# Courses

Get a feel for the exciting variety of courses taught at Marlboro.

This is a list of courses that faculty felt was representational of the courses offered. It is not a complete list of courses, some courses are offered yearly, while others are infrequent. A course may be inspired by events or strong interests and taught only once.

Most advanced work is in the form of tutorials on specific subjects, a collaboration between one faculty member and one student or a handful of students.

## Mathematics

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A Whirlwind Tour of Mathematics
(3.00 Credits — Introductory)

Do you want a thorough understanding of the most important and deep theorems in every branch of mathematics? Do you want to achieve this in a three credit course from a standing start? Good luck with that - you won't manage it in this course. Instead, we'll look at six to ten topics, chosen for their accessibility and beauty, and drawn from a broad range of subdisciplines of math. Possibilities include: irrational and imaginary numbers, the infinite, chaos and fractals, Fermat's Last Theorem, the Platonic solids, the fourth dimension, the combinatorial explosion, P vs. NP, the Four Color Theorem, non-Euclidean geometry, logical paradoxes, and many others. No prior mathematical experience is expected. Prerequisite: None

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Algebraic Structures
(4.00 Credits — Intermediate)

An introduction to the theory of groups, rings and fields. Advanced topics covered will depend on the interests of the participants and might include Sylow theory, combinatorial or number theoretic connections, error-correcting codes, modules or Galois theory.

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ASPECTS OF GEOMETRY
( Variable Credits — Introductory)

Throughout the history of geometry, great advances have been made through radical reconceptualizations of the entire subject: Euclid's axiomatic geometry; the analytic geometry of Descartes, et. al.; the projective geometry stemming from Renaissance art; the unification of geometry and number through complex numbers, quaternions and linear algebra; the discovery of non-Euclidean geometries by Gauss, Lobachevsky and Bolyai; the group theoretical synthesis of geometry of Felix Klein's 1872 Erlanger Programm. We will focus on conceptual aspects of these points of view and see how each shift addressed fundamental issues left unresolved by existing theories. Prerequisite: None

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Calculus
(4.00 Credits — Introductory)

A one semester course covering differential and integral calculus and their applications. This course provides a general background for more advanced study in mathematics and science.

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Calculus II
(4.00 Credits — Intermediate)

We build on the theory and techniques developed in Calculus (NSC515). Topics include techniques and applications of integration, complex numbers, power series, parametric equations and differential equations.

- Calculus or permission of instructor

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Calculus III
(4.00 Credits — Intermediate)

Calculus III continues the development of the techniques of Calculus into multi-variable and vector-valued functions.

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Combinatorics Study Group
(2.00 Credits — Intermediate)

Combinatorics is a broad subfield of math concerned with discrete, often finite, structures. It is unusual in that it is possible to engage seriously with difficult questions in the field without an extensive list of prerequisites. That's exactly what we'll do here. Likely objects of study include graphs, Latin squares (sudoku puzzles being a well-known example of these) and combinatorial designs, enumeration problems and integer sequences. The goal of the course is not to give an account of the main tools or topics of combinatorics, although we'll do some of this in passing, but to get a taste for some of the many aspects involved in the creation of mathematics, including imagination, frustration, collaboration, bewilderment, hard work, insight, luck and maybe even joy. May be repeated for credit. Prerequisite: Previous math courses, ideally including Discrete Math or something similar. Programming experience is useful but not required.

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COMPLEX VARIABLES
(4.00 Credits — Advanced)

Prerequisite: Calculus II

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Differential Equations
(3.00 Credits — Advanced)

Differential equations is the mathematics of changing systems. It has wide-ranging applications, including biology, physics, and economics. This course is an introduction to ordinary differential equations, with an emphasis on finding and applying techniques to solve first-order and linear higher-order differential equations. Prerequisite: Calculus II

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Discrete Mathematics
(4.00 Credits — Introductory)

Discrete math is the study of mathematical objects on which there is no natural notion of continuity. Examples include the integers, networks, permutations and search trees. After an introduction to the tools needed to study the subject, the emphasis will be on you *doing* mathematics. Series of problems will lead gradually to proofs of major theorems in various areas of the discipline. This course is recommended for those intending to do advanced work in math or computer science. Prerequisite: None

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Dynamical Systems and Chaos Theory
(4.00 Credits — Intermediate)

Our goal for this class will be to study dynamical systems. To study dynamical systems, we will be using the software Mathematica (no initial computer programming knowledge is required). We will not only learn some of the theory behind dynamical systems, but we will also experiment on the computer. We will look at simple dynamical systems, use graphical analysis to help describe the behavior of a system, symbolic dynamics, examine fractals and look at the Mandelbrot set and Julia sets. Prerequisite: Linear Algebra or instructor's approval.

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Fun with Logic
(4.00 Credits — Multi-Level)

This course is meant to be an introduction to logical systems, their use and applications to all fields of science and humanities. It will be accessible to all levels of students from all fields.

We will be studying several logical systems, such as propositional and classical logic, intuitionistic logic and modal logic.

Prerequisite: None

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Group Theory and Rubik's Cube
(2.00 Credits — Introductory)

This course will explain the basics of a branch of mathematics called Group Theory by examining Rubik's Cube and other similar puzzles. My hope is that the puzzles will motivate the ideas behind Group Theory. Although this is an introductory course and does not depend on any previous math (we will, for example, hardly use numbers at all), students should be comfortable with abstract thought. Prerequisite: Some facility with abstract concepts

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Group Tutorial: Real Analysis
(4.00 Credits — Intermediate)

Real Analysis is the study of the real number system. In this tutorial we look at how the real numbers are built and put the results developed in the Calculus sequence on a more rigorous footing. As this is a tutorial, responsibility will fall mostly on the students to choose the precise topics and navigate a route through the material. Prerequisite: Calculus 2 and permission of instructor

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Linear Algebra
(4.00 Credits — Intermediate)

Linear Algebra is important for its remarkable demonstration of abstraction and idealization on the one hand, and for its applications to many branches of math and science on the other. This tutorial covers linear algebra in n-dimensional space. Matrices, vector spaces and transformations are studied extensively.

