Mathematics
Professors M. ANDERSON, M. SIDDOWAY (associate dean, chair); Associate Professors D. BROWN, A. BRUDER (associate chair), J. BURGE, S. ERICKSON, J. McDOUGALL, M. WHITEHEAD; Assistant Professors B. MALMSKOG, M. MORAN, F. SANCIERBARBOSA, YLVISAKER; Visiting Assistant Professors S. AGRAWAL, D. ELLSWORTH, K. HOGENSON, A. KELLEY, N. NGUYEN; Visiting Professor R. WELLMAN.
The study of mathematics has always been central to the liberal arts, and mathematics has never been more vital for understanding our world than it is today. At Colorado College, mathematics courses emphasize both the practical applications of the subject and its inherent beauty. Majors and minors in mathematics receive a broad perspective on mathematical ideas, working closely with their peers and faculty in small classes. The department is known for being inclusive and welcoming, and it takes pride in helping each student achieve his or her potential. Students in our department participate in many enriching activities, including the Budapest Semester in Mathematics, the Mathematical Contest in Modeling, the Putnam Exam, Research Experiences for Undergraduates (REUs), and our own blockly pizza problems.
Faculty in the department maintain active research programs, and students have ample opportunity to work on research projects in diverse areas of pure and applied mathematics and statistics. Our alumni have gone on to graduate school and academic careers in mathematics and other disciplines, as well as careers in finance, K12 education, medicine, law, engineering, and information technology. A degree in mathematics opens many doors, and closes none.
Major Requirements
In addition to the All College Requirements, a major in mathematics must complete:
 MA126 Calculus 1 and MA129 Calculus 2 (or equivalent)
 MA204 Calculus 3, MA220 Linear Algebra, and MA251 Number Theory (must be completed before declaring the major)
 MA321 Abstract Algebra 1 and MA375 Real Analysis 1
 One of the following (which should be taken in the same year as its prerequisite whenever possible):
 Three other 300400 level oneunit courses, OR two other 300400 level oneunit courses AND two other 200 level one unitcourses not meeting one of the requirements above, (excluding Independent Studies: MA255, MA355, MA455). EC404 can also count as a 200level elective.
A student majoring in mathematics must also:
 Attend at least four departmental Fearless Friday talks or departmentapproved talks after declaring the major, and submit a summary of each talk on Canvas within three weeks of the seminar. These writeups must be submitted by the beginning of Block 7 of the student's senior year.
 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 300400 level electives needed for the major.
 Complete a senior thesis. The student must enroll in one block of MA499 (Senior Thesis) with his or her thesis advisor during the senior year. Students must submit proposals for their thesis topics in the spring semester of their junior year.
To be considered for graduation with Distinction in Mathematics, a student must complete three courses with a 300level prerequisite, one of which must be MA410. In addition, such students must complete a senior thesis as their capstone experience. Awarding of distinction is approved by a vote of the department faculty, who consider the depth and quality of the student's academic work and the quality of their senior thesis, among other factors. Further information is available from the department's website.
Minor Requirements
To minor in mathematics, a student must either:
Successfully complete one of the eight designated “options for a minor in mathematics” listed below.
OR
Successfully complete a mathematics minor designed in consultation with a department member and approved by the department. A plan for a minor must be approved by the department by the end of the first block of the student’s senior year.
 Statistics: MA204, (MA217 or MA218/EV228), MA220, MA313, and MA417
 Applied Variations: MA204, MA220, MA315, and two of (MA313, MA416, MA311, or MA418)
 Including the MA311 option would constitute a mathematical physics minor.
 Analysis Variations: MA204, MA220, MA251, MA375, and (MA315, MA475, or MA417)
 Discrete: (MA201 or MA202), MA220, MA251, MA325, and (CP222, or MA321, CS majors must select the MA321 option)
 Algebra: MA204, MA220, MA251, MA321, and MA421
 Pure: MA204, MA220, MA251, MA300, and (MA321 or MA375)
 Mathematical Biology: MA204, MA256, MA220, MA315, and (MA313 or MA416)
 Modeling: MA204, MA217, MA220, MA313, and MA315
Courses
Mathematics
 MA110 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. (Not offered 201718).
Prerequisite: Not recommended for Math majors.
.5 to 1 unit
 MA117 Probability & Statistics
An introduction to the ideas of probability, including counting techniques, random variables and distributions. Elementary parametric and nonparametric statistical tests with examples drawn from the social sciences and life sciences. (No credit if taken after any other collegelevel 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 — Brown, Erickson, Gordon, Hogenson
 MA123 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. Meets the Critical Perspectives: Quantitative Reasoning requirement.
Prerequisite: consent of instructor.
1 unit — Bush
 MA125 PreCalculus & Calculus
This course covers the same material as 126 together with one block of content 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.
Prerequisite: consent of instructor.
2 units — Howland, Kelley
 MA126 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 — Abrams, Anderson, Bruder, Kelley
 MA127 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 201718).
Prerequisite: One year of high school calculus and consent of instructor.
1 unit
 MA129 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 — Anderson, Erickson, Glen, Malmskog
 MA130 Viruses: The Biology and Mathematical Modeling of Epidemics
Basic overview of viral infections, symptoms, mutations, and viral life cycles, and how politics, history, and culture can affect the spread of viral epidemics. Second block will provide a meaningful research experience using techniques from differential calculus to model viral epidemics and provide a deeper understanding of how calculusbased ideas fit into a biological context. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement.
Prerequisite: High School Calculus, Mathematics 125, Mathematics 126, or consent of instructor.
2 units — Bertrand, Brown
 MA151 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. (Not offered 201718).
2 units
 MA155 Independent Study:

