2011-2012 Mathematics Courses

An introduction to the concepts, techniques, and reasoning that are central to the understanding of data, this lecture course focuses on the fundamental ideas of statistical analysis used to gain insight into diverse areas of human interest.  The use, abuse, and misuse of statistics will be the central focus of the course. Topics of exploration will include the core statistical topics in the areas of experimental study design, sampling theory, data analysis, and statistical inference. Applications will be drawn from current events, business, psychology, politics, medicine, and other areas of the natural and social sciences. Statistical software will be introduced and used extensively in this course, but no prior experience with the software is assumed. This seminar is an invaluable course for anybody planning to pursue graduate work and/or research in the natural or social sciences. No college-level mathematical knowledge is required.

Our world is dominated by motion and change. The Earth spins on its axis, as it rotates around the Sun. Stock prices rise and fall. An apple, acting in accordance with the laws of physics, falls onto the head of a modern day Newton. Calculus is the intriguing branch of mathematics whose primary goal is the understanding of the laws governing motion and change. The sum of the calculus—its methods, tools, and ideas—is often cited as one of the greatest intellectual achievements of humanity. Though just a few hundred years old, the calculus has become an indispensable research tool in both the natural and the social sciences. Our study begins with the central concept of the calculus, the limit, and proceeds to explore the dual notions of differentiation and integration. Numerous applications of the theory will be examined. The minimum required preparation for successful study of the calculus is one year each of high-school algebra and geometry. The precalculus topics of trigonometry and analytic geometry will be developed as the need arises. For conference work, students may choose to undertake a deeper investigation of a single topic or application of the calculus or conduct a study in some other branch of mathematics. This seminar is intended for students interested in advanced study in mathematics or science, for students preparing for careers in the health sciences, and for any student wishing to broaden and enrich the life of the mind.

The purpose of this course is to explore various systems of geometry, as well as different approaches to these systems. A brief review of high-school geometry (including an exposition of logical objections to it) will be the starting point for branching out into other areas. Problem solving will play a central role in the development and exposition of much of the material in the course. Topics may be chosen from analytic, neutral, non-Euclidean (Lobechevskian and Riemannian), and incidence geometries.

This course will build upon and continue to develop the study of the differential calculus as it was developed in Calculus I. It will include the definitions of antiderivatives and integrals (including the fundamental theorems of both integal and differential calculus). We will develop and study exponential, logarithmic, and inverse trigonometric functions. Much effort will be devoted to studying various techniques and applications (geometric and physical) of integration. As time permits, some elementary differential equations and basic infinite series may be included. Prerequisite: Calculus I

This highly abstract course will be directed toward the axiomatic development of basic algebraic systems. Both mathematical and nonmathematical models will be used to illustrate these systems. Topics will be chosen from the theories of groups, rings, fields, and matrices. Although there are no prerequisites and no prior experience with the material is necessary, some mathematical sophistication is essential. Individual weekly conferences will be used to reinforce the class work when necessary and for independent study projects otherwise.

There is a world of mathematics beyond what students learn in high-school algebra, geometry, and calculus courses. This seminar serves as an introduction to this realm of elegant mathematical ideas. With an explicit goal of improving students’ mathematical reasoning and problem-solving skills, this seminar provides the ultimate intellectual workout. Five important themes are interwoven in the course: logic, the nature of proof, combinatorial analysis, discrete structures, and mathematical philosophy. For conference work, students may choose to undertake a deeper investigation of a single topic or application of discrete mathematics or to conduct a study in some other branch of mathematics. This seminar is a must for students interested in advanced mathematical study and highly recommended for students with an interest in computer science, law, or philosophy. Some prior study of calculus is required.

The world and our lives are fundamentally multivariate. Tomorrow’s weather forecast is based on today’s solar wind velocity, heat transfer rates, pressure, and humidity levels, among other factors. The price to the consumer of a commercial flight is dependent partly on market demand, travel distance, cost of fuel, and governmental taxes. Multivariable calculus addresses the mathematics of functions such as these that depend on several variables. Specific topics to be addressed include vectors, partial derivatives, gradients, multiple integration, line and surface integrals, and their diverse applications. For conference work, students may choose to undertake a deeper investigation of a single topic or application of the calculus or to conduct a study in some other branch of mathematics. Prerequisite: Two semesters of college-level calculus.

This course is devoted to the study of the integers. Although the approach will be mainly axiomatic, consideration will be given to historical aspects of the subject. Special attention will be given to problem solving, both as a central device for exposing the development of the theory of the course and for its own sake. Topics will include divisibility properties of integers, prime numbers, modular arithmetic, Diophantine equations, and special-number theoretic functions. No prior experience with this material is necessary, although mathematical sophistication would be important.