Mathematics

http://www.stolaf.edu/depts/math

Chair, 2000-01: Bruce H. Hanson, complex analysis

Faculty, 2000-01: Richard J. Allen, logic programming, intelligent tutoring systems; Peder A. Bolstad, precalculus, graph theory; Pam Brethorst, visiting master teacher; Richard A. Brown, computer science, distributed systems; Judith N. Cederberg, geometry; Clifton E. Corzatt, number theory, combinatorics; Jill M. Dietz, algebraic topology; Phil Gloor, harmonic analysis; Paul D. Humke, real analysis, dynamical systems; Michael Kahn, probability, applied statistics; Steven McKelvey, operations research, wildlife modeling; David Molnar, ergodic theory; Miriam Newton, statistics; Arnold M. Ostebee, applied mathematics; Matthew Richey, mathematical physics, computational mathematics; Kay E. Smith, logic, discrete mathematics; Lynn Steen, analysis, education; Theodore A. Vessey, probability, complex analysis; Martha Tibbetts Wallace, mathematics education; Paul Zorn, complex analysis, computer algebra systems

Mathematics, as the study of patterns and order, is a creative art, a language, and a science. The practice of mathematics combines the aesthetic appeal of creating patterns of ideas with the utilitarian appeal of applications of these same ideas. Known as the language of physical science, mathematics is also used increasingly to model phenomena in the biological and social sciences. As technology becomes more pervasive, mathematical literacy is an indispensable skill in today's society. A knowledge of mathematics has become the key to the fastest growing careers.

Mathematics at St. Olaf is interesting, exciting, accessible, and an appropriate area of study for all students. Each year, approximately seven to 10 percent of graduating seniors have 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.

REQUIREMENTS FOR THE MAJOR

Students arrange a major in Mathematics by developing individualized contracts tailored to their particular interests. These contracts should include courses that represent a complete, coherent program of study consistent with the goals of the individual student.

A contract normally includes two semesters of elementary calculus, linear algebra and seven or more intermediate or advanced mathematics courses, including 244, 252, at least one applied course (266, 312, 316, 330, 384, seminar) and at least one course core course (340, 344, 348, 352, 356, 364, 370, or seminar).

Courses in other departments and programs that make extensive use of mathematical techniques may be allowed as substitutes for mathematics courses. These include, but are not limited to, courses from Economics, Physics, Chemistry, and Statistics.

Mathematics majors who intend to teach secondary school mathematics must meet the above requirements (see also the Education Department description and the Mathematics Licensure Adviser). Their contracts must include Mathematics 232, 244, 252, 262, 356, a course in statistics, and Education 350 in order to meet the State of Minnesota licensure requirements. Students wishing a teaching minor should also submit a contract. These should emphasize breadth and will normally include the equivalent of six courses, in addition to Education 350.

COURSES

109 Calculus with Algebra I

The first in a two-course sequence that integrates precalculus and first-semester calculus topics, this course seeks 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. The course is designed for students not ready to begin Mathematics 120; it does not satisfy the general education requirement. Prerequisite: Mathematics Placement Recommendation. Fall Semester only.

114 Finite Mathematics

Students explore mathematical modeling by looking at problems in behavioral and life sciences from both geometric and quantitative points of view. For example, linear programming uses 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, Markov chains, and several computer applications. Prerequisite: Mathematics Placement Recommendation. GE: MAR. Interim only. Offered periodically.

117 Gateways to Mathematics

Students learn principles of mathematical thinking by investigating a particular mathematical topic. Recent topics included dynamic geometry, mathematics of games, and cryptology. Students investigate ideas through technical and non-technical reading and problem solving, introducing them to mathematical literature and exposition. Offered both semesters. The course is intended for students with standard precalculus preparation. GE: MAR.

119 Calculus with Algebra II

This course is the continuation of Mathematics 109, completing preparation for Mathematics 126. Prerequisite: Mathematics 109. GE: MAR. Spring Semester only.

120 Calculus I

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. GE: MAR. Always offered Fall Semester.

122 Mathematical Analysis I

This introductory honors course in calculus is open only by invitation to registrants with superior preparation and ability. The course 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. GE: MAR. 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. GE: MAR. 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. GE: MAR. 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. GE: MAR. Offered 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. GE: MAR. 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. Offered both semesters.

230 Introduction to Differential Equations

This course introduces differential equations and analytical, numerical and graphical techniques for their analysis. First and second order differential equations and linear systems will be studied. 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. Offered both semesters.

