Mathematics and Statistics

Mathematics is an art, a pure science, a language, and an analytical tool for the natural and social sciences, a means of exploring philosophical questions, and a beautiful edifice that is a tribute to human creativity. The mathematics curriculum is designed to provide essential skills for students in a variety of disciplines and to provide mathematics majors with a deep understanding of mathematics as it has evolved over the past two thousand years and how it is practiced today. The statistics curriculum provides students with opportunities to analyze real data and enhance their communication skills.

Students who wish to major in both Mathematics and Statistics should note the College policy that double majors may count no more than four courses toward both majors. Courses for which a student earns AP Credit, such as calculus, are included among these four courses.

Mathematics Skills Center:

This Center offers extra assistance to students in lower-level mathematics courses and other courses requiring basic mathematical skills.

 

Requirements for the Mathematics Major

The Mathematics major requires 72 credits:

 A. Required Core Courses:

  • Mathematics 101 or 111, 120 or 121, 210 or 211 Calculus
  • Mathematics 232 Linear Algebra
  • Mathematics 236 Mathematical Structures

 B. Electives (36 credits): Six courses from among:

  • Mathematics courses numbered above 236
  • Computer Science 252, 254, 352
  • Statistics 250, 320, 340

At least four of these electives must be Carleton courses with a MATH designation. At least three of the following five areas of mathematics must be represented by the six electives (36 credits).

  • Algebra: Mathematics 312, 332, 342, 352
  • Analysis: Mathematics 251, 261, 321, 331, 361
  • Applied Mathematics: Mathematics 240 (formerly Mathematics 265), 241, 341, Statistics 250 (formerly Mathematics 275), 320 (formerly Mathematics 315), 340 (formerly Mathematics 315)
  • Discrete Structures: Mathematics 333, Computer Science 252, 254, 352
  • Geometry and Topology: Mathematics 244, 344, 354

Of the six advanced courses, at least four must be Carleton courses with a Mathematics designation. Advanced courses substituted for Mathematics 232 or Mathematics 236 must also be Carleton courses with a Mathematics designation.

In addition, each senior major must complete an integrative exercise, Mathematics 400 (6 credits) which can be either a group or individual project. Majors must also accumulate eight talk credits during their junior and senior year by attending colloquia and the comps talks of their fellow mathematics or statistics majors. Students who major in both Mathematics and Statistics must accumulate a total of thirteen talk credits. We encourage majors to participate in the numerous activities that take place in the department.

Potential majors with especially strong preparation may petition the department for permission to substitute an advanced course for Mathematics 232 and/or for Mathematics 236. Advanced courses substituted for Mathematics 232 or Mathematics 236 must also be Carleton courses with a Mathematics designation.

There are many patterns of courses for the major depending upon a student's mathematical interests and career goals. A guide for majors, which supplies information about suitable patterns of courses, is available on the Mathematics and Statistics Department web site.

Major under Combined Plan in Engineering:

In addition to completing requirements for the mathematics major listed above including Mathematics 241 and 341, the student should take the following courses required for admission to engineering schools: Two terms of 100-level Physics, Chemistry 123, 224, and Computer Science 111.

Requirements for the Statistics Major

The requirements for the Statistics Major are 74 credits:

 A.  Supporting Courses (30 credits):

  • Mathematics 101 or 111, 120 or 121, 210 or 211 Calculus
  • Mathematics 232 Linear Algebra
  • Computer Science 111 Introduction to Computer Science

 B.  Required Core (18 credits): All of the following, of which at least two must be taken at Carleton

  • Statistics 230 (formerly Mathematics 245) Applied Regression Analysis
  • Mathematics 240 (formerly Mathematics 265) Probability
  • Statistics 250 (formerly Mathematics 275) Introduction to Statistical Inference

