Carleton College

MATH 100. Mathematics and Democracy Mathematics has a substantive role to play in the implementation of democracy. How do we ensure equitable representation? How do we fairly divide finite resources (and share responsibilities and burdens)? How do we ensure that the results of our elections reflect the popular will? Social scientists and mathematicians have turned some powerful mathematical tools onto the investigation of such questions in recent years. We will study some of that work with the dual goals of gaining appreciation for the power and elegance of the mathematical approach to problem-solving and understanding at a deeper level how to construct a just society. 6 cr., S/CR/NC, MS, Fall—S. Kennedy

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). Not open to students who have received credit for Math 111. 6 cr., MS, Fall—E. Egge

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, Spring—M. Krusemeyer

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. 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, Fall,Winter,Spring—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. Not open to students who have already received credit for Math 211, Math 215 or Psychology 200/201. 6 cr., MS, Fall,Spring—L. Chihara, K. St. Clair

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, 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, 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 131 or placement via Calculus Placement Exam #3. 6 cr., MS, 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, sampling distributions, estimation, hypothesis testing, and two-way tables. Not open to students who have already received credit for Math 115 or Math 275. Students who have received MS credit for Psychology 200/201 cannot receive MS credit for Math 215. Students who have taken Math 211 are encouraged to consider the more advanced Math 265-275 probability-statistics sequence. 6 cr., MS, Fall,Winter,Spring—Staff

MATH 232. Linear Algebra Vector spaces, linear transformations, determinants, inner products and orthogonality, eigenvectors and eigenvalues; connections with multivariable calculus. Prerequisite: Mathematics 211. 6 cr., MS, Fall,Winter,Spring—J. Armel, J. Goldfeather, S. Patterson, H. Wong

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, Fall,Winter,Spring—D. Haunsperger, E. Egge, M. Krusemeyer

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. Prerequisites: Mathematics 232 or permission of the instructor. 6 cr., MS, Winter—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, Offered in alternate years. Fall—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. Prerequisites: Mathematics 215 (or equivalent) or 275. 6 cr., MS, Winter,Spring—L. Chihara, R. Dobrow

MATH 251. Chaotic Dynamics An exploration of the behavior of non-linear dynamical systems. Topics include one-dimensional dynamics, Sarkovskii's Theorem, chaos, symbolic dynamics, fractals, Mandelbrot and Julia sets. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, Offered in alternate years. Not offered in 2009-2010.

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, Fall—L. Chihara, K. St. Clair

MATH 275. Introduction to Statistical Inference Introduction to mathematical statistics. The mathematics underlying fundamental statistical concepts will be covered as well as applications of these ideas to real-life data. Topics include: confidence intervals, hypothesis testing, parameter estimation, maximum likelihood, goodness of fit tests and regressions. A statistical software package will be used to analyze data sets. Prerequisite: Mathematics 265. 6 cr., MS, Winter—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 numbers, transfinite induction, or consistency/independence proofs. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, Spring—G. Nelson

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, Offered in alternate years. Not offered in 2009-2010.

MATH 315. Topics in Probability & Statistics: Introduction to Stochastic Processes Random walk, Markov chains, Poisson process, Brownian motion, with applications. Prerequisite: Mathematics 265. 6 cr., MS, Spring—R. Dobrow

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, 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, Offered in alternate years. Not offered in 2009-2010.

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, Offered in alternate years. Fall—J. Goldfeather

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, Offered in alternate years. Not offered in 2009-2010.

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, Fourier transforms, and uniqueness of solutions. Prerequisite: Mathematics 241. 6 cr., MS, 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 consent of the instructor. 6 cr., MS, Winter—E. Egge

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. Riemannian geometry. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, Offered in alternate years. Not offered in 2009-2010.

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, Not offered in 2009-2010.

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, Offered in alternate years. Spring—S. Patterson

MATH 352. Abstract Algebra II An intensive study of one or more of the types of algebraic systems studied in Mathematics 342. Prerequisite: Mathematics 342 or consent of the instructor. 6 cr., MS, Offered in alternate years. Spring—M. Krusemeyer

MATH 354. Topology An introduction to the topology of surfaces. We will cover basic point-set, geometric and algebraic topology. Topics include continuity, connectedness and compactness; triangulations and classification of surfaces; topological invariants (Euler characteristic); homology. Prerequisite: Mathematics 236. 6 cr., MS, Offered in alternate years. Winter—H. Wong

MATH 395. Seminar in Mathematics: Functional Analysis Selected topics in Functional Analysis, the study of infinite dimensional vector spaces. Possible topics include topologies and convergence, Banach and Hilbert spaces, dual spaces, linear operators, compact self-adjoint operators, and distributions. Prerequisite: Math 321. 6 cr., MS, Winter—J. Armel

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