Carleton College

MATH 100. Combinatorial Games Combinatorial games are 2-player games in which players take turns making moves, there are no hidden elements, and there is no element of chance. Famous combinatorial games include Chess, Checkers, Go, Dots and Boxes, Hex, Hackenbush, and Nim. In this seminar we will study a variety of combinatorial games, as well as their connections with alternative number systems and mathematical methods of transmitting information over a noisy channel. No particular mathematical background is assumed, but we will spend some time learning to write mathematical arguments about games. 6 cr., WR; AI, WR1, Fall—E. Egge

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., FSR, Fall—T. Occhipinti

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., FSR, Not offered in 2014-2015.

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., 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., FSR, QRE, Fall,Spring—Staff

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., 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, 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., 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, Fall,Winter,Spring—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—E. Egge, J. Goldfeather, R. Jones, S. Kennedy

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., FSR, Fall,Winter,Spring—J. Goldfeather, T. Occhipinti, H. Wong

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, Not offered in 2014-2015.

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., FSR, Winter,Spring—S. Kennedy, 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., FSR, Offered in alternate years. Not offered in 2014-2015.

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., 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., FSR, Offered in alternate years. Winter—S. Patterson

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 211. 6 cr., FSR, Offered in alternate years. Spring—Staff

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., FSR, Fall—L. Chihara, R. Dobrow

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., FSR, QRE, Winter—K. St. Clair

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, FSR, QRE, Fall,Winter,Spring—L. Chihara

MATH 295. Differential Forms and Vector Calculus Differential Forms provide a modern approach to a classical topic: Vector Calculus. They find applications in such diverse fields as geometry, algebra, engineering, electricity and magnetism, and general relativity. This course will rigorously develop differential forms then apply them to classical topics including divergence, gradient, and curl. A primary focus of the course will be the proof of the generalized Stokes' Theorem which is a general n-dimensional form of the familiar Fundamental Theorem of Calculus. Modern treatments of other topics from advanced calculus will be considered as time permits. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., FSR, Spring—S. Patterson

MATH 295. Cryptography, Coding Theory, and Compression This course will cover cryptosystems, error correcting codes, and compression methods, including RSA, Diffie-Hellman, Hamming codes, and Huffman coding. The course will also cover methods of breaking codes, including the quadratic sieve. The mathematics behind these methods will be emphasized, including linear algebra, number theory, and probability. The course will include light programming in the freely available software package SAGE, but no knowledge of programming is required. Prerequisite: Mathematics 232 and (Mathematics 236 or Computer Science 202). 6 cr., NE, Spring—T. Occhipinti

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, NE, Fall,Winter,Spring

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., FSR, Offered in alternate years. Fall—R. Jones

MATH 315. Topics in Probability and Statistics: 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 265. Mathematics 232 may be waived with consent of instructor. 6 cr., FSR, QRE, 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., 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., FSR, 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., FSR, Offered in alternate years. Not offered in 2014-2015.

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., FSR, Offered in alternate years. Spring—M. Krusemeyer

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—A. Tanguay

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., FSR, 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. Prerequisite: Mathematics 236 or permission of the instructor. 6 cr., FSR, Offered in alternate years. Fall—S. Patterson

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 permission of the instructor. 6 cr., NE, Spring—D. Haunsperger

MATH 351. Functions of a Complex Variable Cauchy-Riemann equations, analytic functions, complex integration, series, residues, applications. Prerequisite: Mathematics 211. 6 cr., FSR, Not offered in 2014-2015.

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., FSR, Spring—E. 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 321 or permission of the instructor. 6 cr., FSR, Offered in alternate years. Not offered in 2014-2015.

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 265. 6 cr., FSR, QRE, Offered in alternate years. Spring—B. Dobrow

MATH 395. Senior Seminar in Knot Theory Introduction to the mathematical theory of knots. Basic properties and methods of description. Techniques to distinguish knots, especially algebraic and combinatorial invariants. Applications to biology, chemistry, and physics. Prerequisite: Permission of instructor. 6 cr., NE, Winter—H. Wong

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. 6 cr., S/NC, NE, Fall—Staff