C. Electives (18 credits): Three electives, of which at least two must be Carleton courses with a Statistics designation

  • Statistics 260 (formerly Mathematics 255) Introduction to Sampling Techniques
  • Statistics 220 (formerly Mathematics 285) Introduction to Data Science
  • Statistics 320 (formerly Mathematics 315) Time Series Analysis
  • Statistics 330 (formerly Mathematics 345) Advanced Statistical Modeling
  • Statistics 340 (formerly Mathematics 315) Bayesian Statistics
  • Mathematics 295 Numerical Analysis
  • Computer Science 314 Data Visualization
  • Computer Science 324 Data Mining
  • Computer Science 362 Computational Biology

 D. Statistical Practice (2 credits):

  • Statistics 285 (formerly Mathematics 280) Statistical Consulting 

In addition, each senior major must complete an integrative exercise. Statistics 400 (6 credits), which can be either a group or individual project. Majors must accumulate eight talk credits during their junior and senior year by attending department colloquia and the comps talks of their fellow mathematics or statistics majors. Students who major in both Mathematics and Statistics must accumulate a total of thirteen talk credits. We encourage majors to participate in the numerous activities that take place in the department.

We recommend statistics majors also take courses in a discipline in which statistics can be applied. Students interested in data science should consider taking additional computer science courses.

Students considering graduate school in statistics or biostatistics are strongly encouraged to take Mathematics 236 (Mathematical Structures) and Mathematics 321 (Real Analysis). Consult a statistics faculty member for more information specific to your choice of program.

 

Requirements for the Mathematics Minor

To earn a minor in Mathematics, a student must earn 42 credits from courses taken in the Department of Mathematics and Statistics at Carleton. (Students who place out of courses based on work done outside of Carleton are still required to earn 42 credits from courses taken in the Department of Mathematics and Statistics at Carleton.) At least 36 of the required 42 credits must come from courses with a Mathematics designation. In addition, the only Statistics courses which can be counted toward the Mathematics minor are Statistics 250, 320 and 340.

Students who wish to major in Statistics and minor in Mathematics should note the College policy that a student may not fulfill more than half the credits for a minor from the courses counted toward their major or majors.

