2007–2008 Mathematics Courses
Number Theory
Level: Open
Semester: Spring
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, diaphantine equations, and special number theoretic functions. There are no course prerequisites, but permission of the instructor is required.An Introduction to Statistical Methods and Analysis
Level: Open,Intermediate
Semester: Fall
An introduction to the concepts, techniques, and reasoning 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 experimental study design, sampling design, data analysis (visual devices, measures of center, measures of spread, normal distributions, correlation, and regression); sampling distributions and the Central Limit Theorem; estimation (confidence intervals); and hypothesis testing (z-test, t-test, chi-square, ANOVA). Applications will be drawn from current events, business, psychology, politics, medicine, and other areas of the life and social sciences. Statistical software will be introduced and used extensively in this course, but no prior experience is assumed. An invaluable course for students planning graduate study in the natural or social sciences.
Calculus and Differential Equations
Level: Open
Semester: Year
This course is designed primarily for students planning work in science or further work in mathematics, but it can serve as a capstone to solid work in mathematics for those who do not plan more than one year of study in the field. Topics from trigonometry and analytic geometry will be developed as the need for them arises. Calculus is a systematic development of mathematical relations among errors that arise in the use of approximations in computations of a theoretical nature, and, though its primary application is to the physical sciences, its recent applicability is far wider. To begin, the notions of function and limit will be explored, and these will lead to definitions of the derivative and the integral that generalize the notions of velocity and area, respectively. After investigating elementary applications of these notions, the course will give special attention to trigonometric, exponential, and logarithmic functions and the differential equations that define them. As time permits, the second semester will deal with further applications, with techniques of integration, infinite series, and improper integrals. Emphasis in conference work will be on both technical and theoretical content to develop one or the other more fully and to explore other areas closely related to calculus. For most students, a significant portion of conference time will be used to develop assurance in handling problems assigned weekly.
Abstract Algebra
Level: Open
Semester: Fall
The course will be directed toward the axiomatic development of basic abstract algebraic systems. Both mathematical and nonmathematical models will be used to illustrate these systems, and applications will be studied when appropriate. Topics will be chosen from the theories of groups, rings, fields, and matrices. There are no prerequisites, and no experience with the material is necessary although some mathematical sophistication is essential. Individual weekly conferences will be used to reinforce the course work, when necessary, and for independent study projects otherwise.
Open with permission from the instructor.
Mathematical Modeling for the Natural Sciences
Level: Open
Semester: Fall
Mathematics not only has the extraordinary power to reduce complicated problems to simple rules and procedures; it also serves as a tool to understand the world around us. In this seminar, students will learn to create mathematical models that will help them understand and analyze real-life phenomena in new and fruitful ways and to apply mathematics to a wide range of practical problems. The central theme of the course is functions as models of change. Linear, exponential, logarithmic, and trigonometric functions will be utilized to model physical phenomena. Conference time will emphasize the development and mastery of the technical and theoretical course content as well as the exploration of additional mathematical topics. Although the course is designed primarily for those students planning further study in the mathematical, physical, or natural sciences, this course may benefit anyone seeking to develop mathematical reasoning and problem-solving skills.
Game Theory: The Mathematics of Strategy and Conflict
Level: Open
Semester: Spring
Warfare, elections, auctions, labor-management negotiations, inheritance disputes, even divorce—these and many other conflicts can be successfully understood and studied as games. A game, in the minds of social scientists and mathematicians, is any situation involving two or more participants (players) each capable of rationally choosing among a set of possible actions (strategies) which, in turn, leads to some final result (outcome). Game theory is the interdisciplinary study of conflict whose primary goal is the answer to a single, simply stated, but surprisingly complex question: What is the best thing to do? Although the principles of game theory have been widely applied throughout the social and biological sciences, its greatest impact has been felt in the fields of economics and political science. This seminar represents a survey of the basic techniques and principles in the theory of games and strategy. Of primary interest will be the applications of the theory to real-world conflicts of historical or current interest.
No college level mathematical knowledge is required.
Discrete Mathematics: A Gateway to Advanced Mathematics
Level: Intermediate
Semester: Fall
There is a world of mathematics beyond what students learn in high school algebra, geometry, and calculus courses. This seminar will serve as an introduction to the world of elegant mathematical ideas and to the unspoken logic and reasoning that underlie mathematical thought. With an emphasis on mathematical reasoning and problem-solving skills, this seminar will provide the ultimate intellectual workout. Five central themes are interwoven in the course: logic, the nature of proof, combinatorial analysis, discrete structures, and mathematical philosophy. Conference time will be allocated to clarifying course ideas and to the study of additional mathematical topics. This seminar is highly recommended for students interested in subsequent advanced mathematical study and/or for students with an interest in computer science, natural science, or philosophy.
Prior study of calculus is required.