# Mathematics

www.coloradocollege.edu/academics/dept/mathematics/

Professors M. ANDERSON, JANKE, M. SIDDOWAY, TINSLEY; Associate Professors BROWN (chair), McDOUGALL, A. TAYLOR (associate chair); Assistant Professors BRUDER, ERICKSON, WHITEHEAD; Visiting Assistant Professors MALMSKOG, PENN. Visiting Lecturer JAMES.

**A student majoring in mathematics must:**

- Attend at least four departmental seminars or department-approved talks after declaring the major, and submit a brief (one or two pages) summary of each talk to the associate chair within two weeks of the seminar. This should be completed by the end of Block 7 of the student’s senior year. Consult the department for further details on what constitutes an approved talk.
- Complete a capstone experience, intended to give the student an opportunity to engage mathematics in a deep and meaningful way. The capstone will challenge the student to read, write, and think about mathematics, drawing on the knowledge and skills that they have acquired throughout their studies. There are two ways to complete the capstone requirement:

- Pass MA408 (History of Mathematics) during the senior year. In this case, MA408 will not count as one of the 300-400 level electives needed for the major.
- Complete a senior thesis. The thesis will involve either original research or substantial expository work. The student must have a thesis advisor in the department, and submit a thesis proposal to the department by the end of Block 3 of his or her senior year. The student must enroll in at least one block of MA499 (Senior Thesis) with his or her thesis advisor during the senior year. Consult the department for further details.

To be considered for graduation with Distinction in Mathematics, a student must complete three courses with a 300-level prerequisite, one of which must be MA410. In addition, the student must select the thesis option of the capstone requirement, and successfully complete a thesis of sufficient scope and depth. Engagement in the intellectual and social life of the department is also considered in the awarding of distinction, which is decided by a vote of the department faculty. Consult the department for further details.

**THE MINOR — REQUIREMENTS:**

To minor in mathematics a student must either:

- Successfully complete one of the “options for a minor in mathematics” listed on the department webpage www.coloradocollege.edu/academics/dept/mathematics/
- Successfully complete a mathematics minor designed in consultation with a department member and approved by the department. A plan for a minor not covered in #1 must be approved by the department by the end of the first block of the student’s senior year.

## Mathematics Courses

### 110 Mathematical Explorations:

An introduction to mathematical thinking through specified topics drawn from number theory, geometry, graph theory, algebra or combinatorics. The course will focus on giving students the opportunity to discover mathematics on their own. No previous mathematical background is required, but students will be expected to come with curiosity and a willingness to experiment. Not recommended for math majors. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement.

*Prerequisite:* Not recommended for Math majors.

1 unit —

### 117 Probability & Statistics

An introduction to the ideas of probability, including counting techniques, random variables and distributions. Elementary parametric and non-parametric statistical tests with examples drawn from the social sciences and life sciences. (No credit if taken after any other college-level statistics course.) Not recommended for mathematics majors. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement.

*Prerequisite:* No credit after BY220/EC200.

1 unit —

### 123 Mathematics for Elementary Educators

Skillful teaching of mathematics requires the teacher to understand the material from a variety of perspectives, and with greater depth than his or her students. This course helps to prepare future elementary teachers by exploring some of the deeper structure of elementary mathematics. Topics will include: counting and cardinality, ratio and proportional relationships, elementary number theory, operations and algebraic thinking, and the role of axioms, deduction, examples, and counterexamples.

*Prerequisite:* consent of instructor.

1 unit —

### 125 Pre-Calculus & Calculus

The same calculus as 126 together with materials from algebra, trigonometry, analytic geometry and the study of functions. Intended solely for students not sufficiently prepared for 126. (Fulfills one unit of the divisional requirement in the natural sciences.) Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement.

1 or 2 units —

### 126 Calculus 1

Differential and integral calculus of algebraic and transcendental functions and applications. Students normally begin the calculus sequence with this course. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement.

1 unit —

### 127 Calculus 1 & 2 Accelerated

An accelerated review of differential and integral calculus of one variable, including a study of the differential calculus for functions of several variables. Designed for students who have already been exposed to topics traditionally included in two semesters of calculus. MA 127 fulfills all requirements met by MA 129; no credit after MA 128 or MA 129. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement. (Not offered 2013-14).

