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 mathematic 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.

There are two tracks in the major: Mathematics and Mathematics/Statistics (a double major is not allowed in the two tracks). Students choose from the same integrative exercise choices.

Requirements for the Mathematics Track

The course requirements are Mathematics 101 or 111, 121, 211, 232, 236 and six advanced courses from among: Mathematics courses numbered above 236 and Computer Science 252, 254. Potential majors with especially strong preparation may petition the department for exemption from the Mathematics 232 and/or 236 requirement(s). Mathematics majors are encouraged to take Computer Science 111.

At least three of the following five areas of mathematics must be represented by the six advanced courses.

Algebra: Mathematics 312, 332, 342, 352

Analysis: Mathematics 251, 261, 321, 331, 361

Applied Mathematics: Mathematics 241, 245, 255, 265, 275, 315, 341, 365

Discrete Structures: Mathematics 333, Computer Science 252, 254

Geometry and Topology: Mathematics 244, 344, 354

Of the six advanced courses, at most two may be from outside the Carleton Department of Mathematics.

In addition, each senior major must complete an integrative exercise which consists of a group or original research project. Majors are required to participate in the mathematical life of the department by attending colloquia, comps presentations, and other activities.

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 department web site. Those planning to attend graduate school should consider acquiring a reading knowledge of at least one of the following languages: French, German or Russian.

In order to meet State of Minnesota certification requirements, prospective secondary school teachers must take Mathematics 265, 275, 244 (recommended) or 344, and 349. A computer science course is also strongly recommended.

Requirements for the Mathematics/Statistics Track

A Carleton student following this option must take the following courses: Mathematics 101 or 111, 121, 211, 232, 236, 245, 265, 275 and one of 255, 315 or 365, plus two Mathematics courses above 236 (one of which may be Computer Science 324); at least one of these two courses must be taken outside of the Applied Mathematics area. In addition, the Senior Integrative Exercise is required.

It strongly recommended that students on this track take CS 111 and engage in some data analysis learning experience outside of the classroom such as an internship involving data analysis, a research experience with a statistician, either on or off campus, or a comps project that is explicity statistical in nature.

Students interested in graduate school in statistics should consider taking Mathematics 321 (Real Analysis I).

Major under Combined Plan in Engineering (see Engineering in index):

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, 230, and Computer Science 111.

Mathematics Skills Center:

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

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. (Meets Monday through Friday). Prerequisite: Not open to students who have received credit for Mathematics 111. 6 credits; FSR; Fall; Deanna Haunsperger

MATH 106 Introduction to Mathematics This course is designed to provide an understanding of fundamental concepts, and examples of applications, of mathematics. It attempts to provide insights into the nature of mathematics and its relation to other branches of knowledge, and helps students develop skill in mathematical reasoning. No prerequisites. 6 credits; FSR; Not offered 2016-17

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; Peri Shereen, Rita B Thompson

MATH 115 Statistics: Concepts and Applications Introduction to statistical concepts with emphasis on understanding and interpretation of statistical information, especially in the context of media reports and scholarly articles. Examples taken from a wide-range of areas such as public policy, health and medicine, and the social and natural sciences. Computationally less intensive than Math 215. Students will learn how to use statistical software. Topics include: Uncertainty and variability, statistical graphs, types of studies, correlation and linear regression, two-way tables, and inference. Prerequisite: Not open to students who have already received credit for Mathematics 211, Mathematics 215, Psychology 200/201, Statistics 120 or Sociology/Anthropology 239. 6 credits; FSR, QRE; Fall; Bob Dobrow

MATH 121 Calculus II Integration techniques, improper integrals, the calculus of the logarithmic, exponential and inverse trigonometric functions, applications, Taylor polynomials and infinite series. Prerequisite: Mathematics 101, 111 or placement via Calculus Placement Exam # 2. 6 credits; FSR; Fall, Winter, Spring; Liz Sattler, Eric S Egge, Rob Thompson, Rafe Jones

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; Mark Krusemeyer

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; Eric S Egge, Mark Krusemeyer, Sam Patterson, Helen M Wong, Peri Shereen, Liz Sattler, Josh Davis

