as modeling revenue, cost and profit with functions) used to solve problems in business and economics. Assumes familiarity with the basic properties of linear, polynomial, exponential, and logarithmic functions. Topics include the method of least squares, regression curves, solving equations involving functions, compound interest, amortization, and other consumer finance models. (Graphing calculator required, see instructor for make and model.) *Prereq. Not open to students in the College of Computer and Information Science.*

**MATH 1180. Statistical Thinking. 4 Hours.**

Introduces statistical thinking to students without using any sophisticated mathematics. Uses extensive class discussion and homework problems to cover statistical reasoning and to evaluate critically the usage of statistics by others. Readings from a wide variety of sources are assigned. Topics include descriptive statistics, sampling theory, and fundamentals of statistical inference (confidence intervals and hypothesis testing). *Prereq. Not open to students in the College of Computer and Information Science.*

**MATH 1213. Interactive Mathematics. 4 Hours.**

Develops problem-solving skills while simultaneously teaching mathematics concepts. Each unit centers on a particular applied problem, which serves to introduce the relevant mathematical topics. These may include but are not limited to polling theory, rate of change, the concepts behind derivatives, probability, binomial distributions, and statistics. The course is not taught in the traditional lecture format and is particularly suited to students who work well in collaborative groups and who enjoy writing about the concepts they are learning. Assessment is based on portfolios, written projects, solutions to “problems of the week,” and exams. *Prereq. Not open to students in the College of Computer and Information Science.*

**MATH 1215. Mathematical Thinking. 4 Hours.**

Focuses on the development of mathematical thinking and its use in a variety of contexts to translate real-world problems into mathematical form and, through analysis, to obtain new information and reach conclusions about the original problems. Mathematical topics include symbolic logic, truth tables, valid arguments, counting principles, and topics in probability theory such as Bayes’ theorem, the binomial distribution, and expected value.

**MATH 1216. Recitation for MATH 1215. 0 Hours.**

Provides small-group discussion format to cover material in MATH 1215.

**MATH 1220. Mathematics of Art. 4 Hours.**

Presents mathematical connections and foundations for art. Topics vary and may include aspects of linear perspective and vanishing points, symmetry and patterns, tilings and polygons, Platonic solids and polyhedra, golden ratio, non-Euclidean geometry, hyperbolic geometry, fractals, and other topics. Includes connections and examples in different cultures.

**MATH 1225. Game Theory. 4 Hours.**

Uses the unifying theme of game theory to explore mathematical techniques for gaining an understanding of real-world problems. Includes matrix algebra, linear programming, probability, trees, von Neumann’s minimax theorem, and Nash’s theorem on equilibrium points. Considers zero-sum and non-zero-sum games, multiperson games, and the prisoner’s dilemma. Explores the applications of game theory, including conflict analysis, and various issues in psychology, sociology, political science, economics, and business. *Prereq. Mathematics SAT of at least 600 or permission of instructor.*

**MATH 1231. Calculus for Business and Economics. 4 Hours.**

Provides an overview of differential calculus including derivatives of power, exponential, logarithmic, logistic functions, and functions built from these. Derivatives are used to model rates of change, to estimate change, to optimize functions, and in marginal analysis. The integral calculus is applied to accumulation functions and future value. Emphasis is on realistic business and economics problems, the development of mathematical models from raw business data, and the translation of mathematical results into verbal expression appropriate for the business setting. Also features a semester-long marketing project in which students gather raw data, model it, and use calculus to make business decisions; each student is responsible for a ten-minute presentation. (Graphing calculator required, see instructor for make and model.).

**MATH 1241. Calculus 1. 4 Hours.**

Serves as both the first half of a two-semester calculus sequence and as a self-contained one-semester course in differential and integral calculus. Introduces basic concepts and techniques of differentiation and integration and applies them to polynomial, exponential, log, and trigonometric functions. Emphasizes the derivative as rate of change and integral as accumulator. Applications include optimization, growth and decay, area, volume, and motion. *Prereq. Not open to students in the College of Computer and Information Science.*

**MATH 1242. Calculus 2. 4 Hours.**

Continues MATH 1241. Introduces additional techniques of integration and numerical approximations of integrals and the use of integral tables; further applications of integrals. Also introduces differential equations and slope fields, and elementary solutions. Introduces functions of several variables, partial derivatives, and multiple integrals. *Prereq. MATH 1231, MATH 1241, or MATH 1341; not open to students in the College of Computer and Information Science.*

**MATH 1251. Calculus and Differential Equations for Biology 1. 4 Hours.**

Begins with the fundamentals of differential calculus and proceeds to the specific type of differential equation problems encountered in biological research. Presents methods for the solutions of these equations and how the exact solutions are obtained from actual laboratory data. Topics include differential calculus: basics, the derivative, the rules of differentiation, curve plotting, exponentials and logarithms, and trigonometric functions; using technology to understand derivatives; biological kinetics: zero- and first-order processes, processes tending toward equilibrium, bi- and tri-exponential processes, and biological half-life; differential equations: particular and general solutions to homogeneous and nonhomogeneous linear equations with constant coefficients, systems of two linear differential equations; compartmental problems: nonzero initial concentration, two-compartment series dilution, diffusion between compartments, population dynamics; and introduction to integration.