Mathematics Courses

MATH 101 Calculus with Problem Solving An introduction to the central ideas of calculus with review and practice of those skills needed for the continued study of calculus. Problem solving strategies will be emphasized. In addition to regular MWF class time, students will be expected to attend two problem-solving sessions each week, one on Monday or Tuesday, and one on Wednesday or Thursday.  Details will be provided on the first day of class. Prerequisite: Not open to students who have received credit for Mathematics 111. 6 credits; FSR; Fall; Deanna B Haunsperger
MATH 111 Introduction to Calculus An introduction to the differential and integral calculus. Derivatives, antiderivatives, the definite integral, applications, and the fundamental theorem of calculus. Prerequisite: Requires placement via the Calculus Placement Exam 1, see Mathematics web page. Not open to students who have received credit for Mathematics 101. 6 credits; FSR; Fall, Winter; Kate J Meyer, Caroline L Turnage-Butterbaugh
MATH 120 Calculus 2 Inverse functions, integration by parts, improper integrals, modeling with differential equations, vectors, calculus of functions of two independent variables including directional derivatives and double integrals, Lagrange multipliers. Prerequisite: Mathematics 101, 111, score of 4 or 5 on Calculus AB Exam or placement via a Carleton placement exam. Not open to students who have received credit for Mathematics 211 or have a score of 4 or 5 on the AP Calculus BC exam. 6 credits; FSR; Fall, Winter, Spring; Steve T Scheirer, MurphyKate Montee, Rafe F Jones, Owen D Biesel
MATH 206 A Tour of Mathematics A series of eight lectures intended for students considering a Mathematics major. The emphasis will be on presenting various striking ideas, concepts and results in modern mathematics, rather than on developing extensive knowledge or techniques in any particular subject area. 1 credit; S/CR/NC; NE; Winter; Kate J Meyer
MATH 207 Communicating Mathematics An introduction to communicating mathematics to a general audience in both writing and speaking. Students will gain practice in presenting their ideas and receive feedback.  Students will use LaTeX and Beamer. Prerequisite: Mathematics 236 or 240 (formerly Mathematics 265) or instructor permission. 2 credits; NE; Not offered 2020-21
MATH 210 Calculus 3 Vectors, curves, calculus of functions of three independent variables, including directional derivatives and triple integrals, cylindrical and spherical coordinates, line integrals, Green's theorem, sequences and series, power series, Taylor series. Prerequisite: Mathematics 120. This course cannot be substituted for Mathematics 211. 6 credits; FSR; Fall, Winter, Spring; Alex J Barrios, Alexander Garver, Rob C Thompson, Caroline L Turnage-Butterbaugh
MATH 211 Introduction to Multivariable Calculus Vectors, curves, partial derivatives, gradient, multiple and iterated integrals, line integrals, Green's theorem. Prerequisite: Score of 4 or 5 on the AP Calculus BC exam, or placement via Calculus Placement Exam #3. 6 credits; FSR; Fall, Winter, Spring; Alexander Garver, Eric S Egge, Owen D Biesel
MATH 232 Linear Algebra Vector spaces, linear transformations, determinants, inner products and orthogonality, eigenvectors and eigenvalues. Prerequisite: Mathematics 120 or Mathematics 211. 6 credits; FSR; Fall, Winter, Spring; Rafe F Jones, Caroline L Turnage-Butterbaugh, Alexander Garver
MATH 236 Mathematical Structures Basic concepts and techniques used throughout mathematics. Topics include logic, mathematical induction and other methods of proof, problem solving, sets, cardinality, equivalence relations, functions and relations, and the axiom of choice. Other topics may include: algebraic structures, graph theory, and basic combinatorics. Prerequisite: Mathematics 232 and either Mathematics 210 or Mathematics 211. 6 credits; FSR; Fall, Winter, Spring; Deanna B Haunsperger, MurphyKate Montee, Alex J Barrios
MATH 240 Probability (Formerly Mathematics 265) Introduction to probability and its applications. Topics include discrete probability, random variables, independence, joint and conditional distributions, expectation, limit laws and properties of common probability distributions. Prerequisite: Mathematics 120 or Mathematics 211. 6 credits; FSR; Fall, Winter; Adam Loy, Josh R Davis, Alexander Garver
