2013-2014 Mathematics Courses
First-Year Studies: Mathematics in Context: Philosophy, Society, Culture, and Conflict
Mathematics has been an undeniably effective tool in humanity’s ongoing effort to understand the nature of the world around us, yet the mantra of high-school students is all too familiar: What is math good for anyway? When am I ever going to use this stuff? What serves to explain the puzzling incongruity between the indisputable success story of mathematics and students’ sense of the subject’s worthlessness? Part of the explanation resides in the observation that all too many mathematics courses are taught in a manner that entirely removes the subject matter from its proper historical, social, and cultural context—naturally leaving students with the distinct impression that mathematics is a dead subject, one utterly devoid of meaningfulness and beauty. In reality, mathematics is one of the oldest intellectual pursuits, its history a fascinating story filled with great drama, extraordinary individuals, and astounding achievements. This seminar focuses on the role played by mathematics in the emergence of civilization and follows their joint evolution over nearly 5,000 years to the 21st century. We will explore some of the great achievements of mathematics and examine the full story behind those glorious achievements. The ever-evolving role of mathematics in society and the ever-intertwined threads of mathematics, philosophy, religion, and culture provide the leitmotif of the course. Specific topics to be explored include the early history of mathematics, logic and the notion of proof, the production and consumption of data, the analysis of conflict and strategy, and the concept of infinity. Readings will be drawn from a wide variety of sources (textbooks, essays, articles, plays, and fictional writings), connecting us to the thoughts and philosophies of a diverse set of scholars; a partial list includes Pythagoras, Euclid, Galileo, René Descartes, Isaac Newton, Immanuel Kant, Lewis Carroll, John Von Neumann, John Nash, Kurt Gödel, Bertrand Russell, Jorge Luis Borges, Kenneth Arrow, and Tom Stoppard.
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.
Calculus II: Modeling With Differential Equations
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: Calculus I (differential calculus in either a high-school or college setting).
Calculus I: The Study of Motion and Change
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 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 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 precalculus is highly recommended. No previous calculus experience is necessary or desired.
Discrete Mathematics: A Gateway to Advanced Mathematics
This seminar is an introduction to the world of elegant mathematics, beyond that encountered in high school, under the guise of an introductory survey course in discrete mathematics. We will touch on the tips of many icebergs! The subject of discrete mathematics houses the intersection of mathematics and computer science; it is an active area of research that includes combinatorics, graph theory, geometry, and optimization. The topics in this course are selected to give an idea of the types of thinking used in a variety of discrete mathematics research areas. Learning the facts and techniques of discrete mathematics is inextricably intertwined with reasoning and communicating about discrete mathematics. Thus, at the same time as surveying discrete mathematics, this course is an introduction to rigorous reasoning and to writing convincing arguments. These skills are necessary in all of mathematics and computer science and very applicable to law and philosophy. Conference work will explore additional mathematical topics. The seminar is essential for students planning advanced study in mathematics and highly recommended for students studying computer science, law, or philosophy or who seek to enhance their logical thinking and problem-solving skills. Prerequisite: Prior study of Calculus or equivalent preparation.
Topics in Multivariable and Vector Calculus
Compared to the familiar single-variable territory of Calculus I and II, multivariable calculus is a foreign land. Imagine, if you will, that instead of a function taking a single input and producing a single output, we either use one input and get multiple outputs (vector functions) or use several inputs and get one output (multivariable functions). And yes, there are even functions that have several inputs and multiple outputs! In this new realm, we will investigate lines and planes, curves and surfaces, and multidimensional generalizations of these objects, with a focus on those functions that can be visualized in three dimensions. For both vector and multivariable functions, we will address the basic questions of calculus: How do we measure rates of change? How do we find areas and volumes? How can we interpret derivatives and integrals both geometrically and for practical purposes? Fascinatingly, each of these questions has more than one answer. We will examine gradients and directional derivatives, maxima and minima and saddle points, double and triple integrals, integrals taken along curves, and more—as time permits. This seminar is essential for students intending to pursue engineering, physics, mathematics, graduate study in economics, or rocket science and is recommended for students pursuing chemistry or computer science. Prerequisites: Calculus I and Calculus II.
Topology: The Nature of Shape and Space
Topology, a modernized version of geometry, is the study of the fundamental, underlying properties of shapes and spaces. In geometry, we ask: How big is it? How long is it? But in topology, we ask: Is it connected? Is it compact? Does it have holes? To a topologist there is no difference between a square and a circle and no difference between a coffee cup and a donut because, in each case, one can be transformed smoothly into the other without breaking or tearing the mathematical essence of the object. This course will serve as an introduction to this fascinating and important branch of mathematics. Conference work will be allocated to clarifying course ideas and exploring additional mathematical topics. Successful completion of a yearlong study of Calculus is a prerequisite and completion of an intermediate-level course (e.g., Discrete Mathematics, Linear Algebra, Multivariable Calculus, or Number Theory) is strongly recommended.