Mathematics

OVERVIEW

Mathematics at St. Olaf is interesting, exciting, accessible, and an appropriate area of study for all students. Over the past years, more than 10 percent of graduating seniors have had mathematics majors. The department offers courses in several mathematical sciences -- pure mathematics, applied mathematics, statistics, computer science, operations research and mathematics education.

In addition, special concentrations in computer science and statistics may be earned in conjunction with any major. For further information on these concentrations, consult the Index.

General Education Credit

Distribution Credit

Prerequisites

Prerequisites for mathematics courses are found in the course descriptions.

REQUIREMENTS FOR THE MAJOR

A contract for a mathematics major normally includes two semesters of elementary calculus, linear algebra and seven or more intermediate or advanced mathematics courses, including Mathematics 244, 252, at least one applied course (266, 270, 312, 316, 330, 390, or appropriate seminar) and at least one course from the core of classical mathematics (340, 344, 348, 352, 356, 364, 370, or approp riate seminar). Every contract must contain intermediate or advanced course work in each of the general areas of analytic, axiomatic, and applied mathematics. Courses in other departments that make extensive use of mathematical techniques may be allowed as substitutes for mathematics courses when preparing a contract. In addition to course work, each mathematics major should engage in some independent mathematical activity, e.g., problem solving, tutoring, work as a course assistant, independent study, an internship, or a special research project.

Mathematics majors who intend to teach secondary school mathematics must meet the above requirements. Their contracts must include Mathematics 244, 252, 262, 356, a course in Foundations, and Education 350 in order to meet the State of Mi nnesota certification requirements. Students wishing a teaching minor should also submit a contract. These should emphasize breadth and will normally include the equivalent of six courses, including Mathematics 244, in addition to Education 350.

Placement

COURSES

109 Calculus with Algebra I

The first in a two-course sequence that integrates precalculus and first-semester calculus topics. The quest is for command of the words, graphs, and symbols that are the world's basic vocabulary for quantifying and communicating astronomical, chemical, meteorological, economic, biological, and many other rates of change. Designed for students not ready to begin Mathematics 120. Prerequisite: Mathematics Placement Recommendation. Does not satisfy distribution requirement. Fall Semester only.

112 Elementary Statistics

An introduction to concepts in statistics using a more mathematical approach than Statistics 110. Topics include descriptive measures, probability, random variables, binomial and normal distributions, estimation and hypothesis testing, contingency tables, analysis of variance, regressions and correlation. Computer applications using Minitab are integrated throughout. Designed for behavioral and health science students. Prerequisite: Mathematics Placement Recommendation. Offered both semesters.

114 Finite Mathematics

This course focuses on mathematical modeling by looking at problems in behavioral and life sciences from both geometric and quantitative points of view. For example, linear programming, a technique for solving optimization problems, makes use of a sophisticated symbolic algorithm that can be understood by considering the process from a geometric point of view. In addition to linear programming, the course introduces linear models, matrix theory, combinatorics, probability, statistics, and Markov chains, as well as several computer applications. Prerequisite: Mathematics Placement Recommendation. Interim only.

117 Gateways to Mathematics

Learn the principles of mathematical thinking by investigating a particular mathematical topic. Topics vary among sections but samples include dynamic geometry, puzzles and recreational mathematics, and cryptology. Students will investigate ideas through technical and non-technical reading, and problem solving, introducing them to mathematical literature and exposition. Intended for students with standard precalculus preparation. Fall offering open only to first-year students; spring offering open to first- and second-year students. Prerequisite: Mathematics Placement Recommendation. Offered both semesters.

119 Calculus with Algebra II

The continuation of Mathematics 109, completing preparation for Mathematics 126. Prerequisite: Mathematics 109. Spring Semester only.

