Courses in Mathematics

MATH 116 - Introduction to Statistical Science:This is a course in how to draw conclusions from real-world data. We'll explore graphical data analysis in a statistical computing package. We'll also learn the standard tools of statistical inference: confidence intervals and hypothesis tests. You'll realize that these concepts appear everyday in the news. What does the "margin of sampling error" statement for an opinion poll mean? What does it mean to say results from a clinical trial of a new medical treatment were "statistically significant?" You'll also discover that these concepts apply in your other science and social science courses at Middlebury; for example, when you're asked to plan a survey or an experiment, or to fit a model to data you've collected from laboratory or field measurements, or to interpret the results of a published research paper.

MATH 120 - Extended Calculus A course for students with only two or three years of secondary school mathematics. This course integrates the material prerequisite to calculus with calculus itself. Algebra, graphs, and properties of continuous functions are studied as these topics relate to the calculus. Trigonometric functions, analytic geometry, logarithms, and exponential functions will be integrated into the course as the need arises. Extended Calculus is a two-semester course that begins in the fall and continues in the spring.

MATH 121 - Calculus I: Calculus is the mathematics of motion and change. Where geometry studies the properties of static figures and where elementary algebra may study the slope of a given line, in calculus we study the position of a moving object at different times and discuss the nature of slopes of tangent lines to curves for changing arguments. Calculus is to pre-calculus as motion pictures is to still photography. Specifically, MATH 121 covers the differential calculus of algebraic and trigonometric functions, optimization and related rate problems, the definite and indefinite integral, areas under curves, and volumes of solids of revolution.

MATH 122 - Calculus II: MATH 122 is a continuation of MATH 121. Topics include the logs and exponential functions, techniques of integration, graphing and areas in polar coordinates, improper integrals, infinite series and Taylor series, and the rudiments of differential equations. Applications of calculus range from modeling population growth to describing the curve formed by telephone lines.

MATH 145 - Mathematical Foundations of Computing: What does it mean to say that one algorithm is "better" than another algorithm? How can we construct a network of N computers to minimize the number of direct connections while guaranteeing that a message from one computer to any other takes no more than logN hops? Is it possible to prove how much time or memory will be used by an algorithm? Analyzing and solving these kinds of questions requires an understanding of the mathematical foundations of computer science. MATH 145 provides an introduction to the methods of formal reasoning and the mathematical theory essential to the discipline of computer science.

MATH 200 - Linear Algebra:Vectors, polynomials, continuous functions, matrices, and solutions to certain types of differential equations all have something in common. Specifically, each of these sets has the property that adding or multiplying by scalars does not take you out of the set. The idea of Linear Algebra is to isolate this fundamental property by assuming we have a set of abstract objects (called a "vector space") with this property. We then prove theorems about their structure and the structure of functions that act on them. The subject matter of this course is really geometry. Every theorem says something plausible about geometric objects like points, lines and planes. The magic occurs when you see how this geometric intuition can be applied to other examples of vector spaces that occur in areas such as statistics, probability, and differential equations.

MATH 223 - Multivariable Calculus: All the functions you've studied in calculus so far live on a flat piece of paper. But you live in (at least!) three dimensions. Now you certainly know that calculus was invented to solve problems about the physical world, so we're going to have to move off that flat paper at some point. MATH 223 is where it happens. The key is the concept of a vector. If you've had a little bit of physics, you may have heard a vector is an object having direction and magnitude. In MATH 223, we'll tighten that definition up, and study functions whose domains and ranges consist of vectors. Can limit, derivative and integral make sense out here? The answer is yes, and when you're through you'll know how Newton's calculus--the greatest intellectual achievement of mankind! -- made sense of Kepler's empirical observations about the motion of the planets -- the greatest scientific discovery of all time! Come to think of it, maybe this course should be required for graduation...

MATH 225 - Topics in Linear Algebra and Differential Equations:Differential equations are best understood as a topic in linear algebra, and so we spend the semester looking at different kinds of differential equations from a linear algebra perspective. The course is quite applied and practical. You can take this course after linear algebra since multivariable calculus is not a prerequisite.