232 Discrete Mathematics

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. Prerequisite: Mathematics 119, 120, or 122, or permission of the instructor. Offered alternate years.

234 The Structure of Higher Mathematics (abroad)

This course provides students with a transition from calculus and linear algebra to more advanced courses in theoretical mathematics. The unique feature of this course is that it is taught with a cultural context in Budapest, Hungary. This course not only supplies a bridge from beginning to advanced mathematics, but also allows the participants to encounter one of the important world-wide centers of mathematics. Prerequisite: Mathematics 220. Offered Interim only.

244 Elementary Real Analysis

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

248 Knot Theory

Knots are formally presented in a mathematical context. Techniques from algebra and linear algebra are applied to study properties of knots and their classification. The course culminates in scientific applications involving chirality of molecules in chemistry and DNA supercoiling in biology. Prerequisite: Mathematics 220 Elementary Linear Algebra. Offered Interim only, 2000-01, and then alternate years.

252 Abstract Algebra I

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 study structures known as groups, rings, and fields. Prerequisite: Mathematics 220. 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. Prerequisite: Mathematics 244. Offered periodically, usually during Interim.

262 Probability Theory

This course is an introduction to the mathematics of randomness and games of chance. Topics include combinatorial analysis, elementary probability measures, conditional probability, random variables, special distributions, expectations, generating functions and limit theorems. Prerequisite: Mathematics 126 or 128. Offered both semesters.

266 Operations Research

Students are introduced to modeling and mathematical optimization techniques (e.g., linear programming, network flows, discrete optimization, constrained and unconstrained nonlinear programming, queuing theory). The course emphasizes applications, but prior computer experience is not assumed. Prerequisites: Mathematics 126 or 128 and 220. Recommended: Mathematics 226 and/or 262. Spring Semester only.

294 Internship

298 Independent Study

312 Mathematical Statistics

This 20th-century material has rapidly become a cornerstone of many disciplines. Students examine sampling distributions, point and interval estimation, hypothesis testing, analysis of variance, and correlation and regression. The course utilizes 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. Students 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 and diffusion. Mathematical computing plays an important role. Prerequisite: Mathematics 230. Formerly offered as Mathematics 326. Offered periodically.

340 Complex Analysis

Complex analysis is, roughly speaking, the calculus of complex-valued functions of a complex variable. Familiar words and ideas from ordinary calculus (limit, derivative, integral, maximum and minimum, infinite series) reappear in the complex setting. Topics include complex mappings, derivatives, and integrals; applications focus especially on the physical sciences. Recommended for mathematicians, physicists, and engineers.

344 Real Analysis

Main topics of this course are measure theory on the real line and the Lebesgue integral (an improved version of the standard Riemann integral from calculus). Prerequisite: Mathematics 244, or permission of instructor. Offered in 1999-2000 and then alternate years in Spring Semester.

348 Topology

This course is an introduction to topological spaces and their structure from point-set, differential, and algebraic points-of-view. Topics may include separation axioms, compactness, connectedness, classification of surfaces, homology, fundamental group, and others. Prerequisite: Math 244 or 252. Offered in 2000-2001 and then alternate years in Spring Semester.

352 Abstract Algebra II

This course offers a continuation of group theory and field theory, including group actions, Sylow theory, and Galois theory. Other topics may include representation theory, module theory, and more, depending on the instructor. Prerequisite: Math 252. Offered in 2000-2001 and then alternate years in Spring Semester.

356 Geometry

Properties of axiomatic systems are illustrated with finite geometries and applied in a synthetic examination of Euclid's original postulates, well-known Euclidean theorems, and non-Euclidean geometries. Euclidean, similarity and affine transformations are studied analytically. These transformations are generalized to obtain results in projective geometry or used to generate fractals in an exploration of fractal geometry. Dynamic geometry software and hands-on labs are used to explore both the transformations and properties of these geometries. Prerequisite: Mathematics 220 and 244 or 252. Interim only.

364 Combinatorics

This course covers basic enumeration, including generating functions, recursion, inclusion-exclusion, Polya theory, etc. Students 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 1999-2000 and then alternate years in Fall Semester.

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 intelligent computer systems are based. Initially, students 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. Later, they explore applications of logic to computer science. Prerequisite: Mathematics 244 or 252. Offered in 2000-2001 and then alternate years in Fall Semester.

382 Topics in Analytic Mathematics

Students work intensively 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

Students work intensively 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

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