120 Calculus I

Calculus is among the most important developments in human history. It is not only the language of physical science but is also used increasingly to model phenomena in the biological and social sciences. This course introduces differential and integral calculus of functions of a single real variable, including trigonometric, exponential, and logarithmic functions. Derivatives and integrals are explored graphically, symbolically, and numerically. Applications of the derivative are included. Prerequisite: Mathematics Placement Recommendation. Credit may be earned for either Mathematics 120 or 122 but not both. Offered both semesters.

122 Mathematical Analysis I

An introductory honors course in calculus open only by invitation to registrants with superior preparation and ability. Covers the subject matter of Mathematics 120 in greater depth and includes supplementary material. Prerequisite: Mathematics Placement Recommendation. Credit may be earned for either Mathematics 120 or 122, but not both. Fall Semester only.

126 Calculus II

This continuation of Mathematics 120 concentrates on methods and applications of integration and infinite sequences and series. May also include elementary differential equations and multiple integrals. Prerequisite: Mathematics 119 or 120 or 122, or Mathematics Placement Recommendation. Credit may be earned for either Mathematics 126 or 128, but not both. Offered both semesters.

128 Mathematical Analysis II

This continuation of Mathematics 122 covers the material in Mathematics 126 in greater depth and includes supplementary material. Prerequisite: Mathematics 122 or Mathematics 119 or 120 and permission of the instructor, or Mathematics Placement Recommendation. Credit may be earned for either Mathematics 126 or 128 but not both. Offered both semesters.

210 Principles of Mathematics

Selected topics demonstrate the scope and power of mathematics. Students will work on creative problem-solving, placing mathematical ideas within the context of their historical development, and making connections to other disciplines.Intended for students with weak backgrounds in mathematics. Not open to first-year students. (Formerly offered as Mathematics 110.) Prerequisite: Mathematics Placement Recommendation. (Interim only.)

220 Elementary Linear Algebra

This course beautifully illustrates the nature of mathematics as a blend of technique, theory, abstraction, and applications. The important problem of solving systems of linear equations leads to the algebra of matrices, determinants, vector spaces, bases and dimension, linear transformations, and eigenvalues. Prerequisite: Mathematics 114, 119, 120, or 122. Credit may be earned for either Mathematics 220 or 222, but not both. Offered both semesters.

222 Elementary Linear Algebra

This introductory honors course in linear algebra covers the material in Mathematics 220 in greater depth and includes supplementary material. Prerequisite: Mathematics 114, 119, 120, or 122. Credit may be earned for either Mathematics 220 or 222, but not both. Offered both semesters.

226 Multivariable Calculus

This course extends important ideas of single-variable calculus (derivatives, integrals, graphs, approximation, optimization, fundamental theorems, etc.) to higher-dimensional settings. These extensions make calculus tools far more powerful in modeling the (multi-dimensional) real world. Topics include partial derivatives, multiple integrals, transformations, Jacobians, line and surface integrals, and the fundamental theorems of Green, Stokes, and Gauss. Prerequisites: Mathematics 126 or 128, and 220 or 222. Offered both semesters.

230 Introduction to Differential Equations

In the 17th century, Isaac Newton formulated the laws that govern motion and invented a new mathematical language with which to express these laws, namely differential equations. Differential equations are now essential tools for describing a wide variety of phenomena. This course introduces differential equations and analytical, numerical and graphical techniques for their analysis. Applications will be selected from areas such as biology, chemistry, economics, ecology, and physics. Students will use computers extensively to calculate and visualize results. Prerequisite: Mathematics 126 or 128 and 220 or 222. Offered both semesters.

232 Discrete Mathematics

Prepare for the 21st century by studying the branch of mathematics that models the world of information processing and social decision-making. Discrete (noncontinuous) mathematics has become increasingly important as more situations are investigated, represented, and solved using computers (essentially discrete machines). Students will explore finite graphs, recurrence relations, and combinatorial optimization using problem solving techniques and algorithm design strategies. Formerly offered as Mathematics 214 Prerequisite: Mathematics 119, 120, or 122, or permission of the instructor. Interim only, 1995-96 and alternate years.