*Prerequisite:* One year of high school calculus and consent of instructor.

1 unit

### 129 Calculus 2

Techniques of integration, applications of the definite integral, differential equations, Taylor polynomials, vectors in two and three dimensions, differential calculus of functions of several variables. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement.

*Prerequisite:* Mathematics 125 or 126. No credit after Mathematics 128.

1 unit —

### 151 The World of Numbers: From Euclid to the Information Age

People have been writing numbers for as long as they have been writing. This course traces the use of numbers from ancient civilizations to modern times and examines how our view of numbers has changed over that period: natural numbers, prime numbers, rational numbers, Fibonacci numbers, real numbers and complex numbers, as well as the way in which our ability to calculate has evolved. Meets the Critical Perspectives: Quantitative Reasoning requirement. Meets the Critical Perspectives: The West in Time requirement.

2 units —

### 155 Independent Study:

*Prerequisite:* consent of instructor.

.5 to 1 unit —

### 161 Mathematics in a Cultural Context

(Not offered 2013-14).

*Prerequisite:* First Year Experience Course. 1st Years Only.

2 units

### 175 Chaos under Control: Computation, Calculus and Order Within Chaotic Systems

Traces the evolution of geometry and dynamics from antiquity to the present, while following the thread of developing technology . Geometry in Euclid’ s time and Aristotle’ s dynamics are inadequate for the study of natural objects such as fern leaves or the weather . Examines how the development of calculating machines has affected and deepened understanding of the natural world. Following the development of early calculating machines into modern day computers, we will see how Newton’ s and Leibniz’ s calculus laid the foundations for the study of differential equations, chaotic and nonlinear dynamics, fractals, and the butterfly effect. First Year Experience course; first year students only . Prerequisite: Calculus 1 from high school, or COI (Not offered 2013-14).

1 unit

### 201 Foundations of Discrete Mathematics

An introduction to sets and logic, relations and functions, combinatorics, graphs, recursion, and algorithms. The topics are fundamental for the study of many areas of mathematics as well as for the study of computer science. Meets the Critical Perspectives: Quantitative Reasoning requirement. (Not offered 2013-14).

*Prerequisite:* Any 100 level Mathematics course or consent of instructor.

1 unit

### 202 Foundations of Discrete Mathematics: A Cross-Cultural Approach

Opportunity to study new mathematical ways of thinking in a cultural context. Much like the division between plants and animals in biology, mathematics can be divided into continuous mathematics (e.g. calculus) and discrete mathematics, the latter of which is the subject of this course. Includes concepts that are fundamental to modern mathematics and computer science. We will also introduce mathematics with important applications to the social sciences. Mathematical topics will be illuminated by examining their treatment in a variety of non-Western cultures, both historical and traditional. (Not offered 2013-14).

*Prerequisite:* 1 high school course in calculus or computer science.

1 to 2 units

### 203 Calculus 3

Vectors in two and three dimensions, and the calculus of functions of several variables. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. (Not offered 2013-14).

*Prerequisite:* (No credit if taken after Mathematics 204.).

1 unit

### 204 Calculus 3

Sequences and infinite series, non-Cartesian coordinate systems, integral calculus for functions of several variables, and the calculus of vector valued functions. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement.

*Prerequisite:* Mathematics 129. (No credit if taken after MA203).

1 unit —

### 217 Probability and Statistical Modeling

Introduction to probability distribution theory and statistical inference. Descriptive methods for building models with emphasis on linear regression models including variance and covariance. Analysis of model fit and discussion of modern robust techniques. (This course is an appropriate first course in statistics for students with stronger mathematical backgrounds.) Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement.

*Prerequisite:* Mathematics 117 or 126.

