Morrissey College of Arts and Sciences

# Course Descriptions for Majors and Minors

## department of mathematics

This page contains both required and elective courses for Majors and Minors.

Majors must complete:

• MATH 2210
• MATH 2216
• MATH 3310
• MATH 3320

Minors must complete:

• MATH 2210

MATH 2210 Linear Algebra (Fall/Spring: 3)
Corequisite: MATH 2202/3 Multivariable Calculus

This course is an introduction to the techniques of linear algebra in Euclidean space. Topics covered include matrices, determinants, systems of linear equations, vectors in n-dimensional space, complex numbers, and eigenvalues. The course is required of mathematics majors and minors, but is also suitable for students in the social sciences, natural sciences, and management.

MATH 2211 Linear Algebra (Honors) (Spring: 3)
Prerequisite: MATH 2203 Multivariable Calculus (Honors) or permission of the Assistant Chair for Undergraduates

This course is intended for students with strong preparation and high motivation. Topics covered include matrices, linear equations, determinants, eigenvectors and eigenvalues, vector spaces and linear transformations, inner products, and canonical forms. The course will include significant work with proofs.

MATH 2216 Introduction to Abstract Mathematics (Fall/Spring: 3)
This course is designed to develop the student's ability to do abstract mathematics through the presentation and development of the basic notions of logic and proof. Topics include elementary set theory, mappings, integers, rings, complex numbers, and polynomials.

MATH 3305 Advanced Calculus for Science Majors (Spring: 4)
Prerequisite: MATH 2202. Cannot be used for Major credit.

MATH 3305 is required for Geology-Geophysics, Geophysics, and Physics majors. It is also recommended for Chemistry majors.

Topics include linear second order differential equations, series solutions of differential equations including Bessel functions and Legendre polynomials, and solutions of the diffusion and wave equations in several dimensions.

MATH 3310 Introduction to Abstract Algebra (Fall/Spring: 3)
Prerequisites: MATH 2210 Linear Algebra and MATH 2216 Introduction to Abstract Mathematics.

This course studies four fundamental algebraic structures: groups, including subgroups, cyclic groups, permutation groups, symmetry groups and Lagrange's Theorem; rings, including subrings, integral domains, and unique factorization domains; polynomials, including a discussion of unique factorization and methods for finding roots; fields, introducing the basic ideas of field extensions and ruler and compass constructions.

MATH 3311 Algebra I (Fall: 3)
MATH 3312 Algebra II (Spring: 3)

Prerequisites: MATH 2210 Linear Algebra and MATH 2216 Introduction to Abstract Mathematics.

This year-long sequence studies the basic structures of abstract algebra. Topics include groups, subgroups, normal subgroups, factor groups, Lagrange's Theorem, the Sylow Theorems, rings, ideal theory, integral domains, field extensions, and Galois theory.

Note: Students may not take both MATH 3310 and MATH 3311. With the permission of the Assistant Chair for Undergraduates, students who have taken MATH 3310 may be allowed to take MATH 3312. However, they may need to do additional work on their own in order to make that transition. Students considering a B.S. in Mathematics are strongly encouraged to take MATH 3311.

MATH 3320 Introduction to Analysis (Fall/Spring: 3)
Prerequisites: MATH 2202 Multivariable Calculus and MATH 2216 Introduction to Abstract Mathematics.

The purpose of this course is to give students the theoretical foundations for the topics taught in MATH 1102-1103. It will cover algebraic and order properties of the real numbers, the least upper bound axiom, limits, continuity, differentiation, the Riemann integral, sequences, and series. Definitions and proofs will be stressed throughout the course.

MATH 3321 Analysis I (Fall: 3)
MATH 3322 Analysis II (Spring: 3)

Prerequisites: MATH 2210 Linear Algebra and MATH 2216 Introduction to Abstract Mathematics.

This year-long sequence studies the basic structure of the real numbers. Topics include the least upper bound principle, compactness of closed intervals (the Heine-Borel theorem), sequences, convergence, the Bolzano-Weierstrass theorem, continuous functions, boundedness and intermediate value theorems, uniform continuity, differentiable functions, the mean value theorem, construction of the Riemann integral, the fundamental theorem of calculus, sequences and series of functions, uniform convergence, the Weierstrass approximation theorem, special functions (exponential and trig), and Fourier series. As time permits, other topics may include metric spaces, calculus of functions of several variables, and an introduction to measure and integration.