238 Elementary Number Theory

The great mathematician Gauss called number theory "the Queen of Mathematics." An ancient area of mathematics which continues to be studied by some of today's best mathematicians, number theory is the study of the properties of the integers and related number systems. Part of the appeal of number theory is that many extremely difficult problems can be stated in very simple terms. Topics include divisibility, distribution of primes, and modular arithmetic. Prerequisite: Mathematics 214, 220 or 222. Interim only, 1994-95 and alternate years.

244 Elementary Real Analysis

This course introduces students to the theory of calculus and to tools for communicating these ideas with technical accuracy and sophistication. The goal is mastery of the concepts (e.g., limit, continuity, derivatives, and integrals) necessary to verify the fundamental results of the subject. Highlights include proofs of the Fundamental Theorem of Calculus and the continuity of the uniform limit of continuous functions. Includes historically important results, such as the Bolzano-Weierstrass Theorem, and their context in the theory of analysis. Prerequisite: Mathematics 126 or 128. Offered both semesters.

252 Abstract Algebra I

In recent years the study of abstract algebra has become increasingly important not only in mathematics but also in fields like physics, chemistry, and computer science. Algebra is concerned with sets of objects and operations on these sets. In an axiomatic, or abstract treatment one assumes basic properties and then deduces many other properties. Using this method we will study structures known as groups, rings, and fields. Prerequisite: Mathematics 220 or 222 Offered both semesters.

260 Masterpieces of Mathematics

Students learn about "great novels" in English, "great symphonies" in music and "great leaders" in history. This course pursues an analogous goal in mathematics: to study some of the "great theorems" from a historical and mathematical viewpoint. Theorems considered "masterpieces" and studied include Euclid's proof of the Pythagorean theorem, Newton's approximation to pi and Euler's works on infinite series and number theory. Counts as a "foundations" course for Mathematics Education students. Prerequisite: Mathematics 244. Interim only, 1995-96 and alternate years.

262 Probability Theory

"It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge," wrote Laplace in Théorie Analytique des Probabilités. This course combines theory with applications, covering topics in combinatorial analysis, elementary probability measures, conditional probability, random variables, special distributions, mathematical expectations, and limit theorems. Prerequisite: Mathematics 126 or 128. Offered both semesters.

266 Operations Research

An introduction to the mathematics of optimization, this course considers the optimal utilization of valuable resources and the intelligent management of complex systems. Students will thoroughly examine linear programming from both theoretical and applied perspectives. We will discuss problems involving inventory management, transportation, project scheduling, endangered species recovery, resource allocation, and interaction with external markets. We will also consider nonlinear programming, queueing theory, and game theory. Students learn LINGO and make extensive use of several software packages. No prior computer experience is assumed. Prerequisites: Mathematics 126 or 128 and 220 or 222. Spring Semester only.

270 Numerical Analysis

Students will examine solutions of equations, error analysis, solutions of simultaneous linear equations, numerical differentiation and integration, interpolation, least squares approximations, and numerical solutions to ordinary differential equations. Prerequisites: Mathematics 126 or 128, and 220 or 222. Offered in 1995-96 and alternate years.

294 Internship

298 Independent Study

312 Mathematical Statistics

This 20th-century material has rapidly become a cornerstone of many disciplines. We will examine sampling distributions, point and interval estimation, hypothesis testing, analysis of variance, and correlation and regression. Computer applications using Minitab. Prerequisite: Mathematics 262. Offered both semesters.

316 Linear Models and Data Analysis

This course introduces students to statistics as the art of data analysis via exploratory graphical and numerical methods using current data sets from a variety of disciplines. Topics will be chosen from multiple regression/linear model theory; diagnostic analysis and outlier detection; and log-linear models/logistic regression analysis. Prerequisite: Mathematics 312. Fall Semester only.