**MATH 1252. Calculus and Differential Equations for Biology 2. 4 Hours.**

Continues MATH 1251. Begins with the integral calculus and proceeds quickly to more advanced topics in differential equations. Introduces linear algebra and uses matrix methods to analyze functions of several variables and to solve larger systems of differential equations. Advanced topics in reaction kinetics are covered. The integral and differential calculus of functions of several variables is followed by the study of numerical methods in integration and solutions of differential equations. Provides a short introduction to probability. Covers Taylor polynomials and infinite series. Special topics include reaction kinetics: Michaelis-Menten processes, tracer experiments, and inflow and outflow through membranes. *Prereq. MATH 1251.*

**MATH 1260. Math Fundamentals for Games. 4 Hours.**

Discusses linear algebra and vector geometry in two-, three-, and four-dimensional space. Examines length, dot product, and trigonometry. Introduces linear and affine transformations. Discusses complex numbers in two-space, cross product in three-space, and quaternions in four-space. Provides explicit formulas for rotations in three-space. Examines functions of one argument and treats exponentials and logarithms. Describes parametric curves in space. Discusses binomials, discrete probability, Bézier curves, and random numbers. Concludes with the concept of the derivative, the rules for computing derivatives, and the notion of a differential equation.

**MATH 1340. Intensive Calculus for Engineers. 6 Hours.**

Contains the material from the first semester of MATH 1341, preceded by material emphasizing the strengthening of precalculus skills. Topics include properties of exponential, logarithmic, and trigonometric functions; differential calculus; and introductory integral calculus.

**MATH 1341. Calculus 1 for Science and Engineering. 4 Hours.**

Covers definition, calculation, and major uses of the derivative, as well as an introduction to integration. Topics include limits; the derivative as a limit; rules for differentiation; and formulas for the derivatives of algebraic, trigonometric, and exponential/logarithmic functions. Also discusses applications of derivatives to motion, density, optimization, linear approximations, and related rates. Topics on integration include the definition of the integral as a limit of sums, antidifferentiation, the fundamental theorem of calculus, and integration by substitution.

**MATH 1342. Calculus 2 for Science and Engineering. 4 Hours.**

Covers further techniques and applications of integration, infinite series, and introduction to vectors. Topics include integration by parts; numerical integration; improper integrals; separable differential equations; and areas, volumes, and work as integrals. Also discusses convergence of sequences and series of numbers, power series representations and approximations, 3D coordinates, parameterizations, vectors and dot products, tangent and normal vectors, velocity, and acceleration in space. *Prereq. MATH 1341 or permission of head mathematics advisor.*

**MATH 1343. Calculus 2 for Engineering Technology. 4 Hours.**

Builds upon the differential and integral calculus topics in MATH 1341 to develop additional tools such as partial derivatives and multiple integrals needed by students of engineering technology. This course is not equivalent to MATH 1342. *Prereq. MATH 1341.*

**MATH 1352. Recitation for MATH 1342. 0 Hours.**

Provides small-group discussion format to cover material in MATH 1342.

**MATH 1365. Introduction to Mathematical Reasoning. 4 Hours.**

Covers the basics of mathematical reasoning and problem solving to prepare incoming math majors for more challenging mathematical courses at Northeastern. Focuses on learning to write logically sound mathematical arguments and to analyze such arguments appearing in mathematical books and courses. Includes fundamental mathematical concepts such as sets, relations, and functions.

**MATH 1990. Elective. 1-4 Hours.**

Offers elective credit for courses taken at other academic institutions.

**MATH 2201. History of Mathematics. 4 Hours.**

Traces the development of mathematics from its earliest beginning to the present. Emphasis is on the contributions of various cultures including the Babylonians, Egyptians, Mayans, Greeks, Indians, and Arabs. Computations and constructions are worked out using the techniques and notations of these peoples. The role of mathematics in the development of science is traced throughout, including the contributions of Descartes, Kepler, Fermat, and Newton. More modern developments are discussed as time permits.

**MATH 2210. Foundations of Mathematics. 4 Hours.**

Investigates the modern revolutions in mathematics initiated by Cantor, Gödel, Turing, and Robinson in the fields of set theory, provability, computability, and analysis respectively, as well as provides background on the controversy over the philosophy and underlying logic of mathematics.