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Map Coloring, Graph Theory and the Four Color Theorem
(4.00 Credits — Intermediate)

Every map, no matter how the countries are laid out, can be colored using just four colors in such a way each pair of adjacent countries are different colors (there are some minor, natural restrictions). This is the celebrated Four Color Theorem. Conjectured to be true in the 1850s and subject to many failed attempts at proof, it was controversially settled in 1976. The controversy comes from the fact that the proof relies on a computer calculation; no human has (or could) check all of the details. This result lies within the field of Graph Theory, one of the most vibrant subfields of math of the last 100 years (and still so today). This course will take us through the methods used in the proof of the Four Color Theorem by way of many discursions into Graph Theory. Topics to be covered include chromatic polynomials, hamiltonicity, planarity, graph decompositions and classifying polyhedra. We'll also investigate related problems: What if each country has a lunar colony that must be colored with the same color as that country? How many colors would we need if we lived on a torus?

Prerequisite: Discrete Math or permission of instructor

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Multivariable Calculus
(4.00 Credits — Intermediate)

An extension of the ideas from Calculus and Calculus 2 to multivariable and vector functions. Topics covered include the geometry of 3-dimensional space, partial derivatives, multiple integrals and higher dimensional analogues of the fundamental theorem of calculus. Prerequisite: Calculus II or equivalent

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Number Theory
(3.00 Credits — Introductory)

An introduction to number theory, from its inconspicuous beginnings on a Babylonian clay tablet almost four thousand years ago to its mature majesty in the 19th century as "the Queen of Mathematics" (in the words of Gauss), and beyond. Topics include the infinitude of primes, modulo arithmetic, the RSA cryptosystem, Pell's equation, Fermat's Last Theorem, and much more. Prerequisite: None

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Probability
(4.00 Credits — Intermediate)

Probabilities pop up everyday like "There's a 30% chance of rain" or "The probability of being dealt a full house in stud poker is approximately 0.00144." Our main goal for the class will be developing various tools to calculate probabilities. Topics include axioms of probability, counting techniques, conditional probability, discrete and continuous random variables, special discrete and continuous distributions and joint distributions. Prerequisite: Calculus I

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PUZZLED?
(2.00 Credits — Introductory)

This course will give students a chance to test and develop their puzzle-solving ingenuity. We'll attack a series of puzzles, going from Lewis Carroll's logic problems via the classic "recreational math" puzzles of Lucas, Loyd and Dudeney to modern crazes such as the sudoku. Pass/Fail grading. Prerequisite: None

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Set Theory & Logic
(3.00 Credits — Advanced)

After a review of various notions from logic and set theory that will be mostly familiar from previous courses, we study a variety of topics that form the foundation of mathematics. The exact list of topics depends on the interests of the class, but some natural candidates are: cardinals and ordinals, propositional and first-order logic, axiomatic set theory, models, constructions of number systems, and categories. Students may take this course for three or four credits. Those students taking the class for four credits will undertake an extensive investigation into an additional topic.

Prerequisite: Several math courses, preferably including either Real Analysis (NSC626) or Formal Languages and the Theory of Computation (NSC543)

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Statistics
(4.00 Credits — Introductory)

Statistics is the science--and art--of extracting data from the world around us and organizing, summarizing and analyzing it in order to draw conclusions or make predictions. This course provides a grounding in the principles and methods of statistics as commonly used in the natural and social sciences. Topics include: probability theory, data collection, description, visualization, probability, hypothesis testing, correlation, regression and analysis of variance. We will use the open source statistical computing package R (no prior computing experience is assumed).

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Statistics Workshop
( Variable Credits — Multi-Level)

A follow-up to Statistics (NSC123) in which students acquire and hone the statistical skills needed for their work on Plan or simply pursue more advanced topics within the field. Course content is driven by the interests and requirements of those taking the class. Variable credit (1-4). May be repeated for credit.

- Statistics (NSC123) or permission of the instructor

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Topics in Algebra, Trigonometry and Pre-Calculus
(3.00 Credits — Introductory)

This course covers a wide range of math topics prerequisite for further study in mathematics and science and of interest in their own right. The course is divided into 10 units, listed on the course web page. One credit will be earned for each unit completed. Students select units depending on their interest and need. The course is especially designed for students who plan to study calculus or statistics, would like to prepare for the GRE exam or who just want to learn some math. Over the semester, 3-4 units will be offered in the timetabled sessions. Individual tutorial-style arrangements can be made with students who want to study the non-timetabled units, or who want to study units at their own pace. Prerequisite: None

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Writing Math
(1.00 Credit — Multi-Level)

We will study the writing and presentation of mathematics. All skills needed for writing Plan-level math will be discussed, from the overall structure of a math paper down to the use of the typesetting package LaTeX. Much of the time will be spent working on writing proofs. Short papers, based on material in your other math classes, will be read and discussed as a group. May be repeated for credit. Prerequisite: Passed Clear Writing Requirement; concurrent course or tutorial that includes substantial mathematical content

### For Mathematics offerings, also see:

• Algorithms• Formal Languages and the Theory of Computation

• How Environmentally Sustainable is Marlboro College?

• Information Theory

• Introduction to Cartography: History, Theory and Practice