# 2015-2016 Mathematics Courses

#### Multivariable Modeling II: Differential Equations

Many laws governing physical and natural phenomena and, of late, a growing number of theories describing social phenomena are expressed in terms of the rates of change (derivatives) of interrelated variables. Differential equations, the branch of mathematics that explores these important relationships, provides a collection of tools and techniques fundamental to advanced study in engineering, physics, economics, and applied mathematics. The investigation of such equations and their applications will be the focus of this second half of the Multivariable Modeling two-course sequence. Conference work will involve a concentrated investigation of one application of multivariable mathematics. *Prerequisite: successful completion of Multivariable Modeling I or the equivalent (college-level courses in Multivariable Calculus and Linear Algebra). *

#### 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 (*payoff or utility*) 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, the greatest impact has been felt in the fields of economics, political science and biology. This course represents a survey of basic techniques and principles. 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. *

#### Multivariable Modeling I: Vectors, Functions, and Matrices

Rarely is a quantity of interest (tomorrow’s temperature, unemployment rates across Europe, the cost of a spring break flight to Denver) a simple function of just one other variable. Reality, for better or worse, is mathematically multivariable. This course provides an introduction to an array of topics and tools used in the mathematical analysis of multivariable functions. The intertwined theories of vectors, functions, and matrices will be the central theme of exploration in this first semester of the Multivariable Modeling two-course sequence. Specific topics to be covered include the algebra and geometry of vectors in two, three, and higher dimensions; dot and cross products and their applications; equations of lines and planes in higher dimensions; solutions to systems of linear equations using Gaussian elimination; theory and applications of determinants, inverses, and eigenvectors; volumes of solids in three dimension via integration; and spherical and cylindrical coordinate systems. Conference work will involve a concentrated investigation of one application of multivariable mathematics. *Prerequisite: successful completion of Calculus II or the equivalent (a score of 4 or 5 on the Calculus BC Advanced Placement exam). *

#### 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. Spreadsheet statistical software will be introduced and used extensively in this course, but no prior experience with the software is assumed. Conference work will involve working in a small group to conceive and execute a small-scale research study. 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 I: The Study of Motion and Change

Our existence lies in a perpetual state of change. An apple falls from a tree. Clouds move across expansive farmland blocking out the sun for days. Meanwhile, satellites zip around the Earth, transmitting and receiving signals to our cell phones. The calculus was invented to develop a language to accurately describe and study the change that we see. The ancient Greeks began a detailed study of change but were scared to wrestle with the infinite, so it was not until the 17th century that Isaac Newton and Gottfried Leibniz, among others, tamed the infinite and gave birth to this extremely successful branch of mathematics. Though just a few hundred years old, the calculus has become an indispensable research tool in both the natural and social sciences. Our study begins with the central concept of the limit and proceeds to explore the dual of differentiation and integration. Numerous applications of the theory will be examined. 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, students preparing for careers in the health sciences or engineering, and any student wishing to broaden and enrich the life of the mind. *Prerequisites: The minimum required preparation for successful study of the Calculus is one year each of high school algebra and geometry. Students concerned about meeting the course prerequisites are encouraged to contact the instructor as soon as possible. *

#### Calculus II: Further Study of Motion and Change

This course continues the thread of mathematical inquiry, following an initial study of the dual topics of differentiation and integration (see Calculus I course description). Topics to be explored in this course include the calculus of exponential and logarithmic functions, applications of integration theory to geometry, alternative coordinate systems, and power series representations of functions. 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, students preparing for careers in the health sciences or engineering, and any student wishing to broaden and enrich the life of the mind. The theory of limits, differentiation, and integration will be briefly reviewed at the beginning of the term. *Prerequisites: One year of high school calculus or one semester of college-level calculus. Students concerned about meeting the course prerequisites are encouraged to contact the instructor as soon as possible. *

#### Topology

First known as the geometry of position, topology is the study of spatial properties that do not depend on distance or angle measure. Such properties include order, continuity, configuration, boundary, and dimension. Gottfried Leibniz was probably the first mathematician to recognize the need for such methods when he wrote, “I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry…we need yet another kind of analysis…which deals directly with position, as algebra deals with magnitude.” The unusual task of measuring space without distance, let alone coordinates, is achieved through the language of sets; and the seminar is primarily concerned with so-called point-set topology. Today, the field of topology is an active area of mathematics research, and we will discuss the questions that motivate its various branches, including geometric, algebraic, and differential topology. Conference work will be allocated to clarifying course ideas and exploring additional mathematical topics. *Successful completion of a yearlong study of calculus is required; completion of an intermediate-level course (e.g., Discrete Mathematics, Linear Algebra, Multivariable Calculus, or Number Theory) is strongly recommended. *

#### Discrete Mathematics: A Gateway to Advanced Mathematics

Your voice will produce a mostly continuous sound signal when you read this sentence out loud. As it appears on the page, however, the previous sentence is composed of 79 distinct characters, including letters and a punctuation mark. Measuring patterns—whether continuous or discrete—is the raison d'être of mathematics, and different branches of mathematics have developed to address the two sorts of patterns. Thus, a course in calculus treats motion and other continuous rates of change. In contrast, discrete mathematics addresses problems of counting, order, computation, and logic. We will explore these topics and their implications for mathematical philosophy and computer science. The form of this seminar will be that of a (mathematical) writing workshop. We will work collaboratively to identify and reproduce the key formal elements of mathematical exposition and proof as they appear in both the discrete mathematics literature and in each other’s writing. This seminar is designed for students interested in advanced mathematical study and highly recommended for students with an interest in computer science, law, logic, or philosophy.

#### First-Year Studies: The New Elements: Mathematics and the Arts

This seminar will explore the bearing of modern mathematical ideas on 20th-century creative and performing arts. Euclid’s collection of geometric propositions and proofs, entitled *The Elements,* is an archetype of logical reasoning that, since antiquity, has had a broad influence beyond mathematics. The non-Euclidean revolution in the 19th century initiated a radical re-conception of not only geometry but also mathematics as a whole. We will investigate, on the one hand, the role of “math as muse,” a source of new forms of expression that include, for example, non-Euclidean geometry, the fourth dimension, set theory, functions, networks, topology, and chance. On the other hand, we will study “math as maker” and the artists and writers who, intentionally or unintentionally, adopt modern mathematical attitudes in an attempt to break with the past. While the seminar will not aim for a comprehensive survey of the entire last century, we will investigate a sequence of case studies, including: Russian Suprematist art; the Bauhaus school in Western European architecture and design; Serialism in Western music; OuLiPo, “a secret laboratory of literary structures” in postwar French literature; and postmodern dance in 1960-70s North America, among others. This course assumes no particular expertise with mathematics or cultural history. Seminar readings, guest speakers, and a program of art and performance viewings will establish a basis for investigating the relevance of fundamental mathematical concepts to modern literature and the arts. One of the primary goals of the seminar is to assess the variety of ways that mathematics and the arts pose and address questions. Conference projects in the fall will focus on one of the elements of modern mathematics; in the spring, on an individual artist, composer, writer, or dancer whose work reflects a mathematical imagination.