2012-2013 Mathematics Courses
An Introduction to Statistical Methods and Analysis
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 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 sciences or social sciences. No college-level mathematical knowledge is required.
Game Theory: The Study of Strategy and Conflict
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 parlance of social scientists and mathematicians, is any situation involving two or more participants (players) capable of rationally choosing among a set of possible actions (strategies) that lead to some final result (outcome) of typically unequal value (payoffs) to the players. Game theory is the interdisciplinary study of conflict whose primary goal is the answer to the single, simply-stated, but surprisingly complex question: What is the best way to “play”? Although the principles of game theory have been widely applied throughout the social and natural sciences, its greatest impact has been felt in the fields of economics and political science. This course represents a survey of the basic techniques and principles in the field. Of primary interest will be the applications of the theory to real-world conflicts of historical or current interest. The minimum required preparation for successful study of game theory is one year each of high-school algebra and geometry. No other knowledge of mathematics or social science is presumed.
Calculus is the study of rates of change of functions (the derivative), accumulated areas under curves (the integral), and how these two ideas are (surprisingly!) related. The concepts and techniques involved apply to medicine, economics, engineering, physics, chemistry, biology, ecology, geology, and many other fields. Such applications appear throughout the course, but we will focus on understanding concepts deeply and will approach functions from graphical, numeric, symbolic, and descriptive points of view. Conference work will explore additional mathematical topics. This seminar is intended for students planning further study in mathematics or in science, medicine, engineering, economics, or any technical field, as well as students who seek to enhance their logical thinking and problem-solving skills. Facility with high-school algebra and basic geometry are prerequisites for this course. Prior exposure to trigonometry and/or pre-calculus is highly recommended.
An infectious disease spreads through a community...what is the most effective action to stop an epidemic? Populations of fish swell and decline periodically...should we change the level of fishing allowed this year to have a better fish population next year? Foxes snack on rabbits...in the long term, will we end up with too many foxes or too many rabbits? Calculus can help us answer these questions. We can make a mathematical model of each situation, composed of equations involving derivatives (called differential equations). These models can tell us what happens to a system over time, which in turn gives us predictive power. Additionally, we can alter models to reflect different scenarios (e.g., instituting a quarantine, changing hunting quotas) and then see how these scenarios play out. The topics of study in Calculus II include power series, integration, and numerical approximation, all of which can be applied to solve differential equations. Our work will be done both by hand and by computer. Conveniently, learning the basics of constructing and solving differential equations (our first topic of the semester) includes a review of Calculus I concepts. Conference work will explore additional mathematical topics. This seminar is intended for students planning further study in mathematics or science, medicine, engineering, economics, or any technical field, as well as students who seek to enhance their logical thinking and problem-solving skills. Prerequisite: Exposure to differential calculus in either a high-school or college setting.
Linear Algebra: The Mathematics of Matrices and Vector Spaces
An introduction to the algebra and geometry of vector spaces and matrices, this course stresses important mathematical concepts and tools used in advanced mathematics, computer science, physics, chemistry, and economics. Systematic methods of solving systems of linear equations is the underlying theme, and applications of the theory will be emphasized. Topics of exploration include Gaussian elimination, determinants, linear transformations, linear independence, bases, eigenvectors, and eigenvalues. Conference time will be allocated to clarifying course ideas and exploring additional applications of the theory. This seminar is intended for students interested in advanced mathematics, computer science, the physical sciences, or economics. Prerequisite: prior study of Calculus or Discrete Mathematics.
Multivariable Calculus and Differential Equations
The laws of the universe are written in the language of mathematics. Most of the quantities that we regularly study in physics, biology, economics, and a variety of other fields are not static. Indeed, many interesting phenomena (both natural and manmade) can be described and studied as functions or equations relating several changing quantities. Multivariable calculus is the branch of mathematics that explores the properties of functions of several variables; differential equations is the study of how these variables change over time and/or space. This seminar will explore the theory and applications of both of these important areas of mathematics. Aimed at students with a primary interest in the natural sciences, economics, or mathematics, this seminar is meant as a follow up to the traditional first-year study of calculus. Conference time will be allocated to clarifying course ideas and to the study of additional mathematical topics. Prerequisite: A yearlong study of Calculus
Abstract Algebra: Theory and Applications
In pre-college mathematics courses, we learned the basic methodology and notions of algebra. We appointed letters of the alphabet to abstractly represent unknown or unspecified quantities. We discovered how to translate real-world (and often complicated) problems into simple equations whose solutions, if they could be found, held the key to greater understanding. But algebra does not end there. Advanced algebra examines sets of objects (numbers, matrices, polynomials, functions, ideas) and operations on these sets. The approach is typically axiomatic: One assumes a small number of basic properties, or axioms, and attempts to deduce all other properties of the mathematical system from these. Such abstraction allows us to study, simultaneously, all structures satisfying a given set of axioms and to recognize both their commonalties and their differences. Specific topics to be covered include groups, actions, isomorphism, symmetry, permutations, rings, and fields. Prerequisites: Calculus I and Discrete Mathematics; enrollment only by consent of the instructor.