MATH 235 - Visible Mathematics:At a fundamental level, mathematics is the study of structure. In this course, we'll look into some topics that you may or may not have seen previously within a mathematical context - topics such as braids, knots, surfaces, and symmetry. Our mathematical exploration will have the theme of simple, observable objects and their structures - but we'll see that there is a great deal going on even in these simple objects that's worthy of mathematical study.

MATH 241 - Elementary Number Theory:Number theory deals with the mathematical properties of the integers. We study the prime numbers (2, 3, 5, 7, 11, ...), Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, ...), triangular numbers (1, 3, 6, 10, 15, ...), perfect numbers (which are the sum of their proper divisors -- like 6 = 1 + 2 + 3), and amicable pairs (like 220 and 284). Some of the questions we address are: How do we determine if a given integer is prime? How many primes are there? Are there any odd perfect numbers? Can we show that every positive integer can be expressed as the sum of at most four squares? The course involves proving important theorems and solving lots of intriguing problems. We also briefly discuss Fermat's Last Theorem, the most important mathematical proof of the 20th century.

MATH 247 - Graph Theory:Graph theory is the study of networks, or points and lines. Perhaps no other subject in mathematics begins with such simplicity. Yet the subject is replete with important applications, sophisticated theorems and unsolved problems. It has applications to chemistry (organic molecules), sociology (kinship structures), city planning (traffic patterns), computer science (the structure of programs), engineering (oil refineries), sports (tournaments), communications (telephone lines), and, of course, mathematics. The course proceeds by discovery. Students will try to find patterns, describe important ideas, formulate mathematical definitions, make conjectures and either prove them or find counterexamples. The course concludes with a proof of the fabulous Four Color Theorem, perhaps the most famous mathematics problem to be solved this century. (Some of the number theory people may dispute this claim, but what do they know?)

MATH 261 - History of Mathematics: The history of mathematics dates back to the stone age with people grappling with the nature of time, movement of the stars, and concepts of magnitude, shape, and measurement. In this course we study Babylonian, Egyptian, and Chinese mathematics, Greek geometry, Hindu and Arabic algebra, and Western mathematics and mathematicians into the 19th century. Some of the questions we address are: When was the concept of zero and negative numbers developed? Who invented calculus? Is there a cubic formula analogous to the familiar quadratic formula? Where did algebra come from? How can mathematics be used to determine the weight of the earth? Why have there been so few women mathematicians until recently? The course involves lectures and fun student projects and presentations.

MATH 302 - Abstract Algebra: Remember the associative laws? There's one for addition and one for multiplication in the real number system, as well as for addition in a system of vectors and multiplication in a system of matrices. In this course we prove useful things about these systems and others all at once by using the power of abstraction (that is, considering only their important common properties apart from individual characteristics; you may have done this with vector spaces). This course could also be called "Groups, Rings, and Fields;" a group is a system with one associative operation and some other properties, a ring has two associative operations with a connecting distributive law, and a field is a special kind of ring.

MATH 310 - Probability: When should you fold with a poker hand? Or take another "hit" when playing blackjack? What are your chances of winning the "suitcase weekend" in Bermuda? If your older sister has two girls and is expecting again, what are the chances she will have yet another girl? In a much more serious vein, what are the chances of getting AIDS if a College senior has unprotected sex? Does it matter if the student is male or female?
This course puts much of what you learned in Calculus I and Calculus II to work on real problems that involve randomness. It combines an axiomatic presentation of an important branch of mathematics, probability, with a focus on interesting and often useful problem-solving strategies. Probability is the cornerstone of statistics, and it is useful in economics, insurance, health sciences, psychology, engineering, computer science, and many other fields.

MATH 311 - Statistics: Is the new diet pill advertised on TV really effective in helping people lose weight? How much credibility should we place in the latest TIME/CBS poll about voter preferences in the next presidential election? Do student evaluations of their college courses really help distinguish effective teaching that leads to successful learning? Does treatment of AIDS with AZT and another drug really work better than treatment with AZT alone? How do we know the answers to questions like these, based on objective empirical data? How confident are we that our best guesses will turn out right?
This course uses the material of MATH 310 Probability to draw inferences, to make predictions, and to reach important decisions. MATH 311 also explores ways of developing and fitting mathematical models that can help us describe random phenomena. It helps us understand the natural variability in many outcomes and enables us to find the real and important messages that are often hidden in a collection of data.