**MATH 2230. Mathematical Encounters. 4 Hours.**

Covers interesting and significant developments in pure and applied mathematics, from ancient times to the present. Fundamental mathematical ideas have a power and utility that are undeniable and a beauty and clarity that can be inspirational. Selected topics may include: prime and irrational numbers, different infinities and different geometries, map coloring, and famous unsolved and recently solved problems. Provides students with an opportunity for hands-on experience actually doing some of the mathematics discussed and to research topics in the library and on the Web. *Prereq. MATH 1120, MATH 1215, MATH 1231, MATH 1241, or MATH 1341; not open to students in the College of Computer and Information Science.*

**MATH 2250. Programming Skills for Mathematics. 2 Hours.**

Introduces basic programming skills for applied mathematics. Also serves as preparation for co-op assignments. Topics include Excel macros, MATLAB programming, and the R statistical package. Every mathematics major or student in a mathematics combined major is required to take this course or an equivalent course in another department. *Prereq. Not open to students in the College of Computer and Information Science.*

**MATH 2280. Statistics and Software. 4 Hours.**

Provides an introduction to basic statistical techniques and the reasoning behind each statistical procedures. Covers appropriate statistical data analysis methods for applications in health and social sciences. Also examines a statistical package such as SPSS or SAS to implement the data analysis on computer. Topics include descriptive statistics, elementary probability theory, parameter estimation, confidence intervals, hypothesis testing, nonparametric inference, and analysis of variance and regression with a minimum of mathematical derivations. *Prereq. Nonmathematics majors only; not open to students in the College of Computer and Information Science.*

**MATH 2285. Introduction to Multisample Statistics. 4 Hours.**

Provides an introduction to statistical techniques, including multisample statistics and regression. Offers an opportunity to learn to choose appropriate statistical data analysis methods for applications in various scientific fields and to learn to use a statistical package to implement the data analysis. Topics include descriptive statistics, elementary probability theory, parameter estimation, confidence intervals, hypothesis testing, analysis of variance, and regression. May also include optimal design. *Prereq. MATH 1120, MATH 1215, MATH 1231, MATH 1241, or MATH 1341; not open to students who have completed MATH 2280; not open to students in the College of Computer and Information Science.*

**MATH 2310. Discrete Mathematics. 4 Hours.**

Provides the discrete portion of the mathematical background needed by students in electrical and computer engineering. Topics include Boolean algebra and set theory, logic, and logic gates; growth of functions, and algorithms and their complexity; proofs and mathematical induction; and graphs, trees, and their algorithms. As time permits, additional topics may include methods of enumeration and finite-state machines.

**MATH 2321. Calculus 3 for Science and Engineering. 4 Hours.**

Extends the techniques of calculus to functions of several variables; introduces vector fields and vector calculus in two and three dimensions. Topics include lines and planes, 3D graphing, partial derivatives, the gradient, tangent planes and local linearization, optimization, multiple integrals, line and surface integrals, the divergence theorem, and theorems of Green and Stokes with applications to science and engineering and several computer lab projects. *Prereq. MATH 1342 or MATH 1252.*

**MATH 2322. Recitation for MATH 2321. 0 Hours.**

Provides small-group discussion format to cover material in MATH 2321.

**MATH 2323. Calculus 3 for Business, Economics, and Mathematics. 4 Hours.**

Covers multivariable calculus with applications from economics and business. Designed for combined majors in business and mathematics and in economics and mathematics, but open to all who have taken first-year calculus. Topics include Gaussian elimination, matrix algebra, determinants, linear independence, calculus of several variables, chain rule, implicit differentiation, optimization, Lagrange multipliers, and integration of functions of several variables with applications to probability. *Prereq. MATH 1342.*

**MATH 2331. Linear Algebra. 4 Hours.**

Uses the Gauss-Jordan elimination algorithm to analyze and find bases for subspaces such as the image and kernel of a linear transformation. Covers the geometry of linear transformations: orthogonality, the Gram-Schmidt process, rotation matrices, and least squares fit. Examines diagonalization and similarity, and the spectral theorem and the singular value decomposition. Is primarily for math and science majors; applications are drawn from many technical fields. Computation is aided by the use of software such as Maple or MATLAB, and graphing calculators. *Prereq. MATH 1242, MATH 1252, MATH 1342, or CS 2800.*

**MATH 2341. Differential Equations and Linear Algebra for Engineering. 4 Hours.**

Studies ordinary differential equations, their applications, and techniques for solving them including numerical methods (through computer labs using MS Excel and MATLAB), Laplace transforms, and linear algebra. Topics include linear and nonlinear first- and second-order equations and applications include electrical and mechanical systems, forced oscillation, and resonance. Topics from linear algebra, such as matrices, row-reduction, vector spaces, and eigenvalues/eigenvectors, are developed and applied to systems of differential equations. *Prereq. MATH 1342.*

**MATH 2342. Recitation for MATH 2341. 0 Hours.**

Provides small-group discussion format to cover material in MATH 2341.

