Course Descriptions

First-Semester Courses

MATH 6  Elements of Statistics
Hartlaub, Jones (1/2 unit)

This is a basic course in statistics. The topics to be covered are the nature of statistical reasoning, statistical description, probability, random variables and probability distributions, binomial and normal distributions, t- and chi-square distributions, sampling, estimation and tests of population proportions, inferences concerning population means, and the difference of two population means. A computer statistical package is used. Enrollment limited.

MATH 10  Pre-calculus
C. Schumacher  (1/2 unit)

This course prepares students for the study of calculus. It is particularly directed to those planning to enter the calculus sequence that begins with MATH 11. Primary emphasis is placed on the study of real valued functions, particularly polynomial, rational, logarithmic, exponential, trigonometric, and inverse trigonometric functions. Conceptual understanding will be emphasized. Computer labs, utilizing graphing programs and a computer algebra system, will be employed. Students with 1/2 unit of credit for calculus may not receive credit for MATH 10.

MATH 11  Calculus A
Jones, C. Schumacher, Slack (1/2 unit)

The first in a three-semester calculus sequence, this course covers the basic ideas, techniques, and applications of differential calculus. Those who have had a year of high-school calculus but do not have AP credit for Math 11 should take the Calculus Placement Exam to determine whether they are ready for MATH 12. Students who have 1/2 unit of credit for calculus may not receive credit for MATH 11. Enrollment limited. Prerequisite: solid groundings in algebra, trigonometry, and elementary functions.

MATH 12  Calculus B
Holdener, Shults (1/2 unit)

The second in a three-semester calculus sequence, this course continues with the calculus of elementary functions, integration and the fundamental theorem, techniques of integration, numerical methods, applications of integration, improper integrals, and additional topics as time permits. Prerequisite: MATH 11 or permission of department. Enrollment limited.

MATH 18  An Introduction to Computer Science
Shults   (1/2 unit)

This course presents an introduction to computer science intended for those planning to take additional courses in computing, for those with a strong foundation in mathematics, and for those intending to major in science or mathematics or one of the social sciences where a strong background in computation is desirable. This course will expose the student to a variety of applications where an algorithmic approach is natural and will include both numerical and non-numerical computation. Instruction in a high-level language will be included, and the principles of structured programming will be emphasized. Enrollment limited.

MATH 21  Calculus C
Slack (1/2 unit)

The third in a three-semester calculus sequence, this course includes the topics of sequences and series, vectors, functions of two or more variables, partial derivatives, multiple integrals, and additional applications. Prerequisite: MATH 12.

MATH 28  Data Structures and Program Design
Shults  (1/2 unit)

This course is intended as a second course in programming, as well as an introduction to the concept of computational complexity and the major abstract data structures (such as arrays, stacks, queues, link lists, graphs, and trees), their implementation and application, and the role they play in the design of efficient algorithms. Students will be required to write several programs using a high-level language. Prerequisite: MATH 18.

MATH 29  Design and Analysis of Experiments
Hartlaub  (1/2 unit)

This course will focus on standard methods of designing and analyzing experiments. Simple comparative designs, factorial designs, block designs, and appropriate post-hoc comparisons will be discussed. These techniques are commonly used by statisticians and experimental scientists in a wide variety of fields. Statistical software will be introduced and heavily used throughout the course. No prior experience with the software is necessary. Each student will be asked to design an experiment, conduct the experiment, and collect and analyze the appropriate data. Prerequisite: MATH 6 or permission of instructor. Enrollment limited.

MATH 32  Vector Analysis
Holdener  (1/2 unit)

Physical and natural phenomena depend on a complex array of factors, and to analyze these factors requires the understanding of geometry in two and three (or more) dimensions. Vector Analysis will continue the study of multivariable calculus begun in MATH 21. Topics of study will include vector fields, line and surface integrals, potential functions, classical vector analysis, and Fourier Series. Computer labs will be incorporated throughout the course, and physical applications will be plentiful. Prerequisite: MATH 21.

