Courses

1301. Euclidean and Non-Euclidean Geometries. Development of the mathematical way of thinking through firsthand experience. Emphasis on the student's strengthening of his or her imagination, deductive powers, and ability to use language precisely and efficiently. Study of Euclid's geometry; Hilbert's axioms; neutral geometry; hyperbolic geometry (non-Euclidean geometry of Gauss, Bolyai, Lobachevsky); the axiomatic method; and consistency, independence and completeness of axiom systems. Historical perspective and philosophical implications are included. Students must prove a significant number of theorems on their own. Fall and Spring.

1302. Elements of Number Theory. Development of the mathematical way of thinking. Emphasis on the student's strengthening of his or her imagination, deductive powers, and ability to use language precisely and efficiently. Study of the properties of the whole numbers; the Euclidean algorithm; prime numbers; divisibility; congruencies; residues; and elementary additive number theory. Students must prove a significant number of theorems on their own. Historical perspective and philosophical implications are included.

1303. Precalculus. Advanced Algebra and Trigonometry needed for Calculus. Solving equations and inequalities; polynomials; functions; trigonometry on the unit circle; parametric and polar coordinates; conic sections; arithmetic and geometric sequences; math induction. Prerequisite: successful placement in algebra. Fall.

1306. The Calculus. A careful study of the slope (derivative), area (integral) and the Fundamental Theorem of Calculus, both theory and applications. For over 2500 years, these three ideas formed the foundation of the Calculus, one of the greatest discoveries of mankind. The course focuses on the confluence of two seemingly disparate notions, slope and area, in one of the surprising theorems of our civilization, The Fundamental Theorem of Calculus. The study is done at a deep level and in a visual way but without most of the notation, which sometimes inhibits a humane understanding of this beautiful subject. Students learn how to use their imaginations and to use language precisely and efficiently. Prerequisite: none. Fall and Spring.

1404. Calculus I. Limits, derivatives, applications of derivatives, integration, logarithm and exponential functions. Prerequisite: Grade of C (2.0) or better in Math 1303, or satisfactory placement. Fall and Spring.

1411. Calculus II. L'Hôpital's Rule, inverse trigonometric and hyperbolic functions, methods of integration, analytic geometry, applications of integrals, sequences and series. Prerequisite: Grade of C (2.0) or better in Math 1404, or satisfactory placement. Spring.

1513. Infinite Processes: Theory and Application. The study of the completeness property, sequences, limits, tangency, derivatives, area, and integration. Applications of derivatives, integrals, and linear and separable differential equations. Mathematical modeling including acquisition of data in real time. Computer algebra systems will be used. Prerequisites: Satisfactory placement and consent of the Chairman.

2107. Mathematics Colloquium. A forum for exposing students to the rich and deep areas of mathematics and its applications not normally seen in the first two years of undergraduate studies. Oral presentations are selected for their interest and accessibility. Speakers include faculty members, visiting lecturers, and students. Highly recommended for majors. Visitors are welcome. Public announcements of speakers will be made. Graded Pass/No Pass. May be repeated. Fall and Spring.

2304. Discrete Mathematics. An introduction to the mathematical foundation of computer science with two co-equal components: a study of combinatorics and graph theory including topics from the theory of computer science, and a development of the imagination and analytical skills required in mathematics and computing science. Students are required to do proofs. Prerequisite: MCS 2410. Spring, odd years.

2305. Introduction to Statistics. Statistics may be broadly defined as the science of making rational decisions in the face of quantifiable uncertainty. This course emphasizes a deep understanding of the fundamental elements of so-called "statistical thinking", including randomness, uncertainty, modeling, and decision processes. The superstructure of statistical methodology, including hypothesis testing, inference, and estimation, using the logical methods of mathematics. A significant amount of instruction is computer-based. Prerequisite: Successful demonstration of algebra abilities. Spring.

2412. Calculus III. Vectors, vector calculus, functions of several variables, multiple integrals. Prerequisite: Grade of C (2.0) or better, Mathematics 1411, or satisfactory placement. Fall.

3107. Mathematics Colloquium. This course is similar to 2107 except that extra work is required to earn junior-level credit. Each student is expected to write a paper and present a talk based on it in addition to fulfilling the other requirements. Graded Pass/No Pass. May be repeated. Fall and Spring.