MATH 215 Introduction to Statistics 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 received credit for Mathematics 115 may petition the department to seek approval to register for Mathematics 215. Students who have taken Mathematics 211 are encouraged to consider the more advanced Mathematics 265-275 Probability-Statistics sequence. Prerequisite: Not open to students who have already received credit for Psychology 200/201, Sociology/Anthropology 239 or Math 275. 6 credits; FSR, QRE; Fall, Winter, Spring; Andy Poppick, Katie St. Clair, Laura M Chihara

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; Peri Shereen, Sam Patterson, Bob Dobrow, Mark Krusemeyer

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 Haunsperger, Rafe Jones, Mark Krusemeyer, Helen M Wong

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 Thompson, Sam Patterson

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 2016-17

MATH 245 Applied Regression Analysis 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 or Statistics 250 (formerly Mathematics 215 or 275). 6 credits; FSR, QRE; Fall, Winter, Spring; Laura M Chihara, Andy Poppick, Katie St. Clair

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 236 or instructor permission. 6 credits; FSR; Fall; Stephen F Kennedy

MATH 255 Introduction to Sampling Techniques 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: Mathematics 215 or 275 or Statistics 120. 6 credits; FSR, QRE; Not offered 2016-17

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; Sam Patterson

MATH 265 Probability 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 211. 6 credits; FSR; Fall, Winter; Bob Dobrow, Laura M Chihara

MATH 275 Introduction to Statistical Inference 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 265. 6 credits; FSR, QRE; Winter, Spring; Bob Dobrow, Andy Poppick

MATH 280 Statistical Consulting 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: Mathematics 245 and instructor permission. 2 credits; S/CR/NC; FSR, QRE; Fall, Winter, Spring; Katie St. Clair

MATH 295 Coding Theory This course is an introduction to error-correcting codes. The course will cover topics including linear codes, Hamming codes and cyclic codes. Additional topics may include low-density parity-check codes and perfect codes. Prerequisite: Mathematics 232. 6 credits; FSR; Spring; Peri Shereen

MATH 295 Seminar in Low-dimensional Topology A combinatorial introduction to the study of manifolds in dimensions less than four, including selected topics in knot theory. Prerequisite: Mathematics 236. 6 credits; FSR; Fall; Helen M Wong

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 St. Clair, Laura M Chihara

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; Mark Krusemeyer

MATH 315 Topics Probability/Statistics: Data Science This course will cover the computational side of data analysis, including data acquisition, management and visualization tools. Topics may include data scraping and manipulation, unstructured data, data visualization using packages such as ggplots, cross-validation, classification, and network analysis. Prerequisite: Mathematics 275. 6 credits; FSR, QRE; Fall; Katie St. Clair

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: math.236 or math.236p. 6 credits; FSR; Winter; Helen M Wong

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; Liz Sattler

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 2016-17

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 Fourier Series and Boundary Value Problems Fourier series and their applications to boundary value problems in partial differential equations. Topics include separation of variables, orthogonal sets of functions, representations of functions in series of orthogonal functions, Sturm-Liouville theory, and Fourier transforms. Prerequisite: Mathematics 241. 6 credits; FSR; Spring; Rob 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; Mark Krusemeyer

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; Rob Thompson

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; Spring; Stephen F Kennedy

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; Eric S Egge

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 2016-17

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 2016-17

MATH 365 Stochastic Processes Introduction to the main discrete and continuous time stochastic processes. Topics include Markov chains, Poisson process, continuous time Markov chains, Brownian motion. Use of R and/or Mathematica. Prerequisite: Mathematics 232 and Mathematics 240 (formerly Mathematics 265). 6 credits; FSR, QRE; Spring; Bob Dobrow

MATH 395 Topics in the Theory of Elliptic Curves Introduction to the geometry and arithmetic of elliptic curves, with selected advanced topics. Introductory topics include the geometry of cubics, the group law on an elliptic curve, points of finite order, the group of rational points, heights and the Mordell-Weil theorem. Students will have the opportunity to explore through group projects advanced topics such as: integral points on elliptic curves; elliptic curves over finite fields; elliptic curves with complex multiplication; and Galois representations on torsion points. Prerequisite: Mathematics 342, an equivalent Budapest or Moscow Semester in Mathematics course or instructor permission. 6 credits; FSR; Spring; Rafe Jones

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; Laura M Chihara, Deanna Haunsperger, Mark Krusemeyer, Sam Patterson, Peri Shereen, Katie St. Clair, Rafe Jones, Liz Sattler, Andy Poppick, Helen M Wong