MATH 35  Abstract Algebra I
Holdener  (1/2 unit)

The phrase "abstract algebra" correctly suggests some sort of a generalization of a topic most of us learned in high school, though it goes very much beyond that, of course. Three of the most important structures in abstract algebra are groups, rings, and fields; all three are, in fact, abstractions of familiar objects--the integers form a group or ring, while the real numbers give us an example of a field. Each of these structures has the property that any two of the subjects in the system may be "combined" in some way to produce a new object in the system. In the system of integers, for example, this "combining" might be addition or multiplication. Groups and rings are fundamental tools for any mathematician and many scientists, but these concepts are beautiful and worthy of study in their own right--group theory and ring theory currently are both very active areas of mathematical research. In this course, the student examines the basics of groups and rings, with emphasis on the many examples of these algebraic structures. A possible example might be a study of symmetry with the aid of group theory. Prerequisite: MATH 22 or permission of instructor.

MATH 36  Probability
Hartlaub  (1/2 unit)

This course provides a mathematical introduction to probability. Topics include basic probability theory, random variables, discrete and continuous distributions, mathematical expectation, functions of random variables, and asymptotic theory. Prerequisite: MATH 21.

MATH 41  Real Analysis I
C. Schumacher  (1/2 unit)

This course is a first introduction to Real Analysis. AReal@ refers to the real numbers. Much of the work in the course will revolve around the real number system, starting with careful consideration of the axioms that describe it. Students will be asked to consider many functions that take on real values---that is, each object in our domain will be associated with a real number. For instance, every point in the plane can be associated with its distance from the origin. Two points in the plane give rise to a real number: the distance betweent them. The concept of distance will be a major theme of the course.

AAnalysis@ is one of the principle branches of mathematics. One often hears that analysis is the theoretical underpinnings of the calculus, but though this has a kernel of truth, it is an answer that misleads by oversimplifying. Certainly analysis had its inception in the attempt to give a careful, mathematically sound explanation of the ideas of the calculus. But over the last century, analysis has grown out of its original packaging and is now much more than simply the theory of the calculus. Analysis is the mathematics of Acloseness@ ---the mathematics of limiting processes. The idea of continuity can be phrased in terms of limits. Both derivatives and integrals are the end results of taking a limit. Compactness is a property of sets that underlies many of the most important theorems encountered in calculus. These and related ideas will be the subject of the course. Prerequisites: MATH 21 and 22 or permission of the instructor.

MATH 93  Individual Study
Staff  (1/2 unit)

This course enables students to study a topic of special interest under the direction of a member of the mathematics department. Prerequisites: permission of instructor and department chair.

MATH 97  Senior Honors
Slack, Staff  (1/2 unit)

The content of this course is variable and adapted to the needs of senior candidates for honors in mathematics. Prerequisite: permission of department.

MATH 6  Elements of Statistics
Hartlaub, Jones  (1/2 unit)

See first-semester course description.

MATH 11  Calculus A
Holdener  (1/2 unit)

See first-semester course description.

MATH 12  Calculus B
C. Schumacher, Slack (1/2 unit)

See first-semester course description.

MATH 18  An Introduction to Computer Science
Shults (1/2 unit)

See first-semester course description.

MATH 21  Calculus C
Holdener  (1/2 unit)

See first-semester course description.

MATH 22  Foundations of Analysis
Shults  (1/2 unit)

This course introduces students to mathematical reasoning and rigor in the context of set-theoretic questions, analysis, and geometry. The course will cover fundamental aspects of set theory, such as those related to countable and uncountable sets, fundamental properties of the Euclidean line, the geometry of metric spaces, and aspects of the topology of Euclidean spaces. Emphasis will be placed on helping students in reading, writing, and understanding mathematical reasoning. Students will be actively engaged in creative work in mathematics.

The course should be taken no later than the spring semester of the sophomore year. First-year students interested in mathematics are encouraged to consider this course for the second semester of their first year. (Please see a member of the math faculty if you think you might want to do this.) Prerequisite: credit for at least one Kenyon mathematics course numbered 10 or above or permission of instructor.

MATH 24  Linear Algebra I
Jones  (1/2 unit)

Linear algebra grew out of the study of the problem of organizing and solving systems of equations. Today, ideas from linear algebra are highly useful in most areas of higher-level mathematics. Moreover, there are numerous uses of linear algebra in other disciplines, including computer science, physics, chemistry, biology, and economics.