3310. Linear Algebra. Geometry of R² and R³ including the dot product and parametric equations of lines and planes. Systems of linear equations, matrices, determinants, vector spaces, and linear transformations. Applications to the sciences and economics are included. Prerequisite: Mathematics 1411 or consent of instructor. Fall.

3320. Foundations of Geometry. A systematic development of topics selected from metric and nonmetric geometries, comparison of postulate systems. Prerequisite: Mathematics 1411 or consent of instructor. Spring, even years.

3321. Linear Point Set Theory. Limit points, convergent sequences, compact sets, connected sets, dense sets, nowhere dense sets, separable sets. Prerequisite: Consent of Chairman. Fall.

3322. History and Philosophy of Mathematics. The history of the development of mathematics, the lives and ideas of noted mathematicians. Prerequisite: Consent of instructor.

3324. Differential Equations. First order equations, existence and uniqueness of solutions, differential equations of higher order, Laplace transforms, systems of differential equations. Prerequisite: Mathematics 1411 or consent of instructor. Fall, even years.

3326. Probability. Axioms and basic properties, random variables, univariate probability functions and density functions, moments, standard distributions, Law of Large Numbers, and Central Limits Theorem. Prerequisite: Math 1411. Fall, odd years.

3327. Statistics. Sampling, tests of hypotheses, estimation, linear models, and regression. Prerequisite: Math 3326. Spring, even years.

3338. Numerical Analysis. Zeros of polynomials, difference equations, systems of equations, numerical differentiation and integration, numerical solution of differential equations, eigenvalues and eigenvectors. Prerequisite: Mathematics 3310 and knowledge of a programming language. Spring, even years.

3351. Model Building. Investigation of a series of physical situations for which mathematical models are developed. Emphasis is on the process. Prerequisite: mathematical maturity beyond 1411, or consent of instructor.

3V50. Special Topics. This course is intended to give the student an opportunity to pursue special studies not otherwise offered. Topics in recent years have been chaos, fractals, cellular automata, number theory, and dynamical systems. May be repeated for credit. Prerequisite: consent of Chairman.

4314. Advanced Multivariable Analysis. Continuous and differential functions from R^m into R^m, integration, differential forms, Stokes's theorem. Prerequisite: Mathematics 3310, 2412 or consent of instructor.

4315. Applied Math I. Symmetric linear systems, equilibrium equations of the discrete and continuous cases, Fourier series, complex analysis and initial value problems. Prerequisites: Math 3310, Math 2412. Spring, even years.

4316. Applied Math II. Power series, special functions, partial differential equations of mathematical physics, complex integration, and Fourier transformations. Prerequisite: Math 4315.

4332-4333. Abstract Algebra I, II. Group theory, ring theory including ideals, integral domains and polynomial rings, field theory including Galois theory, field extensions and splitting fields, module theory. Prerequisite: Mathematics 3310 and junior standing, or consent of chairman. Fall, even years (I); Spring, odd years (II).

4334. Topology. Topological spaces, connectedness, compactness, continuity, separation, metric spaces, complete metric spaces, product spaces. Prerequisite: Mathematics 3321 or consent of instructor. Spring, odd years.

4338. Mathematical Logic. Propositional calculus, predicate calculus, first order theories, formal number theory. Prerequisite: consent of instructor.

4339. Axiomatic Set Theory. Axioms, ordinal numbers, finite and denumerable sets, rational and real numbers, the axiom of choice. Prerequisite: consent of instructor.

4341-4342. Analysis I, II. Real number system, topological concepts, continuity, differentiation, the Stieltjes integral, convergence, uniform convergence, sequences and series of functions, bounded variation. Prerequisite: Mathematics 3321 or consent of Chairman. Fall, odd years (I); Spring, even years (II).

4360. Senior Seminar. A study of significant literature with a view toward acquainting the student with the nature of fundamental mathematical research. Many of the important elements of research will be incorporated into this course. Prerequisite: senior standing.

4V43-4V44. Research. Under the supervision of a member of the faculty, the student involves himself or herself in the investigation and/or creation of some areas of mathematics. The research should be original to the student. A paper is required. Prerequisite: junior or senior standing.

4V61. Independent Studies. An opportunity for the student to examine in depth any topic within the field under the guidance of the instructor. For advanced students.