MATH 315 - Mathematical Models in the Social, Life, and Management Sciences: Arms races, rabbits and foxes, chaos, unrequited love, Clinton vs. Dole vs. Perot, the impossibility of fair voting, why "the price is right" in economics, epidemics, cultural stability in Ethiopia, social conformity of college students, flunking out at Middlebury, hospital surgery rooms, and (believe it or not) The Bible. All this, and more, in a mathematics(!) course.

MATH 318 - Operations Research: Operations Research (OR) is a 20th century branch of mathematics, which combines tools from Mathematics, Computer Science and Economics to provide a quantitative basis for making decisions. That's quite a mouthful, but OR techniques have found such wide-ranging applications that it becomes hard to give a precise definition of the field. Still, one recurring theme in OR is the optimal allocation of scarce resources, and the decision criteria in OR problems often have an economics spirit. In the business world, you'll find OR professionals working on problems ranging from manufacturing planning to financial management, and many large companies have their own in-house OR consulting groups. Math majors with an interest in real-world applications, and Economics majors with good analytical skills, should certainly consider taking MATH 318 as an elective.
Most OR applications require the ability to formulate a mathematical model for a decision problem. These models can be large and complex, and the computer becomes an essential tool for solving them. In fact, many OR algorithms are interesting objects of study in their own right. Prime example: the simplex algorithm--invented in 1947 for solving optimization models known as "Linear Programs" -- has direct connections to convexity problems in computational geometry.

MATH 323 - Real Analysis:The calculus that you learned in 121 and 122 is very much in the spirit of what Newton and Leibniz formulated near the end of the 1600's. Specifically, the concepts of differentiation and integration are built on a rather imprecise understanding of limits. But by the 1800's, the expanding ideas of calculus -- most notably in the direction of infinite series -- caused a serious crisis as it became apparent that the intuitive definitions were insufficient for resolving some sticky convergence questions. Cauchy, Dirichlet, Weierstrass, et al., managed with great effort to put the theory of calculus on a rigorous foundation, and in so doing were able to extend its power to new and exciting areas. Some versions of this course include an exploration of a few of these areas (e.g. Fourier series), while others return to the very beginning and actually construct the real numbers on which the theory of limits depends.

MATH 325 - Complex Analysis: Undoubtedly, at some point in your mathematical past, you encountered the complex number system (a+bi , where i=sqrt(-1)). It may seem at first glance that proving the theorems of calculus in this setting would be a predictable exercise, and indeed the path begins with the familiar concepts of limits and the derivative. But subtle warnings begin to arise. The Cauchy-Riemann equations indicate that being differentiable is perhaps a slightly stronger statement than it was in the real setting, and the next thing you know you have shown that if a function is bounded and differentiable, then it is constant! The theory of complex integration is so elegant that it ought to be cross listed in the art department! The theorems are beautiful, surprising, and, despite the fact that we are working with so-called "imaginary numbers", incredibly useful in the real world.

MATH 0335 Differential Geometry
This course will be an introduction to the concepts of differential geometry.  For curves in space, we will discuss arclength parameterizations, Frenet formulas, curvature, and torsion.  On surfaces, we will explore the Gauss map, the shape operator, and various types of curvature.  We will apply our knowledge to understand geodesics, metrics, and isometries of general geometric spaces.  If time permits, we will consider topics such as minimal surfaces, constant curvature spaces, and the Gauss-Bonnet theorem.

MATH 341 - Topics in Number Theory: This course is a continuation of MATH 241, Elementary Number Theory. In fact we address many of the same questions that arise in Elementary Number Theory but use more advanced techniques for solving (or at least trying to solve) them. The particular topics covered and the techniques employed differ depending on the instructor and the interests of the students. In general students are introduced to the concepts of algebraic numbers, number fields, ideals, and algebraic or analytic number theory. Some topics from past courses include, factorization in number fields, binary quadratic forms, Diophantine equations, and elliptic curve theory.