MATH 241 Ordinary Differential Equations An introduction to ordinary differential equations, including techniques for finding solutions, conditions under which solutions exist, and some qualitative analysis. Prerequisite: Mathematics 232 or instructor permission. 6 credits; FSR; Winter, Spring; Rob C Thompson, Kate J Meyer
MATH 244 Geometries Euclidean geometry from an advanced perspective; projective, hyperbolic, inversive, and/or other geometries. Recommended for prospective secondary school teachers. Prerequisite: Mathematics 236. 6 credits; FSR; Not offered 2020-21
MATH 251 Chaotic Dynamics An exploration of the behavior of non-linear dynamical systems. Topics include one and two-dimensional dynamics, Sarkovskii's Theorem, chaos, symbolic dynamics,and the Hénon Map. Prerequisite: Mathematics 232 or instructor permission. 6 credits; FSR; Not offered 2020-21
MATH 261 Functions of a Complex Variable Algebra and geometry of complex numbers, analytic functions, complex integration, series, residues, applications. Not open to students who have already received credits for Mathematics 361. Prerequisite: Mathematics 210 or Mathematics 211. 6 credits; FSR; Spring; Mark Krusemeyer
MATH 295 Combinatorial Games An introduction to the theory and practice of combinatorial games, which are two-player games in which players take turns, both players have complete information about the state of the game at all times, and there is no chance involved. Topics may include impartial games, a complete solution to nim, nim's relationship with other impartial games, the correspondence between games and numbers, sums of games, birthdays of games, hot games, and thermographs of games. Specific games will be studied as examples of the general theory, possibly including hackenbush, domineering, amazons, chomp, the octal games, and variations of these games.  Prerequisite: Mathematics 236 or instructor permission. 6 credits; FSR; Winter; Eric S Egge
MATH 295 Numerical Analysis Methods of numerical appoximation and applications to scientific computing and data analysis. Topics will be selected primarily frm numerical linear algebra and optimization. Theory, implementation and application of numerical algorithms will be emphasized. Prerequisite: Mathematics 232. 6 credits; FSR; Fall; Rob C Thompson
MATH 295 Seminar in Set Theory Introduction to set-theoretic foundations of mathematics. The axiom system of Zermelo-Fraenkel, cardinal and ordinal numbers, and the Axiom of Choice. As time permits, additional topics may include construction of the real number, transfinite induction, or consistency/independence proofs. Prerequisite: Mathematics 236 or instructor permission. 6 credits; FSR; Not offered 2020-21
MATH 297 Assessment and Communication of External Mathematical Activity An independent study course intended for students who have completed an external activity related to the mathematics major (for example, an internship or an externship) to communicate (both in written and oral forms) and assess their mathematical learning from that activity. Prerequisite: Permission of department chair and homework in advance of the external mathematical activity. 1 credit; S/CR/NC; NE; Fall, Winter, Spring; Katie R St. Clair
MATH 312 Elementary Theory of Numbers Properties of the integers. Topics include the Euclidean algorithm, classical unsolved problems in number theory, prime factorization, Diophantine equations, congruences, divisibility, Euler's phi function and other multiplicative functions, primitive roots, and quadratic reciprocity. Other topics may include integers as sums of squares, continued fractions, distribution of primes, integers in extension fields, p-adic numbers. Prerequisite: Mathematics 236 or instructor permission. 6 credits; FSR; Winter; Alex J Barrios
MATH 321 Real Analysis I A systematic study of concepts basic to calculus, such as topology of the real numbers, limits, differentiation, integration, convergence of sequences, and series of functions. Prerequisite: Mathematics 236 or instructor permission. 6 credits; FSR; Fall, Winter; Caroline L Turnage-Butterbaugh, Kate J Meyer
MATH 331 Real Analysis II Further topics in analysis such as measure theory, Lebesgue integration or Banach and Hilbert spaces. Prerequisite: Mathematics 321 or instructor permission. 6 credits; FSR; Spring; Rafe F Jones