Note: Students may not take both MATH 3320 and MATH 3321. With the permission of the Assistant Chair for Undergraduates, students who have taken MATH 3320 may be allowed to take MATH 3322. However, they may need to do additional work on their own in order to make that transition. Students considering a B.S. in Mathematics are strongly encouraged to take MATH 3321.

MATH 4410 Differential Equations (Fall; sometimes Spring: 3)
Prerequisites: MATH 2202 Multivariable Calculus and MT 210 Linear Algebra.

This course is a junior-senior elective intended primarily for the student who is interested in seeing applications of mathematics.

Among the topics covered will be the following: first order linear equations, higher order linear equations with constant coefficients, linear systems, Laplace transforms, and other topics as time permits.

MATH 4412 Partial Differential Equations (Offered Occasionally: 3)
Prerequisite: MATH 4410 Differential Equations.

This course investigates the classical partial differential equations of applied mathematics (diffusion, Laplace/Poisson, and wave) and their methods of solution (separation of variables, Fourier series, transforms, Green's functions, and eigenvalue applications). Additional topics will be included as time permits.

MATH 4414 Numerical Analysis (Spring: 3)
Prerequisites: MATH 2202 Multivariable Calculus, and MATH 2210 Linear Algebra.

Topics include the solution of linear and nonlinear algebraic equations, interpolation, numerical differentiation and integration, numerical solution of ordinary differential equations, approximation theory.

MATH 4426 Probability (Fall/Spring: 3)
Prerequisites: MATH 2202 Multivariable Calculus and familiarity with using a computer.

This course provides a general introduction to modern probability theory.

Topics include probability spaces, discrete and continuous random variables, joint and conditional distributions, mathematical expectation, the central limit theorem, and the weak law of large numbers. Applications to real data will be stressed, and we will use the computer to explore many concepts.

MATH 4427 Mathematical Statistics (sometimes Fall; Spring: 3)
Prerequisites: MT 426 Probability and familiarity with using a computer.

Topics studied include the following: sampling distributions, parametric point and interval estimation, hypothesis testing, goodness-of-fit, parametric and nonparametric two-sample analysis. Applications to real data will be stressed, and the computer will be used to explore concepts and analyze data.

MATH 4430 Introduction to Number Theory (Spring: 3)
Prerequisite: MATH 2216 Introduction to Abstract Mathematics.

Topics covered include divisibility, unique factorization, congruences, number-theoretic functions, primitive roots, diophantine equations, continued fractions, quadratic residues, and the distribution of primes. An attempt will be made to provide historical background for various problems and to provide examples useful in the secondary school curriculum.

MATH 4435 Mathematical Programming (Fall: 3)
Prerequisite: MATH 2210 Linear Algebra.

Topics studied from linear programming include a general discussion of linear optimization models, the theory and development of the simplex algorithm, degeneracy, duality, sensitivity analysis, and the dual simplex algorithm. Integer programming problems, and the transportation and assignment problems are considered,and algorithms are developed for their resolution.

MATH 4440 Dynamical Systems (Offered Occasionally: 3)
Prerequisites: MATH 2202 Multivariable Calculus, MATH 2210 Linear Algebra, and MATH 2216 Introduction to Abstract Mathematics.

This course is an introduction to nonlinear dynamics and their applications, emphasizing qualitative methods for differential equations.

Topics include fixed and periodic points, stability, linearization, parameterized families and bifurcations, and existence and nonexistence theorems for closed orbits in the plane. The final part of the course is an introduction to chaotic systems and fractals, including the Lorenz system and the quadratic map.

MATH 4445 Combinatorics (Fall: 3)
Prerequisites: MT 216 Introduction to Abstract Mathematics and MT 210 Linear Algebra. MT 210 may be taken simultaneously.

This course is an introduction to graph theory and combinatorics, with a strong emphasis on creative problem-solving techniques and connections with other branches of mathematics. Topics will center around the following: enumeration, Hamiltonian and Eulerian cycles, extremal graph theory, planarity, matching, colorability, Ramsey theory, hypergraphs, combinatorial geometry, and applications of linear algebra, probability, polynomials, and topology to combinatorics.

MATH 4450 Advanced Linear Algebra (Offered Occasionally: 3)
Prerequisites: MATH 2210 Linear Algebra and MATH 3310 Introduction to Abstract Algebra.