1 unit —

### 218 Analysis of Environmental Data

This course will focus on the fundamentals of exploratory data analysis, hypothesis testing, and experimental design in the ecological, environmental, and the earth sciences. Topics will include theory and practice of project design, data distribution and description, the central limit theorem, characterization of uncertainty, correlation, univariate hypothesis testing, and multivariate analyses (ANOVA, linear regression). Students will complete a final project using environmental data collected in the field and analyzed using statistical computer software. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement. (Not offered 2013-14).

*Prerequisite:* Mathematics 126 or 125 or 127 or HS equivalent (Calculus I).

1 unit

### 220 Linear Algebra

Matrix algebra and Gaussian elimination. The geometry of vectors in R2, R3 and Rn. Vector spaces and linear transformation. Introduction to orthogonal geometry and eigenvalue problems. Meets the Critical Perspectives: Quantitative Reasoning requirement.

*Prerequisite:* Mathematics 129 or Mathematics 203 or 2 credits of college level calculus with consent of instructor.

1 unit —

### 227 Mathematical Software:

An introduction to one of the major mathematical software packages such as Mathematica or Matlab. Investigation of symbolic computation, numerical algorithms, and graphics as used in these programs. Students may take the course more than once to learn additional software packages, but they may take it a maximum of two times for credit. (May be taught either in the extended format or as a half-block.) (Not offered 2013-14).

*Prerequisite:* Mathematics 128.

.5 unit

### 228 Mathematical Problem Solving Seminar

Students will meet regularly during the semester, in order to learn problem solving techniques as applied to interesting mathematical problems, often drawn from the national William Lowell Putnam competition, or the COMAP Mathematical Modeling Contest. Students may take the course more than once, but at most two times for credit (in different years).

*Prerequisite:* Mathematics 128 or consent of instructor.

.5 unit —

### 229 Seminar in Mathematical Biology

This course will provide a forum for discussing current research and classic papers in mathematical biology. Topics will be chosen that both relate to students' research experiences and broaden their knowledge of mathematical biology. The seminar will also provide a forum for discussing research with visiting scientists. It will meet twice per block for one semester.

*Prerequisite:* Mathematics 126-Calculus I May be taken for credit twice.

.5 unit —

### 240 Topics in Mathematics

Special topics in mathematics not offered on a regular basis. (Not offered 2013-14).

1 unit

### 251 Number Theory

A careful study of major topics in elementary number theory, including divisibility, factorization, prime numbers, perfect numbers, congruences, Diophantine equations and primitive roots. Meets the Critical Perspectives: Quantitative Reasoning requirement.

*Prerequisite:* Mathematics 129 or Mathematics 203 or (Mathematics 128 and Computer Science 222) or 2 credits of college level calculus with consent of instructor.

1 unit —

### 255 Independent Study:

*Prerequisite:* consent of instructor.

1 unit —

### 256 Mathematical Models in Biology

An introduction to selected quantitative models drawn from areas of biology such as ecology, genetics and physiology. For each model, the course includes an investigation of the mathematical methods, an evaluation of the model, and some elementary simulation techniques. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement.

*Prerequisite:* Mathematics 126 & 1 college biology course.

1 unit —

### 300 Geometry

Some current topics in advanced and modern geometry. Topics drawn from linear geometry, affine, inversive and projective geometries, foundations and axiomatics, transformation groups, geometry of complex numbers. (Offered alternate years.)

*Prerequisite:* Mathematics 251.

1 unit —

### 311 Vector Analysis

Vector functions, divergence and curl. Green's and Stoke's theorems, and the properties of three-dimensional curves and surfaces. Related topics from linear algebra and differential equations. (Not offered 2013-14).

*Prerequisite:* Mathematics 203.

1 unit

### 313 Probability

Probability spaces, discrete and continuous random variables, independence, expectation, distribution functions.

*Prerequisite:* Mathematics 203.

1 unit —

### 315 Ordinary Differential Equations

Introduction to methods for finding solutions to differential equations involving a single, independent variable. Topics include linear equations, exact solutions, series solutions. Laplace transforms, Sturm Separation and Comparison Theorems, systems of equations, and existence and uniqueness theorems.

*Prerequisite:* Mathematics 220.