**MATH 2990. Elective. 1-4 Hours.**

Offers elective credit for courses taken at other academic institutions.

**MATH 3000. Co-op and Experiential Learning Reflection Seminar 1. 1 Hour.**

Intended for math majors who have completed their first co-op assignment or other integrated experiential learning component of the NU Core. The goal is to examine the mathematical problems encountered in these experiences and relate them to courses already taken and to the student’s future program. Faculty members and other guests contribute to the discussion. Grades are determined by the student’s participation in the course and the completion of a final paper. *Prereq. Mathematics majors and computer science/mathematics combined majors only.*

**MATH 3081. Probability and Statistics. 4 Hours.**

Focuses on probability theory. Topics include sample space; conditional probability and independence; discrete and continuous probability distributions for one and for several random variables; expectation; variance; special distributions including binomial, Poisson, and normal distributions; law of large numbers; and central limit theorem. Also introduces basic statistical theory including estimation of parameters, confidence intervals, and hypothesis testing. *Prereq. MATH 1242, MATH 1252, or MATH 1342.*

**MATH 3090. Exploration of Modern Mathematics. 4 Hours.**

Offers students a research-minded, elementary, and intuitive introduction to the interplay between algebra, geometry, analysis, and topology using an interactive and experimental approach. *Prereq. MATH 1242, MATH 1252, or MATH 1342; intended for math majors, math combined majors, and students pursuing a math minor; all others should obtain permission of instructor.*

**MATH 3150. Real Analysis. 4 Hours.**

Provides the theoretical underpinnings of calculus and the advanced study of functions. Emphasis is on precise definitions and rigorous proof. Topics include the real numbers and completeness, continuity and differentiability, the Riemann integral, the fundamental theorem of calculus, inverse function and implicit function theorems, and limits and convergence. Required of all mathematics majors. *Prereq. (a) MATH 2321 and (b) MATH 2331 and (c) ENGW 1111, ENGW 1102, ENGL 1111, or ENGL 1102 and (d) junior or senior standing.*

**MATH 3175. Group Theory. 4 Hours.**

Presents basic concepts and techniques of the group theory: symmetry groups, axiomatic definition of groups, important classes of groups (abelian groups, cyclic groups, additive and multiplicative groups of residues, and permutation groups), Cayley table, subgroups, group homomorphism, cosets, the Lagrange theorem, normal subgroups, quotient groups, and direct products. Studies structural properties of groups. Possible applications include geometry, number theory, crystallography, physics, and combinatorics. *Prereq. MATH 2321 and MATH 2331.*

**MATH 3331. Elementary Differential Geometry. 4 Hours.**

Studies differential geometry, focusing on curves and surfaces in 3D space. The material presented here can serve as preparation for a more advanced course in Riemannian geometry or differential topology. *Prereq. MATH 2321 and MATH 2331.*

**MATH 3341. Dynamical Systems. 4 Hours.**

Studies dynamical systems and their applications as they arise from differential equations. Solutions are obtained and analyzed as parameterized curves in the plane and used as a means of understanding the evolution of physical processes. Applications include conservative systems, predator-prey interactions, and cooperation and competition of species. *Prereq. MATH 2341.*

**MATH 3527. Number Theory. 4 Hours.**

Introduces number theory. Topics include linear diophantine equations, congruences, design of magic squares, Fermat’s little theorem, Euler’s formula, Euler’s phi function, computing powers and roots in modular arithmetic, the RSA encryption system, primitive roots and indices, and the law of quadratic reciprocity. As time permits, may cover diophantine approximation and Pell’s equation, elliptic curves, points on elliptic curves, and Fermat’s last theorem. *Prereq. MATH 1242, MATH 1252, or MATH 1342.*

**MATH 3530. Numerical Analysis. 4 Hours.**

Considers various problems including roots of nonlinear equations; simultaneous linear equations: direct and iterative methods of solution; eigenvalue problems; interpolation; and curve fitting. Emphasizes understanding issues rather than proving theorems or coming up with numerical recipes. *Prereq. MATH 2331 or MATH 2341.*

**MATH 3532. Numerical Solutions of Differential Equations. 4 Hours.**

Covers numerical problems in interpolation, differentiation, integration, Fourier transforms, and the solving of differential equations. Emphasizes practical methods and techniques. The heart of the course is a study of modern methods for finding numerical solutions of ordinary differential equations, both initial value problems and boundary value problems. Homework and projects are based on MATLAB. *Prereq. (a) MATH 2331 and (b) MATH 2341 or MATH 2351.*