330 Differential Equations

A sequel to Mathematics 230, this course studies differential equations from a more rigorous mathematical perspective and extends their use as modeling tools. We will examine dynamical systems and their use in the study of chaos and fractals, and partial differential equations and their use to describe complex physical phenomena such as wave motion (essential for electro-magnetics and acoustics) and diffusion (applicable to diverse areas such as heat transport and traffic flow). Mathematical computing will play an important role. Prerequisite: Mathematics 230. Formerly offered as Mathematics 326. Offered in 1995-96 and alternate years.

340 Complex Analysis

The focus of this course is the beautiful Cauchy theory of complex integration and its applications, including Laurent and Taylor series, and conformal mappings. Students of physics, engineering, and theoretical mathematics will find this material of particular importance. Prerequisite: Mathematics 244 or 226. Spring Semester only.

344 Real Analysis

Back in the early m1900s, Henri Lebesgue created an integral which turned out to be superior in many ways to the standard Riemann integral from calculus. In order to accomplish this he needed to develop the theory of measure. The primary goals of this course are to learn the basics of this theory, see how it is used to develop the Lebesgue integral, and appreciate why this integral is better than Riemann's version. Prerequisite: Mathematics 244, or permission of instructor. Spring Semester only.

348 Topology

An introduction to general topological spaces, including separation axioms, compactness, connectedness, and metric and function spaces. We will also study selected topics from differential and algebraic topology. Prerequisite: Mathematics 244, or permission of instructor. Offered in 1996-97 and alternate years.

352 Abstract Algebra II

Topics in finite groups and fields, including Sylow and Galois theory, and field extensions. Prerequisite: Mathematics 252. Offered in 1995-96 and alternate years.

356 Geometry

Students will investigate properties of axiomatic systems by exploring finite geometries, and considering Euclid's axiom system for Euclidean geometry. Changing one of Euclid's axioms gives surprising results in non-Euclidean geometries. Introducing Klein's definition of geometry leads to a transformation approach to Euclidean, similarity, affine and projective geometries (See linear algebra applied!). The concepts of each geometry are explored with dynamic computer geometry programs. Student presentations cover history and applications. Prerequisite: Mathematics 220 or 222 and 244 or 252. Interim only.

364 Combinatorics

This course covers basic enumeration, including generating functions, recursion, inclusion-exclusion, Polya theory, etc. We will also explore topics in graph theory and constructive combinatorics, time permitting. Prerequisite: Mathematics 252; some previous exposure to counting methods is helpful (e.g. Mathematics 262). Offered in 1995-96 and alternate years.

370 Mathematical Logic

Mathematical logic uses mathematical methods to analyze reasoning and to examine what mathematics can and cannot do. It also provides the underlying paradigm on which many developments leading to the present state of intelligent computer systems are based. In the first part of this course we will study the language and rules of inference of predicate logic and investigate the relationships between provability and truth and between a mathematical theory and its models. The second part of the course will introduce applications of logic to computer science. Prerequisite: Mathematics 244 or 252. Offered in 1996-97 and alternate years.

382 Topics in Analytic Mathematics

Intensive work in a special topic of analytic character. Topics vary from year to year and are usually selected from real or complex analysis, differential geometry, or probability theory.

384 Topics in Applied Mathematics

Intensive work in a special topic of an applied character. Topics vary from year to year and are usually selected from statistics, operations research, or differential equations.

388 Seminar

390 Mathematics Practicum

Apply mathematics. Students work in groups on significant problems posed by and of current interest to area businesses and government agencies. The student groups decide on promising approaches to their problem and carry out the necessary investigations with minimal faculty involvement. Each group reports the results of its investigations with a paper and an hour long presentation to the sponsoring organization. Prerequisite: Permission of instructor. (Interim only.)

394 Internship

398 Independent Research

• Mathematics 114 Finite Mathematics

• Mathematics 210 Principles of Mathematics

• Mathematics 232 Discrete Mathematics

• Mathematics 260 Masterpieces of Mathematics

• Mathematics 356 Geometry

• Mathematics 390 Mathematics Practicum

PARACOLLEGE SEMINARS

Applied Mathematics: Real Tools for the Real World

FACULTY