# Mathematics

*Applicable for the 2020-2021 academic year.*

Professors M. ANDERSON (associate chair), M. SIDDOWAY (chair); Associate Professors D. BROWN, A. BRUDER, S. ERICKSON, J. McDOUGALL; Assistant Professors I. AGBANUSI, B. MALMSKOG, M. MORAN, F. SANCIER-BARBOSA; Visiting Assistant Professors S. AGRAWAL, A. KELLEY, N. NGUYEN.

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, K-12 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 student majoring 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):
- One of the following sets of electives (Independent Study courses (MA255, MA355, MA455) cannot be used to satisfy this requirement):
- Three other 300-400 level one-unit courses in Mathematics, not being used to meet one of the requirements above or the capstone requirement
- Two other 300-400 level one-unit courses and two 200-level one unit courses in Mathematics, not being used to meet one of the requirements above or the capstone requirement
- Two other 300-400 level one-unit courses and one 200-level one unit course in Mathematics, not being used to meet one of the requirements above or the capstone requirement, as well as one non-Mathematics course from a list of approved courses on the department’s website.

- All non-transfer students may count at most three 300-400 level courses taken at other institutions (not to include equivalents of MA321 and MA375) toward their Mathematics major, provided that these courses are approved by the department.

**A student majoring in Mathematics must also:**

- Attend at least four departmental Fearless Friday talks or department-approved talks after declaring the major, and submit a summary of each talk on Canvas within three weeks of the seminar. These write-ups 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 300-400 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.

Please visit the department's website for information about Distinction in Mathematics.

## 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

A student minoring in Mathematics must take at least three of the required courses at CC.

## Courses

### Mathematics

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.

An introduction to the ideas of probability, including counting techniques, random variables and distributions. Elementary parametric statistical tests with examples drawn from the social sciences and life sciences. Not recommended for mathematics majors. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement.

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.

This course covers the same material as MA126 together with one block of content from algebra, trigonometry, analytic geometry and the study of functions. Intended solely for students not sufficiently prepared for MA126. (Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement.) Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement.

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.

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 2020-21).

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.

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 calculus-based ideas fit into a biological context. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement. (Not offered 2020-21).

(Summer only 2020-21).

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. (Not offered 2020-21).

(Not offered 2020-21).

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 2020-21).

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.

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 2020-21).

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. Meets the Critical Perspectives: Scientific Investigation of the Natural World requirement. Meets the Critical Perspectives: Quantitative Reasoning requirement.

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.

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 2020-21).

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.

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 2020-21).

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 (Not offered 2020-21).

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 2020-21).

Special topics in mathematics not offered on a regular basis. (Not offered 2020-21).

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.

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.

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 2020-21).

Vector functions, divergence and curl, Green's and Stokes' theorems, and the properties of three-dimensional curves and surfaces. Related topics from linear algebra and differential equations. (Not offered 2020-21).

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

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.

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

A study of graphs as finite mathematical structures. Emphasis on algorithms, optimization and proofs. (Offered alternate years.)

Special topics in mathematics not offered on a regular basis.

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 2020-21).

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.

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 2020-21).

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.)

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.

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

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.

Brief introduction of probability, descriptive statistics, classical and Bayesian statistical inference, including point and interval estimation, hypothesis tests and decision theory. (Offered alternate years.)

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.

Continuation of Mathematics 321. Topics may include Galois theory, commutative algebra, computational algebra, representations of finite groups, or algebraic geometry.

Given on demand for a group of students interested in a topic not included in the regular curriculum. (Not offered 2020-21).

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.

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