**MATH 3533. Combinatorial Mathematics. 4 Hours.**

Introduces techniques of mathematical proofs including mathematical induction. Explores various techniques for counting such as permutation and combinations, inclusion-exclusion principle, recurrence relations, generating functions, Polya enumeration, and the mathematical formulations necessary for these techniques including elementary group theory and equivalence relations. *Prereq. MATH 1242, MATH 1252, or MATH 1342.*

**MATH 3541. Chaotic Dynamical Systems. 4 Hours.**

Presents an experimental study using simple mathematical models of chaotic behavior in dynamical systems. (Such systems are frequently found in science and industry.) Goals include the development of skills of experiment and inquiry, integration of visual and analytical modes of thought, and appreciation of issues of problem formulation and representation. *Prereq. MATH 1342 and two semesters of calculus.*

**MATH 3560. Geometry. 4 Hours.**

Studies classical geometry and symmetry groups of geometric figures, with an emphasis on Euclidean geometry. Teaches how to formulate mathematical propositions precisely and how to construct and understand mathematical proofs. Provides a line between classical and modern geometry with the aim of preparing students for further study in group theory and differential geometry. *Prereq. MATH 2331 or MATH 2341.*

**MATH 3990. Elective. 1-4 Hours.**

Offers elective credit for courses taken at other academic institutions.

**MATH 4000. Co-op and Experiential Learning Reflection Seminar 2. 1 Hour.**

Intended for math majors who have completed their second co-op assignment or other integrated experiential learning component of the NU Core. The goal is to examine the mathematical problems encountered in these experiences and relate them to courses already taken and to the student’s future program. Faculty members and other guests contribute to the discussion. Grades are determined by the student’s participation in the course and the completion of a final paper. *Prereq. Mathematics majors and computer science/mathematics combined majors only.*

**MATH 4020. Research Capstone. 4 Hours.**

Offers students the experience of engaging in mathematical research that builds upon the math courses that they have taken and, possibly, their co-op assignments. Requires students to complete a research project of their own choosing. Focus is on the project and on the students presenting their work. Also requires students to write a reflection paper. Intended for juniors or seniors with experience or interest in mathematics research. *Prereq. (a) MATH 3150 or permission of instructor and (b) MATH 3175 or permission of instructor and (c) junior or senior standing.*

**MATH 4025. Applied Mathematics Capstone. 4 Hours.**

Emphasizes the use of a variety of methods—such as optimization, differential equations, probability, and statistics—to study problems that arise in epidemiology, finance, and other real-world settings. Course work includes assigned exercises, a long-term modeling project on a topic of the student’s choosing, and a reflection paper. *Prereq. MATH 3081 and junior or senior standing.*

**MATH 4525. Applied Analysis. 4 Hours.**

Demonstrates the applications of mathematics to interesting physical and biological problems. Methods are chosen from ordinary and partial differential equations, calculus of variations, Laplace transform, perturbation theory, special functions, dimensional analysis, asymptotic analysis, and other techniques of applied mathematics. *Prereq. (a) MATH 2321 and (b) MATH 2331 and (c) MATH 2341 or MATH 2351.*

**MATH 4535. Mathematical Topics in Computer Vision. 4 Hours.**

Studies topics in computer vision and the mathematical approaches to them. These include but are not limited to detection of object boundaries in images, nonlinear diffusion, optimization, and curve evolution. Students are required to be able to program algorithms that the course develops. *Prereq. MATH 2321 and programming experience with MATLAB or an equivalent computer algebra system; familiarity with matrices and their properties is helpful.*

**MATH 4541. Advanced Calculus. 4 Hours.**

Offers a deeper and more generalized look at the ideas and objects of study of calculus. Topics include the generalized calculus of n-space, the inverse and implicit function theorems, differential forms and general Stokes-type theorems, geometry of curves and surfaces, and special functions. *Prereq. MATH 2321, MATH 2331, and MATH 3150.*

**MATH 4545. Fourier Series and PDEs. 4 Hours.**

Provides a first course in Fourier series, Sturm-Liouville boundary value problems, and their application to solving the fundamental partial differential equations of mathematical physics: the heat equation, the wave equation, and Laplace’s equation. Green’s functions are also introduced as a means of obtaining closed-form solutions. *Prereq. MATH 2351 or MATH 2341.*

**MATH 4555. Complex Variables. 4 Hours.**

Provides an introduction to the analysis of functions of a complex variable. Starting with the algebra and geometry of complex numbers, basic derivative and contour integral properties are developed for elementary algebraic and transcendental functions as well as for other analytic functions and functions with isolated singularities. Power and Laurent series representations are given. Classical integral theorems, residue theory, and conformal mapping properties are studied. Applications of harmonic functions are presented as time permits. *Prereq. MATH 2321.*

