MATHEMATICS AND COMPUTER SCIENCE

Description of

the Mathematics

Curriculum and Major

Grinnell College

 


Mathematics is an important discipline in the liberal arts education, not only as an independent discipline, but also in how it interracts with other communities, providing a common language and set of tools for many disciplines in the natural and social sciences. As a field with a long history, mathematics has many subdisciplines and strands of thought.

At Grinnell, the mathematics curriculum provides appropriate background for students with a wide range of interests. For example, a very large percentage of Grinnell students take statistics, calculus, linear algebra, combinatorics, and/or differential equations as part of their liberal-arts education and because these subjects are vital for their interests in the social sciences and sciences. Mathematics majors and others with strong interests in mathematics take further courses to deepen their understanding and increase the breadth of their knowledge.

Students may enter the study of mathematics at different points, depending upon their background and interests. Those with good preparation normally start in 131, Calculus I; while those with less preparation may start in 123, Functions and Differential Calculus; and those with advanced standing in 133, Calculus II, or 215, Linear Algebra. Thereafter, the student's intellectual curiosity, interests, and abilities and the needs of various disciplines determine the particular mathematics courses selected. Several courses make use of the department's network of PC/Linux workstations for graphics, computation, data analysis, and numeric experimentation.

Mathematics majors pursue many interests. All are encouraged to study in depth at least one field, such as physics or economics, in which mathematics is applied extensively. Some enjoy working on challenging problems, such as those presented in the Putnam Examination or the Mathematical Contest in Modeling, both of which are national mathematics competitions. Many present talks to the Math Journal Club. Films and visiting lecturers extend the curriculum beyond the classroom, as do opportunities for students to do summer research in mathematics.

The mathematics curriculum starts with a two-semester introduction to calculus, Math 131-133. Following this sequence, the Department offers several choices, which students consider in consultation with their advisor. These choices are intended both for prospective majors in mathematics or computer science and for students who may only take an occasional course in mathematics or computer science.

The department's offerings reflect the fascinating breadth of the mathematics. The following list of course topics, discussed later in this document, indicate something of the variety and broad applicability of fields included in Grinnell's mathematics curriculum.

linear algebra differential equations combinatorics
modeling problem-solving applied mathematics
analysis topology probability and statistics
abstract algebra theory of computation geometry

In addition to these formal courses, students may be involved in a wide range of extra-curricular activities. As discussed in the notes that follow, these activities range from some that are quite mathematical to others that are just for fun.

independent projects summer research internships
Math/CS Journal Club contests/competitions social events

Linear Algebra

Linear algebra concerns itself with linear equations and with concepts and techniques for handling many variables simultaneously. Linear algebra is essential, therefore, for solving problems, so common in the sciences and social sciences, involving linear models or the simultaneous analysis of several variables.

Math 215 Linear Algebra stresses linear systems of equations and linear transformations. In this context, the central topics of linear algebra are studied. Applications of these topics are clearly seen in such areas as fitting lines and curve to data, Markov processes, and linear differential equations. Overall, the course combines the techniques and concepts of linear algebra with major areas of applications.

Differential Equations

Differential equations, which are fundamental to the study of most of the sciences, are equations involving rates of change (or derivatives) of unknown functions. For example, force equations in physics are usually differential equations since they involve a second derivative (acceleration) of an unknown function (position). Similarly, an equation involving the rate of change of a chemical's concentration, a population's size, or a commodity's price is a differential equation.

At Grinnell, the study of differential equations begins with systematic and modern investigation of several types of equations, making solid use of linear algebra, computer experimentation, and applications from various disciplines.

Combinatorics

While calculus considers continuous quantities (e.g., the real line contains no ``holes''), combinatorics deals with counting and with entities which are discrete. For example, one might count the number of poker hands which contain exactly two pairs or use a graph (points and line segments) to model a complex system, such as energy use in the United States.