This course involves the study of vector spaces, an appealing geometric way of formulating many of the most important ideas in the subject. Two familiar vector spaces from calculus are the plane and 3-space. In addition, students in MATH 24 examine matrices, which may be thought of as functions between vector spaces. In the past, linear algebra involved tedious calculations. Now we have computers to do this work for us, allowing us to spend more time on concepts and intuition. A computer algebra system such as Maple will likely be used. Prerequisite: MATH 12 or permission of department.

MATH 26  Data Analysis
Hartlaub  (1/2 unit)

This course follows MATH 6 and focuses on (1) additional topics in statistics, including linear regression, nonparametric methods, discrete data analysis, and analysis of variance; (2) efficient use of statistical software in data analysis and statistical inference; and (3) writing and presenting statistical reports, including graphics. The MATH 6; 26 sequence provides a foundation for statistical work in applied fields such as econometrics, psychology, and biology. It also serves as preparation for study of theoretical probability and statistics. Prerequisite: MATH 6.

MATH 33  Differential Equations
Slack  (1/2 unit)

Differential equations arise naturally to model dynamical systems such as occur in physics, biology and economics, and have given major impetus to other fields in mathematics, such as topology and the theory of chaos. This course covers basic analytic, numerical and qualitative methods for the solution and understanding understanding of ordinary differential equations. Computer based technology will be used. Prerequisite or Co-requisite: Math 21.

MATH 46  Mathematical Statistics
Hartlaub (1/2 unit)

This course follows MATH 36 and introduces the mathematical theory of statistics. Topics include sampling distributions, point estimation, interval estimation, and hypothesis testing; these will also be applied to real data sets. Prerequisite: MATH 36.

MATH 47  Mathematical Models
Jones (1/2 unit)

This course introduces students to the concepts, techniques, and power of mathematical modeling. Both deterministic and probabilistic models will be explored, with examples taken from the social, physical, and life sciences. Students engage cooperatively and individually in the formulation of mathematical models and in learning mathematical techniques used to investigate those models. Prerequisites: MATH 6 and MATH 12 or permission of instructor.

MATH 52  Complex Functions
C. Schumacher (1/2 unit)

The course starts with an introduction to the complex numbers and the complex plane. Next students are asked to consider what it might mean to say that a complex function is differentiable (or analytic as it is called in this context) . For a complex function that takes a complex number z to f(z), it is easy to write down (and make sense of) the statement that f is analytic at z if

MATH 94  Individual Study
Staff  (1/2 unit)

This course enables students to study a topic of special interest under the direction of a member of the mathematics department. Prerequisites: permission of instructor and department chair.

MATH 96  Junior Honors Seminar
Shults, Staff  (1/2 unit)

The goals of the Junior Honors Seminar are twofold: to develop a greater understanding of a broad selection of mathematical topics, and to gain the experience of independent exploration in mathematics. Students will work under close supervision of a faculty member on three areas of interest. Topics of study will be chosen by the student. In culmination of the course, each student will write a proposal describing his or her plan of study for Senior Honors. Prerequisite: permission of department.

MATH 98  Senior Honors
Holdener, Staff  (1/2 unit)

The content of this course is variable and adapted to the needs of senior candidates for honors in mathematics. Prerequisite: permission of department.

Our intuitions about sets, numbers, shapes, and logic all break down in the realm of the infinite. The paradoxical facts about infinity are the subject of this course. We will discuss what infinity is, how it has been viewed through history, why some infinities are bigger than others, how a finite shape can have an infinite perimeter, and why some mathematical statements can neither be proved nor disproved. This will very likely be quite different from any math course you have ever taken. Surprises at Infinity focuses on ideas and reasoning rather than algebraic manipulation; a calculator will be entirely useless.

The class will be a mixture of lecture and discussion, based on selected readings. You can expect essay tests and frequent writing assignments. Prerequisites: none.

Part of the appeal of number theory, the study of the properties of the system of whole numbers, is the lure of the unknown: even a beginner can understand problems that the greatest mathematicians in history have been unable to solve. In this course, we will probably not solve them either, but we will learn what they are. We will also learn about such topics as primes and prime factorization, perfect numbers, arithmetic modulon, Diophantine equations, AFermat's Last Theorem,@ and possibly continued fractions or quadratic number fields.