**MATH 4565. Topology. 4 Hours.**

Introduces the student to fundamental notions of topology. Introduces basic set theory, then covers the foundations of general topology (axioms for a topological space, continuous functions, homeomorphisms, metric spaces, the subspace, product and quotient topologies, connectedness, compactness, and the Hausdorff condition). Also introduces algebraic and geometric topology (homotopy, covering spaces, fundamental groups, graphs, surfaces, and manifolds) and applications. Other topics are covered if time permits. *Prereq. MATH 3150.*

**MATH 4571. Advanced Linear Algebra. 4 Hours.**

Provides a more detailed study of linear transformations and matrices: LU factorization, QR factorization, Spectral theorem and singular value decomposition, Jordan form, positive definite matrices, quadratic forms, partitioned matrices, and norms and numerical issues. Topics and emphasis change from year to year. *Prereq. MATH 2331.*

**MATH 4575. Introduction to Cryptography. 4 Hours.**

Presents the mathematical foundations of cryptology, beginning with the study of divisibility of integers, the Euclidian Algorithm, and an analysis of the Extended Euclidian Algorithm. Includes a short study of groups, semigroups, residue class rings, fields, Fermat’s Little Theorem, Chinese Remainder Theorem, polynomials over fields, and the multiplicative group of residues modulo a prime number. Introduces fundamental notions used to describe encryption schemes together with examples, which include affine linear ciphers and cryptanalysis and continues with probability and perfect secrecy. Presents the Data Encryption Standard (DES) and culminates in the study of the Advanced Encryption Standard (AES), the standard encryption scheme in the United States since 2001.

**MATH 4576. Rings and Fields. 4 Hours.**

Introduces commutative rings, ideals, integral domains, fields, and the theory of extension fields. Topics include Gaussian integers, Galois groups, and the fundamental theorem of Galois theory. Applications include the impossibility of angle-trisection and the general insolvability of fifth- and higher-degree polynomials. Other topics are covered as time permits. *Prereq. MATH 3175.*

**MATH 4581. Statistics and Stochastic Processes. 4 Hours.**

Continues topics introduced in MATH 3081. The first part of the course covers classical procedures of statistics including the t-test, linear regression, and the chi-square test. The second part provides an introduction to stochastic processes with emphasis on Markov chains, random walks, and Brownian motion, with applications to modeling and finance. *Prereq. MATH 3081.*

**MATH 4606. Mathematical and Computational Methods for Physics. 4 Hours.**

Covers advanced mathematical methods topics that are commonly used in the physical sciences, such as complex calculus, Fourier transforms, special functions, and the principles of variational calculus. Applies these methods to computational simulation and modeling exercises. Introduces basic computational techniques and numerical analysis, such as Newton’s method, Monte Carlo integration, gradient descent, and least squares regression. Uses a simple programming language, such as MATLAB, for the exercises. *Prereq. (a) PHYS 2303 and (b) MATH 2321 and (c) MATH 2341 or MATH 2351.*

**MATH 4681. Probability and Risks. 4 Hours.**

Reviews main probability and statistics concepts from the point of view of decision risks in actuarial and biomedical contexts, including applications of normal approximation for evaluating statistical risks. Also examines new topics, such as distribution of extreme values and nonparametric statistics with examples. May be especially useful for students preparing for the first actuarial exam on probability and statistics. *Prereq. MATH 3081.*

**MATH 4682. Theory of Interest and Basics of Life Insurance. 4 Hours.**

Reviews basic financial instruments in the presence of interest rates, including the measurement of interest and problems in interest (equations of value, basic and more general annuities, yield rates, amortization schedules, bonds and other securities). Examines numerous practical applications. Also introduces problems of life insurance with examples. May be especially useful for students preparing for the second actuarial exam on theory of interest. *Prereq. MATH 3081.*

**MATH 4683. Financial Derivatives. 4 Hours.**

Presents the mathematical basis of actuarial models and their application to insurance and other financial risks. Includes but is not limited to financial derivatives such as options and futures. Techniques and applications may be useful for students preparing for actuarial Exam 3F (Society of Actuaries Exam MFE). *Prereq. MATH 4681 and MATH 4682.*

**MATH 4970. Junior/Senior Honors Project 1. 4 Hours.**

Focuses on in-depth project in which a student conducts research or produces a product related to the student’s major field. Combined with Junior/Senior Project 2 or college-defined equivalent for 8-credit honors project. *Prereq. MATH 3081.*

**MATH 4971. Junior/Senior Honors Project 2. 4 Hours.**

Focuses on second semester of in-depth project in which a student conducts research or produces a product related to the student’s major field. *Prereq. MATH 4970.*

**MATH 4990. Elective. 1-4 Hours.**

Offers elective credit for courses taken at other academic institutions.