MATH 332 Advanced Linear Algebra Selected topics beyond the material of Mathematics 232. Topics may include the Cayley-Hamilton theorem, the spectral theorem, factorizations, canonical forms, determinant functions, estimation of eigenvalues, inner product spaces, dual vector spaces, unitary and Hermitian matrices, operators, infinite-dimensional spaces, and various applications. Prerequisite: Mathematics 236 or instructor permission. 6 credits; FSR; Not offered 2020-21
MATH 333 Combinatorial Theory The study of structures involving finite sets. Counting techniques, including generating functions, recurrence relations, and the inclusion-exclusion principle; existence criteria, including Ramsey's theorem and the pigeonhole principle. Some combinatorial identities and bijective proofs. Other topics may include graph and/or network theory, Hall's ("marriage") theorem, partitions, and hypergeometric series. Prerequisite: Mathematics 236 or instructor permission. 6 credits; FSR; Spring; Eric S Egge
MATH 341 Partial Differential Equations An introduction to partial differential equations with emphasis on the heat equation, wave equation, and Laplace's equation. Topics include the method of characteristics, separation of variables, Fourier series, Fourier transforms and existence/uniqueness of solutions. Prerequisite: Mathematics 241. 6 credits; FSR; Spring; Rob C Thompson
MATH 342 Abstract Algebra I Introduction to algebraic structures, including groups, rings, and fields. Homomorphisms and quotient structures, polynomials, unique factorization. Other topics may include applications such as Burnside's counting theorem, symmetry groups, polynomial equations, or geometric constructions. Prerequisite: Mathematics 236 or instructor permission. 6 credits; FSR; Fall, Spring; Eric S Egge, MurphyKate Montee
MATH 344 Differential Geometry Local and global theory of curves, Frenet formulas. Local theory of surfaces, normal curvature, geodesics, Gaussian and mean curvatures, Theorema Egregium. Prerequisite: Mathematics 236 or permission of the instructor. 6 credits; FSR; Fall; Josh R Davis
MATH 349 Methods of Teaching Mathematics Methods of teaching mathematics in grades 7-12. Issues in contemporary mathematics education. Regular visits to school classrooms and teaching a class are required. Prerequisite: Junior or senior standing and instructor permission. 6 credits; NE; Not offered 2020-21
MATH 352 Topics in Abstract Algebra An intensive study of one or more of the types of algebraic systems studied in Mathematics 342. Prerequisite: Mathematics 342. 6 credits; FSR; Winter; Mark Krusemeyer
MATH 354 Topology An introduction to the study of topological spaces. We develop concepts from point-set and algebraic topology in order to distinguish between different topological spaces up to homeomorphism. Topics include methods of construction of topological spaces; continuity, connectedness, compactness, Hausdorff condition; fundamental group, homotopy of maps. Prerequisite: Mathematics 236 or instructor permission. 6 credits; FSR; Not offered 2020-21
MATH 361 Complex Analysis The theoretical foundations for the calculus of functions of a complex variable. Prerequisite: Mathematics 321 or instructor permission. Students who have already received credit for Mathematics 261 may only take this course with instructor permission. 6 credits; FSR; Not offered 2020-21
MATH 395 Introduction to Analytic Number Theory An introduction to the techniques and principles of analytic number theory. Topics covered include arithmetical functions, Dirichlet multiplication, averages of arithmetical functions, elementary theorems on the distribution of the primes, and Dirichlet's theorem on primes in arithmetic progressions.  Prerequisite: Mathematics 282 (formerly MATH 312) or Mathematics 342. 6 credits; NE; Spring; Caroline L Turnage-Butterbaugh
MATH 400 Integrative Exercise Either a supervised small-group research project or an individual, independent reading. Required of all senior majors. Prerequisite: Mathematics 236 and successful completion of three courses from among: Mathematics courses numbered above 236, Computer Science 252, Computer Science 254, Computer Science 352, Statistics 250, Statistics 320, Statistics 340. 3 credits; S/NC; Fall, Winter, Spring; Andy N Poppick, Caroline L Turnage-Butterbaugh, Rafe F Jones, Rob C Thompson, Alex J Barrios, Steve T Scheirer, Alexander Garver, Kate J Meyer, Mark Krusemeyer, MurphyKate Montee