Math 218 Combinatorics introduces this field, studying the basic objects, numbers, and techniques of combinatorics. The course stresses problem-solving and considers a rather different kind of reasoning and viewpoint than one finds in other mathematics courses. Some topics discussed in this class also are of considerable importance in the areas of statistics and computer science.

Modeling

Math 306 Mathematical Modeling introduces the process and techniques of modeling real-world situations, using topics from calculus, linear algebra, and differential equations. Appropriate mathematics, including numerical methods, are developed as needed. Specific models and examples are drawn from all the sciences. For example, some questions discussed might include whether the nuclear arms race is inherently unstable or how one might design a snow-removal route. The course includes projects where students construct or study a mathematical model related to a field of their own choosing.

Problem-Solving

Math 271 Problem-Solving Seminar (one credit, pass/fail only) engages students in solving challenging mathematics from many branches of mathematics. Some of the problems are similar to those encountered in the Putnam Examination, for which the seminar helps to prepare students.

Applied Mathematics

Math 314 Topics in Applied Mathematics extends the basic theory of linear algebra and differential equations to study the theory behind many important applications of mathematics. The course itself alternates between two foci: partial differential equations (PDEs) and linear programming. PDEs are central to the derivation and analysis of classical equations encountered in mathematics physics, such as the heat equation and the wave equations. Linear programming provides essential background for a wide range of applications in operations research and game theory.

Analysis

Many disciplines in the social sciences, sciences, and engineering utilize the real numbers and functions involving real numbers as the base for all quantitative work. Math 316 Foundations of Analysis provides an in-depth study of this subject. This work extends and formalizes the study of functions of one and several real variables, first introduced in calculus and differential equations.

Beyond a sophisticated knowledge of the real numbers, many fields of science and engineering require a thorough understanding of functions of complex variables. Math 338 Complex Analysis considers this subject, beginning with a preliminary study of the complex number system, and including topics of analysis (e.g., differentiation and integration). Several important physical applications, such as fluid flow around barriers, are often included.

Topology

Topology uses set theory to generalize the notion of distance. This leads to a formal, axiomatic study of many properties that underlie real and complex numbers and geometric phenomena in many dimensions. Math 331 Topology provides a solid introduction to this foundational field.

Probability and Statistics

Statistics deals with the analysis of data where samples tested may contain variability or where experimental error is likely. At Grinnell, the subject is discussed at three different levels.

Math 115 Introduction to Statistics offers a one semester introduction to statistics, assuming only a background of high school algebra. The course is data-oriented with real-world examples drawn from the social and life sciences, and the computer is used for data analysis and to illustrate probability and statistical concepts. (Note that this course may not be counted toward either the mathematics or the computer science major.)

Math 209 Applied Statistics builds upon the calculus background from Math 131 and 133 to provide a one-semester introduction to statistics. The course covers the application of basic statistical methods such as univariate graphics and summary statistics, basic statistical inference for one and two samples, linear regression, one- and two-way analysis of variation, and categorical data analysis. As with Math 115, students use statistical software to analyze data and conduct simulations.

Math 335--336 Probability and Statistics I and II provide a full year introduction to probability and statistics, including some of the mathematical framework for these subjects. The first semester focuses on probability, while the second uses probability theory to study mathematical statistics, including principles of estimation and testing, regression, sampling distributions, analysis of variance, and non-parametric inferences. The course emphasizes applications of statistics to real-world data.

Abstract Algebra

Abstract algebra studies the structures and operations of algebra in a formal and axiomatic way. The Grinnell curriculum introduces formal algebraic structures with Math 321 Foundations of Abstract Algebra, a one-semester course covering the basics of groups, rings, and fields. Students can build on this base in two important ways:

Math 324 Number Theory applies the basic algebraic structures to explore basic properties, such as congruences, quadratic reciprocity, and unique factorization.

Math 326 Field Theory extends the study of formal algebraic structures, begun in Math 321, by integrating group theory with field theory and the theory of equations. Fundamental topics of algebra include vector spaces and canonical forms, algebraic extensions, finite and cyclotomic fields, geometric constructions, and Galois Theory.