This proof-based course presents a more rigorous approach to Linear Algebra and covers many topics beyond those in MATH 2210. Topics will include Abstract Vector Spaces and Linear Maps over any field, Modules, Canonical Forms and the Geometry of Bilinear Forms. Additional topics, if time permits, could include the basic theorems of Galois Theory, Matrix Factorization, and applications such as Coding Theory, Factor Analysis and Linear Difference Equations.

MATH 4451 Euclidean and Non-Euclidean Geometry (Fall: 3)
Prerequisite: MATH 2216 Introduction to Abstract Mathematics.

This course is an introduction to geometric structure, broadly construed.

Topics may include: Euclidean geometry, hyperbolic and spherical geometry, platonic solids, tilings and wallpaper groups, graph theory, finite geometries, projective geometry, equidecomposition, the isoperimetric problem, surfaces and 3-dimensional manifolds.

MATH 4453 Euclid's Elements (Spring: 3)
Prerequisites: None

This course is a close reading of Euclid's Elements in seminar style, with careful attention to axiomatic reasoning and mathematical constructions that build on one another in a sequence of logical arguments.

We will also emphasize clear and creative communication on mathematical ideas, with some attention to the cultural background of the Elements and its place in a modern education.

MATH 4455 Mathematical Problem Solving (Spring: 3)
Prerequisites: MATH 2202 Multivariable Calculus, MATH 2210 Linear Algebra, and MATH 2216 Introduction to Abstract Mathematics (or equivalent mathematical background). Permission of the instructor required for students outside the LSOE.

This course is designed to deepen students' mathematical knowledge through solving, explaining, and extending challenging and interesting problems. Students will work both individually and in groups on problems chosen from polynomials, trigonometry, analytic geometry, pre-calculus, one-variable calculus, probability, and numerical algorithms. The course will emphasize explanations and generalizations rather than formal proofs and abstract properties. Some pedagogical issues, such as composing good problems and expected points of confusion in explaining various topics will come up, but the primary goal is mathematical insight. The course will be of particular use to future secondary math teachers.

MATH 4460 Complex Variables (Fall/Spring: 3)
Prerequisite: MATH 2202 Multivariable Calculus, and at least one of MATH 2210 Linear Algebra or MATH 2216 Introduction to Abstract Mathematics. Not open to MA students.

This course gives an introduction to the theory of functions of a complex variable, a fundamental and central area of mathematics. It is intended for mathematics majors and minors, and science majors.

Topics covered include: complex numbers and their properties, analytic functions and the Cauchy-Riemann equations, the logarithm and other elementary functions of a complex variable, integration of complex functions, the Cauchy integral theorem and its consequences, power series representation of analytic functions, the residue theorem and applications to definite integrals.

MATH 4461 Stochastic Processes (Spring ; 3)
Prerequisites : MATH 2216 and MATH 4426

A stochastic process describes the evolution of a system that changes over time in a random manner. This course introduces and studies various properties of some fundamental stochastic processes, including Markov chains in discrete and continuous time, renewal processes, and Brownian motion.

MATH 4470 Mathematical Modeling (Fall: 3)
Prerequisites: MATH 2202 Multivariable Calculus and MATH 2210 Linear Algebra.

This is a course primarily for mathematics majors with the purpose of introducing the student to the creation, use and analysis of a variety of mathematical models and to reinforce and deepen the mathematical and logical skills required of modelers.

A secondary purpose is to develop a sense of the existing and potential roles of both small and large scale models in our scientific civilization. It proceeds through the study of the model-building process, examination of exemplary models, and individual and group efforts to build or refine models through a succession of problem sets, laboratory exercises, and field work.

MATH 4475 The History of Mathematics (Alternate Fall semesters: 3)
Prerequisites: MATH 3310 and MATH 3320, one of which may be taken concurrently. Students must be familiar with abstract algebra (groups, rings, fields...) and rigorous analysis (differentiation and integration of real valued functions, sequences and series of functions...).

This course studies the development of mathematical thought, from ancient times to the twentieth century. Naturally, the subject is much too large for a single semester, so we will concentrate on the major themes and on the contributions of the greatest mathematicians. The emphasis in the course will be on the mathematics. Students will follow the historical arguments and work with the tools and techniques of the period being studied.

MATH 4480 Topics in Mathematics (Offered Occasionally: 3)
Topics for this one-semester course vary from year to year according to the interests of faculty and students. With department permission it may be repeated.

MATH 4499 Readings and Research (Fall/Spring: 3)
Department permission is required.

This is an independent study course, taken by arrangement with and under the supervision of a Mathematics Department faculty member.