Carleton College

MATH 100. The Mathematics of Democracy We aspire to live in a democratic society, but what exactly does this mean? How can we decide, for example, which candidate in an election has the support of the people? How can Congressional seats be apportioned to the States "according to their respective numbers?" Are these things even possible? Recently, mathematical analysis has brought new insight to these old questions, often with surprising results. We will study some of this work and its implications for our democratic aspirations, and perhaps gain some appreciation for the power and elegance of mathematics along the way. 6 cr., AI, WR1, Fall—A. Gainer-Dewar

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 Math 111. 6 cr., MS; FSR, Fall—D. 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 cr., MS; FSR, Spring—H. Wong

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 cr., MS; FSR, Fall,Winter—Staff

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 or Psychology 200/201. 6 cr., MS; FSR, QRE, Fall,Spring—K. St. Clair, M. Ott

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 cr., MS; FSR, Fall,Winter,Spring—Staff

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 cr., S/CR/NC, MS; NE, Winter—Staff

MATH 211. Introduction to Multivariable Calculus Vectors, curves, partial derivatives, gradient, multiple and iterated integrals, line integrals, Green's theorem. Prerequisite: Mathematics 121 or placement via Calculus Placement Exam #3. 6 cr., MS; FSR, Fall,Winter,Spring—Staff

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. Prerequisite: Not open to students who have already received credit for Math 115, Psychology 200/201 or Math 275. Students who have taken Math 211 are encouraged to consider the more advanced Math 265-275 probability-statistics sequence. 6 cr., FSR, QRE, Winter—Staff

MATH 232. Linear Algebra Vector spaces, linear transformations, determinants, inner products and orthogonality, eigenvectors and eigenvalues. Prerequisite: Mathematics 211. 6 cr., FSR, Fall,Winter,Spring—Staff

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 or consent of the instructor. 6 cr., MS; FSR, Fall,Winter,Spring—E. Egge, D. Haunsperger, R. Jones

MATH 237. Designing a Curriculum for Math GED We will help local communities respond to the latest changes in GED requirements by observing how GED mathematics is currently taught and preparing new curricular materials to teach it in the future. Prerequisite: Mathematics 236 and permission of the instructor. 2 cr., S/CR/NC, NE, Spring—D. Haunsperger

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 permission of the instructor. 6 cr., MS; FSR, Winter,Spring—M. Krusemeyer

MATH 244. Geometries Euclidean geometry from an advanced perspective; projective, hyperbolic, inversive, and/or other geometries. In addition to foundations, various topics such as transformation and convexity will be treated. Recommended for prospective secondary school teachers. Prerequisite: Mathematics 236. 6 cr., MS; FSR, Offered in alternate years. Winter—S. Kennedy

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: Mathematics 215 (or equivalent) or 275. 6 cr., MS; FSR, QRE, Winter,Spring—L. Chihara, K. 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 permission of the instructor. 6 cr., MS; FSR, Offered in alternate years. Not offered in 2013-2014.

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 211. 6 cr., MS; FSR, Fall—R. Dobrow, M. Ott

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 cr., MS; FSR, QRE, Winter—L. Chihara

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 permission of instructor. 2 cr., S/CR/NC, ND; FSR, QRE, Fall,Winter,Spring—K. St. Clair

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 permission of the instructor. 6 cr., MS; FSR, Spring—G. Nelson

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 cr., S/CR/NC, ND; NE, Fall,Winter,Spring—Staff

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 consent of the instructor. 6 cr., MS; FSR, Offered in alternate years. Not offered in 2013-2014.

MATH 315. Topics in Probability and Statistics: 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. This course will also teach methods for graphing complex survey data and exploring relationships in complex survey data using regression and chi-square tests. Prerequisite: Mathematics 215 (or equivalent) or 275. 6 cr., MS; FSR, QRE, Offered in alternate years. Fall—K. St. Clair

MATH 315. Topics in Probability and Statistics: Statistical Computing Topics include generating random variables, monte carlo integration, markov chains monte carlo. We will use R extensively. Prerequisite: Prerequisite: Mathematics 275. 6 cr., MS; FSR, QRE, Spring—L. Chihara

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 consent of the instructor. 6 cr., MS; FSR, Fall—G. Nelson

MATH 331. Real Analysis II Further topics in analysis such as measure theory, Lebesgue integration or Banach and Hilbert spaces. Prerequisite: Mathematics 321 or consent of the instructor. 6 cr., MS; FSR, Offered in alternate years. Winter—G. Nelson

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 consent of the instructor. 6 cr., MS; FSR, Offered in alternate years. Fall—M. Krusemeyer

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 permission of instructor. 6 cr., MS; FSR, Offered in alternate years. Not offered in 2013-2014.

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 cr., FSR, Spring—S. Patterson

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 permission of the instructor. 6 cr., MS; FSR, Winter—J. Goldfeather

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 cr., MS; FSR, Offered in alternate years. Not offered in 2013-2014.

MATH 349. Methods of Teaching Mathematics Cross-listed with EDUC 350. 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 permission of the instructor. 6 cr., ND; NE, Not offered in 2013-2014.

MATH 351. Functions of a Complex Variable Algebra and geometry of complex numbers, analytic functions, complex integration, series, residues, applications. Prerequisite: Mathematics 211. 6 cr., MS; FSR, Spring—S. Patterson

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 cr., MS; FSR, Spring—M. 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 321 or permission of the instructor. 6 cr., FSR, Offered in alternate years. Spring—H. Wong

MATH 395. Topics in Algebraic Number Theory Study of integers in algebraic extensions of the rationals. Motivated by a failed attempt to prove Fermat's last theorem, we define the ring of integers, examine the failure of unique factorization, and rehabilitate it somewhat by showing that ideals have unique factorization into prime ideals. Further topics may include the finiteness of the class number, units in rings of integers, relations to Galois theory, cyclotomic fields, class number formulas, and the Chebotarev density theorem. Where appropriate for comps projects, links to arithmetic dynamics will be given. Prerequisite: Mathematics 342 and permission of the instructor. 6 cr., FSR, Winter—R. Jones

MATH 395. Combinatorics of Symmetric Functions Study of symmetric functions with an emphasis on the underlying combinatorics. Course opens with several bases for the space of symmetric functions, including elementary symmetric functions, complete homogeneous symmetric functions, power sum symmetric functions, and Schur functions. The rest of the course is devoted to combinatorial answers to algebraic questions; topics may include standard and semistandard tableaux, Kostka numbers, the hook length formula, the Robinson-Schensted-Knuth correspondence, Cauchy's identity, the Pieri rules, lattice paths and the Jacobi-Trudi identities, the Murnaghan-Nakayama rule, the Littlewood-Richardson rule, Knuth equivalence on words, jeu de taquin, and compositions and quasisymmetric functions. Prerequisite: Mathematics 333, an equivalent Budapest Semester in Mathematics course, or consent of the instructor. 6 cr., FSR, Fall—E. Egge

MATH 400. Integrative Exercise A supervised small-group research project for senior mathematics majors. 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. 3 cr., S/NC, ND, Fall,Winter,Spring—Staff