Theory of Computation

Some important questions in both mathematics and computer science concern the classification of problems and their solutions. Math 341 Automata Theory, Formal Languages, and Computational Complexity, also listed as Computer Science 341, provides an introduction to the nature of problems and to theoretical computer science. Abstract machines are studied, and these devices are used to discuss when solutions can or cannot be found to certain problems and when known solutions are feasible.

Geometry

Math 444 Senior Seminar: Topics in Geometry provides students with an opportunity to apply topics from earlier courses. For example, tools of linear algebra (linear transformations and matrices) and abstract algebra (group theory) can be used to discuss and prove geometric theorems.

Altogether, this seminar provides a capstone experience, affording students the chance to work with an advanced topic in mathematics and preparing them for graduate work or any further experiences in mathematics. Students play an active in class sessions, by developing material on a day-to-day basis and by presenting special topics they have researched.

Extra-Curricular Activities

Independent Projects

Some students choose an independent project because they are strongly interested in some area not covered by our curriculum, while others choose a specialty of one of our faculty so they can carry out guided research. In addition, students who want to be well prepared for graduate work are encouraged to take at least one of the following independents:

Our faculty enjoys working with students on many topics and welcomes the opportunity to explore options for independent study, based on particular student interests.

Summer Research

We encourage students to engage in summer research either here with Grinnell faculty or elsewhere. In recent years, students have worked with our mathematics faculty in such areas as tilings and dissections (geometry), word problems (algebra/group theory), iteration problems (number theory), and graph coloring problems (combinatorial graph theory). In addition, the National Science Foundation sponsors many excellent REU's (Research Experiences for Undergraduates) at colleges and universities around the country.

Internships

Students can participate in an internship either during the academic year or during the summer. This can be a good way to find out what a career might be like in a field that uses mathematics. Both our faculty and Grinnell's Office of Career Development are prepared to help students identify such internships.

The Mathematics and Computer Science Journal Club

For many years, the Department has organized a series of talks covering a wide range of topics. For examples, students may describe the results of their summer research or independent projects, students and faculty may identify solutions to challenging mathematical problems, and student teams may discuss their solutions to problems from the national Mathematical Modeling Competition. When possible, we encourage all students to give at least two talks: the first for the experience, the second to benefit from the experience gained in the first talk.

In addition to student presentations, both faculty and outside speakers talk about current topics in mathematics. Some speakers also highlight opportunities for further study at the graduate level or for careers using mathematics in business.

Contests and Competitions

The Putnam Exam is the oldest U.S. mathematics competition. Each student works individually on 12 tough problems on a Saturday in early December. Our 1-credit Problem-Solving Seminar, Math 271, is good preparation.

The Iowa Mathematics Competition is held every spring, but in this one teams of 3 students work together on the problems. (A Grinnell team has won this competition each of the last four years.)

Different yet is the Mathematical Contest in Modeling, held early every second semester over a long weekend. Teams of 3 work together on a single applied problem that usually has no best answer. The quality of the team's written report determines their success. (Twice a Grinnell team has achieved the highest rating, "Outstanding," and once Grinnell's team was flown out to Boston to present their solution at a national meeting.)

Social Events

We have a Math/CS picnic every fall and spring, which always turns out a large crowd. We also have a breakfast for all seniors and their families on graduation day that is immensely popular. Our Student Educational Policy Committee sometimes schedules events, such as the Math/CS Study Breaks this semester.

Conclusions

The mathematics program at Grinnell College provides a breadth of offerings within mathematics, and students may pursue their interests in many areas of mathematics. During your enrollment at Grinnell, you would want to talk with an advisor about which choices are appropriate for you. And, beyond this formal curriculum, the department encourages numerous activities for further study, research, problem solving, and interaction.


This document is available on the World Wide Web as

http://www.math.grinnell.edu/~walker/dept/math-curriculum.html

created January 26, 1998
last revised September 27, 2001

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For more information, please contact Henry M. Walker at (walker@math.grinnell.edu)