MATH 345 - Combinatorics: Combinatorics deals with counting the number of ways a set can be structured in a prescribed fashion and, if possible, exhibiting the structures explicitly. Here are two interesting examples: (1) A group of 15 students take a daily walk in five rows with three abreast. Is there a way to arrange them so that in seven days each student walks once with each other student? (2) In addition to the standard way of numbering a pair of dice, is there another way which has all the same sums and with the same probability distribution?

MATH 351 - Set Theory: So you think the theory of sets can be understood by just drawing a few Venn diagrams? You'll be delighted to know that it's much more pathological than that. The seemingly harmless-looking "Axiom of Choice," which asserts that if you are given a family of non-empty sets then you may form a new set by simply picking an element from each one, leads to wild and ridiculous conclusions. E.g., a sphere can be partitioned into finitely many pieces, and the pieces rearranged to form two spheres of the same size! Hey, it doesn't get any better than this.

MATH 402 - Topics in Algebra: Is there an analog of the quadratic formula for the general fifth-degree polynomial equation? In this course we develop the tools to prove that there is not and, in the process, come to appreciate the genius of Evariste Galois (1811-1832). What Galois did, without the benefit we have of modern definitions, terminology, and theorems, was to see the connection between a field associated with an equation and a group. In addition, we discuss the 19th century resolution of three problems from antiquity: squaring the circle, duplicating the cube, and trisecting an arbitrary angle.

MATH 410 - Stochastic Processes: In MATH 310 you learned that there are laws for randomness. If you followed this up with MATH 311, you studied the implications of randomness when making estimates and predictions from observed data. MATH 410 takes the ideas from MATH 310 in a new direction, looking at the behavior of systems that evolve dynamically in the presence of randomness. Communications delays on the Internet, fluctuations of stock prices, a gambler's fortune at the roulette table, the spread of an epidemic, and gene frequencies in a population represent just a few of the real-world phenomena that can be mathematically modeled as random ("stochastic") processes. The course will be geared to a wide audience. Purists should appreciate the elegance of the theory; students from other majors can expect to see serious applications of probability to their own fields.

MATH 432 - Elementary Topology: At last, a math course with no required exams, no expensive text to buy, and no professorial lectures to sit through. Students form an active research community discovering proofs or counterexamples for purported theorems and then present them to their peers for critical examination. We begin with the cosmic infinite and end with surprisingly deep results about the geometric nature of the line and the plane.

MATH 451 - Mathematical Logic: Starting in the 19th century, a driving theme in mathematics was the search for a bedrock foundation for the different areas of the subject. The idea was to take the results of geometry or calculus and make each into a completely deductive system where all of the theorems could be logically proved from an agreed to set of axioms. By the 20th century, mathematicians had turned their attention toward the process of logic itself - using logic to understand logic as a deductive system. The central goal of this course is to understand ways in which we can formalize - or mechanize - the idea of a proof, and see how this relates to the concept of truth. The class ends with a glimpse at the monumental work of Kurt Godel who demonstrated that, in the end, mathematicians would have to accept a certain amount of incompleteness in their mathematical world.

MATH 480 - Research in Mathematics (CW): The object of this course is active participation in mathematical investigation and exposition. Students work collaboratively on current research questions provided by the instructor. The course includes a review of relevant literature and research methods. Students are required to present their findings both in writing (consistent with the standards of the discipline) and in public presentations. The topic for Spring 2002 will be polynomials over finite fields.

MATH 500 - Advanced Study:It is possible that the normal elective offerings are not what you are looking for at this particular time. Perhaps you have been exposed to an area where there isn't enough interest to justify offering a full course (e.g. set theory, logic), or you have taken a class to which there is no follow up (e.g., topology, real analysis). Discuss with any of your professors the possibility of doing an independent study. It may be that your particular interest is best handled as a thesis topic in MA704, but you should certainly explore your options. If you entered Middlebury with advanced standing, it would seem very likely that you may need to take advantage of this resource.

MATH 704 - Senior Thesis (Fall, Spring): Each student is required to complete and present a major paper on a topic chosen with the advice of a faculty member. In addition, during the academic year, each student is expected to attend a series of lectures designed to introduce and integrate ideas of mathematics and computer science not covered in the previous three years.