Statistics Courses

STAT 120 Introduction to Statistics (Formerly MATH 215) Introduction to statistics and data analysis. Practical aspects of statistics, including extensive use of statistical software, interpretation and communication of results, will be emphasized. Topics include: exploratory data analysis, correlation and linear regression, design of experiments, basic probability, the normal distribution, randomization approach to inference, sampling distributions, estimation, hypothesis testing, and two-way tables. Students who have taken Mathematics 211 are encouraged to consider the more advanced Mathematics 240/Statistics 250 (formerly Mathematics 265 and 275) Probability/Statistical Inference sequence. Prerequisite: Not open to students who have already received credit for Psychology 200/201, Sociology/Anthropology 239 or Statistics 250 (formerly Mathematics 275). 6 credits; FSR, QRE; Fall, Winter, Spring; Andy N Poppick, Samuel D Ihlenfeldt, Owen D Biesel, Steve T Scheirer
STAT 220 Introduction to Data Science (Formerly Mathematics 285) This course will cover the computational side of data analysis, including data acquisition, management, and visualization tools. Topics may include: data scraping, clean up and manipulation, data visualization using packages such as ggplots, understanding and visualizing spatial and network data, and supervised and unsupervised classification methods. We will use the statistics software R in this course. Prerequisite: Statistics 120 (formerly Mathematics 215), Statistics 230 (formerly Mathematics 245) or Statistics 250 (formerly Mathematics 275). 6 credits; FSR, QRE; Winter, Spring; Katie R St. Clair
STAT 230 Applied Regression Analysis (Formerly Mathematics 245) A second course in statistics covering simple linear regression, multiple regression and ANOVA, and logistic regression. Exploratory graphical methods, model building and model checking techniques will be emphasized with extensive use of statistical software to analyze real-life data. Prerequisite: Statistics 120 (formerly Mathematics 215), Statistics 250 (formerly Mathematics 275), Psychology 200, or AP Statistics Exam score of 4 or 5. 6 credits; FSR, QRE; Fall, Winter, Spring; Laura M Chihara
STAT 250 Introduction to Statistical Inference (Formerly Mathematics 275) Introduction to modern mathematical statistics. The mathematics underlying fundamental statistical concepts will be covered as well as applications of these ideas to real-life data. Topics include: resampling methods (permutation tests, bootstrap intervals), classical methods (parametric hypothesis tests and confidence intervals), parameter estimation, goodness-of-fit tests, regression, and Bayesian methods. The statistical package R will be used to analyze data sets. Prerequisite: Mathematics 240 Probability (formerly Mathematics 265). 6 credits; FSR, QRE; Winter, Spring; Andy N Poppick, Laura M Chihara
STAT 260 Introduction to Sampling Techniques (Formerly MATH 255) Covers sampling design issues beyond the basic simple random sample: stratification, clustering, domains, and complex designs like two-phase and multistage designs. Inference and estimation techniques for most of these designs will be covered and the idea of sampling weights for a survey will be introduced. We may also cover topics like graphing complex survey data and exploring relationships in complex survey data using regression and chi-square tests. Prerequisite: Statistics 120 (formerly Mathematics 215), Statistics 230 (formerly Mathematics 245), or Statistics 250 (formerly Mathematics 275). 6 credits; FSR, QRE; Fall; Katie R St. Clair
STAT 285 Statistical Consulting (Formerly MATH 280) Students will apply their statistical knowledge by analyzing data problems solicited from the Northfield community. Students will also learn basic consulting skills, including communication and ethics. Prerequisite: Statistics 230 (formerly Mathematics 245) and instructor permission. 2 credits; FSR, QRE; Fall, Winter, Spring; Katie St. Clair
STAT 297 Assessment and Communication of External Statistical Activity An independent study course intended for students who have completed an external activity related to the statistics major (for example, an internship or an externship) to communicate (both in written and oral forms) and assess their statistical learning from that activity. Prerequisite: Permission of department chair and homework in advance of the external statistical activity. 1 credit; S/CR/NC; Fall, Winter, Spring; Katie R St. Clair
STAT 320 Time Series Analysis (Formerly MATH 315) Models and methods for characterizing dependence in data that are ordered in time. Emphasis on univariate, quantitative data observed over evenly spaced intervals. Topics include perspectives from both the time domain (e.g., autoregressive and moving average models, and their extensions) and the frequency domain (e.g., periodogram smoothing and parametric models for the spectral density). Prerequisite: Statistics 230 and 250 (formerly Mathematics 245 and 275). Exposure to matrix algebra may be helpful but is not required. 6 credits; FSR, QRE; Spring; Andy N Poppick
STAT 330 Advanced Statistical Modeling (Formerly MATH 345) Topics include linear mixed effects models for repeated measures, longitudinal or hierarchical data and generalized linear models (of which logistic and Poisson regression are special cases) including zero-inflated Poisson models. Depending on time, additional topics could include survival analysis, generalized additive models or models for spatial data. Prerequisite: Statistics 230 and 250 (formerly Mathematics 245 and 275) or permission of the instructor. 6 credits; FSR, QRE; Winter; Laura M Chihara
STAT 340 Bayesian Statistics Formerly MATH 315) An introduction to statistical inference and modeling in the Bayesian paradigm. Topics include Bayes’ Theorem, common prior and posterior distributions, hierarchical models, Markov chain Monte Carlo methods (e.g., the Metropolis-Hastings algorithm and Gibbs sampler) and model adequacy and posterior predictive checks. The course uses R extensively for simulations. Prerequisite: Statistics 250 (formerly Mathematics 275). 6 credits; FSR, QRE; Not offered 2020-21
STAT 400 Integrative Exercise Either a supervised small-group research project or an individual, independent reading. Required of all senior majors. Prerequisite: Senior Statistics major. Students are strongly encouraged to complete Statistics 230 (formerly Mathematics 245) and Statistics 250 (formerly Mathematics 275) before starting this course. 3 credits; S/NC; Fall, Winter, Spring; Andy N Poppick, Katie R St. Clair, Rob C Thompson