**MATH 4991. Research. 4 Hours.**

Offers an opportunity to conduct research under faculty supervision.

**MATH 4992. Directed Study. 1-4 Hours.**

Offers independent work under the direction of members of the department on a chosen topic. Course content depends on instructor.

**MATH 4993. Independent Study. 1-4 Hours.**

Offers independent work under the direction of members of the department on a chosen topic. Course content depends on instructor.

**MATH 4994. Internship. 4 Hours.**

Offers students an opportunity for internship work.

**MATH 4996. Experiential Education Directed Study. 4 Hours.**

Draws upon the student’s approved experiential activity and integrates it with study in the academic major. Restricted to those junior and senior mathematics majors who are using it to fulfill their experiential education requirement; for these students it may count as a mathematics elective, subject to approval by instructor and adviser. *Prereq. Junior or senior standing; mathematics majors only.*

**MATH 5050. Advanced Engineering Calculus with Applications. 4 Hours.**

Introduces methods of vector analysis. Expects students to master over thirty predefined types of problems. Topics include analytic geometry in three dimensions, geometric vectors and vector algebra, curves in three-space, linear approximations, the gradient, the chain rule, the Lagrange multiplier, iterated integrals, integrals in curvilinear coordinates, change of variables, vector fields, line integrals, conservative fields, surfaces and surface integrals, the flux and the circulation of a vector field, Green’s theorem, the divergence theorem, and Stokes’ theorem. Illustrates the material by real-world science and engineering applications using the above techniques. *Prereq. Familiarity with single-variable calculus.*

**MATH 5101. Analysis 1: Functions of One Variable. 4 Hours.**

Offers a rigorous, proof-based introduction to mathematical analysis and its applications. Topics include metric spaces, convergence, compactness, and connectedness; continuous and uniformly continuous functions; derivatives, the mean value theorem, and Taylor series; Riemann integration and the fundamental theorem of calculus; interchanging limit operations; sequences of functions and uniform convergence; Arzelà-Ascoli and Stone-Weierstrass theorems; inverse and implicit function theorems; successive approximations and existence/uniqueness for ordinary differential equations; linear operators on finite-dimensional vector spaces and applications to systems of ordinary differential equations. Provides a series of computer projects that further develop the connections between theory and applications. *Prereq. Graduate standing or permission of instructor and head advisor.*

**MATH 5102. Analysis 2: Functions of Several Variables. 4 Hours.**

Continues MATH 5101. Studies basics of analysis in several variables. Topics include derivative and partial derivatives; the contraction principle; the inverse function and implicit function theorems; derivatives of higher order; Taylor formula in several variables; differentiation of integrals depending on parameters; integration of functions of several variables; change of variables in integrals; differential forms and their integration over simplexes and chains; external multiplication of forms; differential of forms; Stokes’ formula; set functions; Lebesgue measure; measure spaces; measurable functions; integration; comparison with the Riemann integral; L2 as a Hilbert space; and Parseval theorem and Riesz-Fischer theorem. *Prereq. Graduate standing or permission of instructor and head advisor.*

**MATH 5104. Basics and Probability and Statistics. 4 Hours.**

Introduces the ideas and the reasoning used in both finite and infinite probabilistic settings. Covers the concepts of sample space, event, and axioms. Studies discrete and continuous probability distributions for one or more random variables, conditional probability, Bayes’s law, independence, and expectation and variance. Explores the use of moments, and the binomial, Poisson, and normal distributions. Examines the law of large numbers, the central limit theorem, and the use of probability in statistical inference including estimation of parameters, confidence interveral, and hypotheses testing. Requires a substantial project that connects the material in this course to the secondary school classroom. *Prereq. MATH 5101; graduate standing or permission of instructor and head advisor.*

**MATH 5105. Basics of Statistics and Stochastic Processes. 3 Hours.**

Focuses on the classical procedures of statistics including the t-test, linear regression, and the chi-square test. Introduces stochastic processes, with an emphasis on Markov chains, random walks, and Brownian motion, with applications to modeling. Requires a substantial project that connects the material in this course to the secondary school classroom. *Prereq. Graduate standing or permission of instructor and head advisor.*

**MATH 5106. Basics of Complex Analysis. 3 Hours.**

Introduces the analysis of functions of a complex variable. Starting with the algebra and geometry of complex numbers, basic derivative and contour integral properties are developed for elementary algebraic and transcendental functions, as well as for other analytic functions with isolated singularities. Gives Power and Laurent series representations. Studies classical integral theorems, residue theory, and conformal mapping properties. Presents applications of harmonic functions as time permits. Requires a substantial project that involves an application of ideas covered in the course. *Prereq. Graduate standing or permission of instructor and head advisor.*

