Course Descriptions For Majors And Minors

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.

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.

Not open to students who have completed MT445 or MC248 or CS245. Cannot be used for Major credit.

This course is intended for computer science majors and introduces the student to the fundamental notions of discrete mathematics, with an emphasis on graph theory and applications.

Topics include the basic notions of set theory and logic, graphs, equivalence relations and partial orderings, basic counting techniques, finite probability, propositional logic, induction, graphs and trees, paths, circuits and cycles, recursion and recurrence relations, and Boolean algebra.

Prerequisite: MT202. Cannot be used for Major credit.

MT305 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.

Prerequisite: MT210 Linear Algebra and MT216 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.

Prerequisite: MT202 Multivariable Calculus and MT216 Introduction to Abstract Mathematics.

The purpose of this course is to give students the theoretical foundations for the topics taught in MT 102-103. 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.

Prerequisites: MT202 Multivariable Calculus and MT210 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, qualitative analysis of non-linear systems, and an introduction to stability and bifurcations.

Prerequisite: MT410 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.

Prerequisites: MT202 Multivariable Calculus, MT210 Linear Algebra, and MT320 Introduction to Analysis (MT320 may be taken concurrently).

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

Prerequisites: MT202 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.

Prerequisites: MT426 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.

Prerequisite: MT216 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.

Prerequisite: MT210 Linear Algebra.

The MT 435-436 sequence demonstrates how mathematical theory can be developed and applied to solve problems from management, economics, and the social sciences.

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. Other topics are drawn from game theory, dynamic programming, Markov decision processes (with finite and infinite horizons), network analysis, and nonlinear programming.

Prerequisites: MT202 Multivariable Calculus, MT210 Linear Algebra, and MT216 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.

Prerequisites: MT202 Multivariable Calculus and at least one of MT210 Linear Algebra or MT216 Introduction to Abstract Mathematics. Not open to students who have completed MT245 or MC248 or CS245.

This is a course in enumeration and graph theory. The object of the course is to develop proficiency in solving discrete mathematics problems.

Among the topics covered are the following: counting methods for arrangements and selections, the pigeonhole principle, the inclusion-exclusion principle, generating functions, recurrence relations, graph theory, trees and searching, and network algorithms. The problem-solving techniques developed apply to the analysis of computer systems, but most of the problems in the course are from recreational mathematics.

Prerequisites: MT210 Linear Algebra and MT310 Introduction to Abstract Algebra.

This proof-based course presents a more rigorous approach to Linear Algebra and covers many topics beyond those in MT 210. 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.

Prerequisite: MT216 Introduction to Abstract Mathematics.

This course surveys the history and foundations of geometry from ancient to modern times.

Topics will be selected from among the following: Mesopotamian and Egyptian mathematics, Greek geometry, the axiomatic method, history of the parallel postulate, the Lobachevskian plane, Hilbert's axioms for Euclidean geometry, elliptic and projective geometry, the trigonometric formulas, models, geometry and the study of physical space.

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.

Prerequisites: MT202 Multivariable Calculus, MT210 Linear Algebra, and MT216 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.

Prerequisite: MT202 Multivariable Calculus, and at least one of MT210 Linear Algebra or MT216 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.

Prerequisites: MT202 Multivariable Calculus and MT210 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.

Prerequisites: MT310 and MT320, 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.

In Spring, 2008, the course will be offered by Professor Bond, and the topic will be Introduction to Cryptography. Prerequisites: MT310 Introduction to Abstract Algebra (or MT816 Introduction to Modern Algebra).

This course is an introduction to Cryptography with a particular emphasis on public key cryptography. Among the topics covered are: affine ciphers, the RSA algorithm, discrete logarithms, digital signatures, primality testing, and methods of factoring large integers. The necessary topics from number theory, especially the theory of congruences, will be discussed as the need arises.

Prerequisites: MT216 or CS245.
This course is cross-listed as CS391 Games and Numbers.

This course is about the mathematical theory of two-person strategy games without chance elements. We examine a large number of such games, find out what mathematics can tell us about finding winning strategies, and a little bit of what Computer Science can tell us about how easy or hard it is for a computer to play such games well. We'll also see how the same ideas can be used to construct numbers, then go back to the world of play and look at the (very different) mathematical ideas behind the solution of puzzles like jump-peg solitaire and Rubik's cube.

Department permission is required.

This is an independent study course, taken by arrangement with and under the supervision of a Mathematics Department faculty member. Interested students should consult with the Undergraduate Vice Chair.