1 unit —

### 316 Partial Differential Equations

Introduction to analytical and numerical methods for finding solutions to differential equations involving two or more independent variables. Topics include linear partial differential equations, boundary and initial value problems, Fourier series solutions, finite element methods, the Laplace equation, the wave equation and the heat equation.

*Prerequisite:* Mathematics 315 or some experience with ordinary differential equations with consent of instructor.

1 unit —

### 318 Numerical Analysis

The development and analysis of algorithms for approximating solutions to mathematical problems. Topics covered include: approximating functions, finding roots, approximating derivatives and integrals, solving differential equations, solving systems of linear equations, and finding eigenvalues.

*Prerequisite:* Mathematics 220.

1 unit —

### 321 Abstract Algebra I

An introduction to the abstract algebraic properties of groups, rings and fields.

*Prerequisite:* Mathematics 220 & 251.

1 unit —

### 322 Abstract Algebra II

Continuation of Mathematics 321.

*Prerequisite:* Mathematics 321.

1 unit —

### 325 Graph Theory

A study of graphs as finite mathematical structures. Emphasis on algorithms, optimization and proofs. (Offered alternate years.) (Not offered 2013-14).

*Prerequisite:* Mathematics 220 & 251.

1 unit

### 340 Topics in Mathematics:

Special topics in mathematics not offered on a regular basis.

1 unit —

### 345 Research in Mathematics

An introduction to the nature of mathematical research. Investigation with a faculty member of current mathematical problems, usually chosen from the field of the faculty member's own research. (Offered in alternate years. May be offered some years as an extended format course for 1/2 unit.) (Not offered 2013-14).

*Prerequisite:* consent of instructor.

.5 to 1 unit

### 355 Independent Study:

*Prerequisite:* consent of instructor.

1 unit —

### 375 Real Analysis l

An introduction to the theoretical basis for the calculus. Sequences and series; topology of the real line; metric spaces; definitions of limit, continuity, compactness.

*Prerequisite:* Mathematics 203 or MA204;MA220; and Mathematics 251.

1 unit —

### 376 Real Analysis ll

Continuation of Mathematics 375. A rigorous treatment of derivatives and integrals, culminating in an introduction to continuity and differentiation of functions of several variables and of vector-valued functions.

*Prerequisite:* Mathematics 375.

1 unit —

### 392 Advanced Topics in Economical Mathematics: Game Theory

Selected topics in the study of Mathematical Economics. Specific content and emphasis are developed by the instructor(s). Topics will meet the ME elective requirement for the Mathematical Economics major. (Not offered 2013-14).

1 unit

### 400 Topology

An introduction to the study of point-set topology. Examples of topological spaces; compactness, connectedness, and continuity; separation axioms. Additional topics chosen from algebraic or geometric topology. (Offered alternate years.) (Not offered 2013-14).

*Prerequisite:* Mathematics 375.

1 unit

### 408 History of Mathematics

A study of selected developments in the history of mathematics and the role of mathematics in different cultures across time. The course often draws on original sources and traces the relationships among different fields within mathematics through the in-depth study of major unifying results. When used to fulfill the capstone requirement for the mathematics department, the course must be taken in the senior year.

*Prerequisite:* Mathematics 321 and 375.

1 unit —

### 410 Complex Analysis

The calculus of functions of a complex variable. Differentiation, contour integration, power-series, residue theory and applications, conformal mapping and applications.

*Prerequisite:* consent of instructor or Mathematics 375.

1 unit —

### 417 Mathematical Statistics

Brief introduction of probability, descriptive statistics, classical and Bayesian statistical inference, including point and interval estimation, hypothesis tests and decision theory. (Offered alternate years.) (Not offered 2013-14).

*Prerequisite:* consent of instructor or Mathematics 313.

1 unit

### 440 Special Topics in Math:

Given on demand for a group of students interested in a topic not included in the regular curriculum. (Not offered 2013-14).

1 unit

### 455 Independent Study:

1 unit —

### 499 Senior Thesis

Advanced work in mathematics on the senior capstone project. Required for all students who are completing their capstone experience through a yearlong project and working towards the required summary seminar and summary paper. This course should be taken in the senior year, during or before Block 6

1 unit —