**MATH 5107. Basics of Number Theory. 3 Hours.**

Introduces number theory. Topics include linear diophantine equations, congruencies, design of magic squares, Fermat’s little theorem, Euler’s formulas, Euler’s phi function, computing powers and roots in modular arithmetic, the RSA encryption scheme, primitive roots and indices, and the law of quadratic reciprocity. As time permits, additional topics may include diophantine approximation and Pell’s equation, elliptic curves, points on elliptic curves modulo, and elliptic curves and Fermat’s last theorem. Requires a substantial project that connects the material in this course to the secondary school classroom. *Prereq. Graduate standing or permission of instructor and head advisor.*

**MATH 5108. Methods for Teaching Math. 3 Hours.**

Explores mathematics teaching methods that are research based, experience based, and grounded in the contemporary theoretical frameworks influencing mathematics education. Emphasis is on issues related to teaching math in an urban school, problem solving, communication, connections, and integrating technology as well as issues of access and equity, assessment, and cross-content teaching strategies. Graduate students are required to demonstrate advanced levels of study and research. *Prereq. Graduate standing or permission of instructor and head advisor.*

**MATH 5111. Algebra 1. 4 Hours.**

Covers vector spaces and linear maps. Topics include row and column operations and their application to normal form; eigenvalues and eigenvectors of an endomorphism; characteristic polynomial and Jordan canonical form; multilinear algebra that covers tensor products, symmetric and exterior powers of vector spaces, and their universality properties; quadratic forms, reduction to diagonal form, and Sylvester theorem; hyperbolic spaces and Witt theorem; the orthogonal group and isotropic subspaces; antisymmetric forms and their reduction to canonical form; the symplectic group; and Pfaffian and Affine geometry, and classification of conic sections. *Prereq. Graduate standing or permission of instructor and head advisor.*

**MATH 5112. Algebra 2. 4 Hours.**

Continues MATH 5111. Topics include groups, such as subgroups, normal subgroups, homomorphism of groups, abelian groups, solvable groups, free groups, finite p-groups, Sylov theorem, permutation groups, and the sign homomorphism; rings, such as homomorphism, ideals, quotient rings, integral domains, extensions of rings, unique factorization domain, Chinese remainder theorem, and Gauss’s lemma; and modules, such as homomorphism, submodules, quotient modules, exact sequence, and structure of finitely generated modules over principal ideal domains. Examples include abelian groups and Jordan canonical form. Also covers representations of finite groups, group rings and irreducible representations, Frobenius reciprocity, Maschke theorem and characters of finite groups, and dual groups. *Prereq. Graduate standing or permission of instructor and head advisor.*

**MATH 5121. Topology 1. 4 Hours.**

Provides an introduction to topology, starting with the basics of point set topology (topological space, continuous maps, homeomorphisms, compactness and connectedness, and identification spaces). Moves on to the basic notions of algebraic and combinatorial topology, such as homotopy equivalences, fundamental group, Seifert-VanKampen theorem, simplicial complexes, classification of surfaces, and covering space theory. Ends with a brief introduction to simplicial homology and knot theory. *Prereq. MATH 5111; graduate standing or permission of instructor and head advisor.*

**MATH 5122. Geometry 1. 4 Hours.**

Covers differentiable manifolds, such as tangent bundles, tensor bundles, vector fields, Frobenius integrability theorem, differential forms, Stokes’ theorem, and de Rham cohomology; and curves and surfaces, such as elementary theory of curves and surfaces in R3, fundamental theorem of surfaces in R3, surfaces with constant Gauss or mean curvature, and Gauss-Bonnet theorem for surfaces. *Prereq. MATH 5101, MATH 5111, and graduate standing or permission of instructor and head advisor.*

**MATH 5131. Introduction to Mathematical Methods and Modeling. 4 Hours.**

Presents mathematical methods emphasizing applications. Uses ordinary and partial differential equations to model the evolution of real-world processes. Topics chosen illustrate the power and versatility of mathematical methods in a variety of applied fields and include population dynamics, drug assimilation, epidemics, spread of pollutants in environmental systems, competing and cooperating species, and heat conduction. Requires students to complete a math-modeling project. *Prereq. Undergraduate-level course work in ordinary and partial differential equations.*

**MATH 5976. Directed Study. 1-4 Hours.**

Offers independent work under the direction of members of the department on chosen topics. *Prereq. Graduate standing or permission of instructor or head advisor.*

**MATH 5978. Independent Study. 1-4 Hours.**

Offers independent work under the direction of members of the department on a chosen topic. Course content depends on instructor. *Prereq. Graduate standing or permission of instructor or head advisor.*

**MATH 5984. Research. 1-4 Hours.**

Offers an opportunity to conduct research under faculty supervision. *Prereq. Graduate standing or permission of head advisor.*