The only prerequisites are a good understanding of high-school algebra and an interest in learning mathematics for its own sake. Prospective majors and students who only plan to take one or two math courses in college are equally welcome. Enrollment limited to first- and second-year students.

Discrete mathematics is concerned with modes of reasoning and mathematical techniques that are useful in investigating questions about large (but finite) sets or intricate relationships among the members of a large set. Such questions abound in the contemporary world. This course focuses on techniques of analysis and problem solving that are especially appropriate for students interested in such studies as computer science, sociology, government, or urban planning. Mathematical topics include Boolean algebra, graphs, trees, combinatorial methods of counting, finite induction, and recursion. Prerequisite: MATH 22 or permission of instructor.

The Elements of Euclid, written over two thousand years ago, is a stunning achievement. The Elements and the Non-Euclidean Geometries discovered by Bolyai and Lobachevsky in the nineteenth century formed the basis of modern Geometry. From this start, our view of what constitutes Geometry has grown considerably. This is due in part to many new theorems that have been proved in Euclidean and non-Euclidean geometry but also to the many ways in which Geometry and other branches of mathematics have come to influence each other over time. Geometric ideas have widespread use in analysis, linear algebra, differential equations, topology, graph theory and computer science, to name just a few. These fields, in turn, affect the way that geometers think about their subject. Students in MATH 30 will consider Euclidean Geometry from an advanced standpoint, but will also have the opportunity to learn about several non-Euclidean geometries such as (possibly) the Poincare plane, geometries relevant to special relativity, or the geometries of Bolyai and Lobachevsky. In addition, the course may take up topics in differential geometry, topology, vector space geometry, mechanics, or others depending on the interests of the students and the instructor. Prerequisite: MATH 22 or permission of instructor.

This course presents a study of the major topics of classical numerical analysis. These include the solution of nonlinear equations, interpolation and approximation, numerical integration, matrices and systems of linear equations, and the solution of differential equations. The course requires extensive use of the computer. Prerequisites: MATH 18 and 21 or permission of department.

Topology is a relatively new branch of geometry that studies very general properties of geometric objects, how these objects can be modified, and the relations between them. Three key concepts in topology are compactness, connectedness, and continuity, and the mathematics associated with these concepts is the focus of the course. Compactness is a general idea helping us to more fully understand the concept of limit, whether of numbers, functions, or even of geometric objects. For example, the fact that a closed interval (or square, or cube, or n-dimensional ball) is compact is required for basic theorems of calculus. Connectedness is a concept generalizing the intuitive idea that an object is in one piece: the most famous of all the fractals, the Mandelbrot Set, is connected, even though its best computer graphics representation might make this seem doubtful. Continuous functions are studied in calculus, and the general concept can be thought of as a way by which functions permit us to compare properties of different spaces or as a way of modifying one space so that it has the shape or properties of another. Economics, chemistry, and physics are among the subjects which find topology useful. The course will touch on selected topics which are used in applications. Prerequisite: permission of instructor.

This is an analysis course with variable content, depending on the needs and interests of the students. Prerequisite: MATH 41.

This course deepens the studies begun in MATH 24. Topics will vary depending on the needs and interests of the students. However, the topics are likely to include some of the following: abstract vector spaces, linear mappings and canonical forms, linear models and eigen vector analysis, inner product spaces. Prerequisite: MATH 24.

This course picks up where MATH 35 ends. In MATH 65, however, the focus is on using the tools considered in Abstract Algebra I. Mathematicians and scientists apply the fundamental algebraic notions of group, ring, and field to a wide variety of mathematical areas and scientific disciplines; in MATH 65, the student explores these applications. The structure will be that of a topics course, the focus being on classical problems that can be solved (and historically were solved) using algebraic structures as tools.

Topics that may be considered include insolvability of a quintic polynomial, the factoring of polynomials (just as in high school, but over arbitrary rings rather than the real numbers), the classification of finite simple groups (something proven very recently--1981), special cases of "Fermat's Last Theorem," Eisenstein's criterion for irreducibility, the beautiful subject of Galois theory, and more. The class may borrow knowledge from subjects including linear algebra, number theory, complex numbers, calculus, and computer programming, though all one needs to know about these subjects will be convered in class. Prerequisite: MATH 35.