Prerequisite: consent of instructor.
.5 to 1 unit
 MA161 Mathematics in a Cultural Context
(Not offered 201718).
Prerequisite: First Year Experience Course. 1st Years Only.
2 units
 MA175 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 Y ear Experience course; first year students only . Prerequisite: Calculus 1 from high school, or COI (Not offered 201718).
1 unit
 MA201 Foundations of Discrete Mathematics
An introduction to combinatorics, graph theory, and combinatorial geometry. The topics are fundamental for the study of many areas of mathematics as well as for the study of computer science, with applications to cryptography, linear programming, coding theory, and the theory of computing. Meets the Critical Perspectives: Quantitative Reasoning requirement.
Prerequisite: Mathematics 129 or Mathematics 204 or 2 credits of college level calculus with consent of instructor.
1 unit — Hogenson
 MA202 Foundations of Discrete Mathematics: A CrossCultural 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 nonWestern cultures, both historical and traditional. (Not offered 201718).
Prerequisite: 1 high school course in calculus or computer science.
1 to 2 units
 MA203 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 201718).
Prerequisite: (No credit if taken after Mathematics 204.).
1 unit
 MA204 Calculus 3
Sequences and infinite series, nonCartesian 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 — Fulghesu, McDougall, Nguyen
 MA217 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 — Glen, Tinsley
 MA218 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 201718).
Prerequisite: Mathematics 126 or 125 or 127 or HS equivalent (Calculus I).
1 unit
 MA220 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 204 or 2 credits of college level calculus with consent of instructor.
1 unit — Hogenson, Kelley, Malmskog, McDougall
 MA227 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 halfblock.) (Not offered 201718).
Prerequisite: Mathematics 128.
.5 unit
 MA228 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). Pass/Fail grade only. .5 units
Prerequisite: Mathematics 128 or consent of instructor.
.5 unit — Bruder, Erickson, Hogenson, Kelley
 MA229 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. (Not offered 201718).
Prerequisite: Mathematics 126Calculus I May be taken for credit twice.
.5 unit
 MA240 Topics in Mathematics
Special topics in mathematics not offered on a regular basis.
1 unit — Anderson, Thakur
 MA251 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 204 or 2 credits of college level calculus with consent of instructor.
1 unit — Erickson, Watkins
 MA255 Independent Study:

Prerequisite: consent of instructor.
.5 to 1 unit
 MA256 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 — Bruder
 MA300 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.) (Not offered 201718).
Prerequisite: Mathematics 220 and 251.
1 unit
 MA311 Vector Analysis
Vector functions, divergence and curl, Green's and Stokes' theorems, and the properties of threedimensional curves and surfaces. Related topics from linear algebra and differential equations. (Not offered 201718).
Prerequisite: Mathematics 204.
1 unit
 MA313 Probability
Probability spaces, discrete and continuous random variables, independence, expectation, distribution functions
Prerequisite: Mathematics 204 and 220.
1 unit — Glen, Kelley
 MA315 Ordinary Differential Equations
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 204 and 220.
1 unit — Bruder, Nguyen
 MA321 Abstract Algebra I
An introduction to the abstract algebraic properties of groups, rings and fields.
Prerequisite: Mathematics 220 & 251.
1 unit — Malmskog, Smith
 MA325 Graph Theory
A study of graphs as finite mathematical structures. Emphasis on algorithms, optimization and proofs. (Offered alternate years.)
Prerequisite: Mathematics 220 & either 251 or 201.
1 unit — Hogenson
 MA340 Topics in Mathematics:
Special topics in mathematics not offered on a regular basis.
1 unit — Moran
 MA345 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 201718).
Prerequisite: consent of instructor.
.5 to 1 unit
 MA355 Independent Study:

Prerequisite: consent of instructor.
.5 to 1 unit
 MA375 Real Analysis l
An introduction to the theoretical basis for the calculus, with an emphasis on rigorous proof. Properties of the real number system; sequences and series; continuity; elementary topology of the real line, Euclidean space and metric spaces; compactness; pointwise and uniform convergence.
Prerequisite: Mathematics 204, 220 and 251.
1 unit — Anderson
 MA392 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 201718).
1 unit
 MA400 Topology
An introduction to the study of pointset topology. Examples of topological spaces; compactness, connectedness, and continuity; separation axioms. Additional topics chosen from algebraic or geometric topology. (Offered alternate years.) (Not offered 201718).
Prerequisite: Mathematics 375.
1 unit
 MA408 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 indepth 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 — Wilson
 MA410 Complex Analysis
The calculus of functions of a complex variable. Differentiation, contour integration, powerseries, residue theory and applications, conformal mapping and applications.
Prerequisite: consent of instructor or Mathematics 375.
1 unit — McDougall
 MA416 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 — Brown
 MA417 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 201718).
Prerequisite: consent of instructor or Mathematics 313.
1 unit
 MA418 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 315 or some experience of ordinary differential equations with consent of instructor.
1 unit — Glen
 MA421 Abstract Algebra ll
Continuation of Mathematics 321. Topics may include Galois theory, commutative algebra, computational algebra, representations of finite groups, or algebraic geometry.
Prerequisite: Mathematics 321.
1 unit — Anderson
 MA440 Special Topics in Math:
Given on demand for a group of students interested in a topic not included in the regular curriculum. (Not offered 201718).
1 unit
 MA455 Independent Study:

1 unit — Erickson, McDougall, Moran
 MA475 Real Analysis ll
Continuation of Mathematics 375. A rigorous treatment of derivatives and integrals of a single variable. Other topics, chosen by the instructor, may include a rigorous approach to multivariable calculus; the implicit and inverse function theorems; analysis on manifolds; dynamical systems; measure theory and the Lebesgue integral; functional analysis.
Prerequisite: Mathematics 375.
1 unit — Moran
 MA499 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 — Anderson, Brown, Bruder, Erickson