Mathematics and Computer Science
Chair: Professor Samuel E. Patterson
Professors: David F. Appleyard, Jack Goldfeather, Stephen F. Kennedy, Mark Krusemeyer, Richard W. Nau, Gail S. Nelson, Jeffrey R. Ondich, Samuel E. Patterson
Associate Professors: Laura M. Chihara, Robert P. Dobrow, Deanna B. Haunsperger
Assistant Professors: Amy Csizmar Dalal, Eric S. Egge, David Liben-Nowell, David R. Musicant
Senior Lecturer: Cris T. Roosenraad
Mathematics and Computer Science...both subjects embody the spirit of the liberal arts. 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. Computer Science, a newer discipline, has roots in mathematics, physics, engineering, philosophy, biology, psychology, linguistics, and art. A single department offering both majors is able to take maximum advantage of the expertise and interests of both mathematicians and computer scientists in developing a diverse and flexible program for students.
Requirements for a Mathematics Major:
The course requirements are Mathematics 110 or 111, 121, 211, 232, 236 and six advanced courses from among: Mathematics courses numbered above 236 and Computer Science 227, 237. Potential majors with especially strong preparation may petition the department for exemption from the Mathematics 232 and/or 236 requirement(s). Mathematics majors are strongly encouraged to take Computer Science 117, preferably during their first two years. Concepts and skills from Computer Science 117 can be particularly valuable in advanced mathematics courses.
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, 311, 321, 331, 351
Applied Mathematics: Mathematics 241, 265, 275, 341
Discrete Structures: Mathematics 333, Computer Science 227, 237
Geometry and Topology: Mathematics 244, 344, 354
In addition, each senior major must complete an integrative exercise which consists of a senior lecture and a written comprehensive examination, and majors must attend a total of twelve other senior lectures during the junior and senior years.
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 web at http://www.mathcs.carleton.edu. 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, 342, 244 (recommended) or 344, and 349, and a computer science course is strongly recommended.
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: Physics 113 or 114, 115, 128, Chemistry 123, 230, and Computer Science 117.
Requirements for a Computer Science Major:
The course requirements are Computer Science 117, 127; Mathematics 110 or 111, 121; Computer Science 177 or Mathematics 236; Computer Science 207, 217, 227, 237; and two additional courses from among: Computer Science courses numbered 240 or above, Mathematics 311, Physics 247 or 343. Although they are not required for the major, Mathematics 232 and a probability and statistics course are recommended. In addition, each senior major must complete an integrative exercise: during fall and winter terms of the senior year, the student will participate on a team of four to seven students working on a faculty-specified project. Potential majors should take Computer Science 127 before the end of the sophomore year.
Students contemplating graduate study in computer science should consider taking additional courses in both computer science and mathematics. Those interested in computer engineering should consider taking physics courses through Electricity and Magnetism and Electronics.
A guide for majors on the web is available at http://cs.carleton.edu/.
Mathematics Skills Center:
This Center offers extra assistance to students in lower-level mathematics courses.
Mathematics Courses (MATH)
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 106. Introduction to Mathematics This course is designed to provide students with an understanding of fundamental concepts and 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—G. Nelson
MATH 109. Calculus I with Review, Part 1 Mathematics 109 and 110 cover in two terms the material covered in Mathematics 111. In addition, topics from precalculus mathematics are reviewed and practiced as needed. Precalculus topics include: algebra and analytic geometry; linear, quadratic, polynomial and rational functions; and trigonometric functions. Enrollment is restricted to those who were advised to take Math 109-110 on the basis of the department's Diagnostic Examination. The two-term Math 109-110 sequence serves as an alternate prerequisite for all college courses requiring Mathematics 111. Meets five days a week. 6 cr., S/CR/NC, ND, Fall—S. Patterson
MATH 110. Calculus I with Review, Part 2 This course continues the study of calculus begun in Mathematics 109. Review of precalculus mathematics continues as needed. Prerequisite: Mathematics 109 or consent of the instructor. Meets five days a week. 6 cr., MS, Winter—S. Patterson
MATH 111. Calculus I An introduction to the differential and integral calculus. Derivatives, antiderivatives, the definite integral, applications, and the fundamental theorem of calculus. Prerequisite is one of the following: admission by way of the department's decision tree to be found in the New Student Course Description booklet, or a satisfactory grade on the Diagnostic Examination. Not open to students who have received credit for Mathematics 110. 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 124.
6 cr., MS, Fall,Winter—L. Chihara, R. Dobrow
MATH 121. Calculus II Integration techniques, improper integrals, the calculus of the exponential, logarithmic, and inverse trigonometric functions, applications, indeterminate forms, Taylor polynomials, infinite series. Prerequisite: Mathematics 110 or 111 or placement by or consent of the department. 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. Calculus III Introduction to multivariable calculus: vectors, curves, partial derivatives, gradient, multiple and iterated integrals, line integrals, Green's theorem. Prerequisite: Mathematics 121 or placement by or consent of the department. 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 124-126 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 216. Seminar: History of Mathematics This seminar will focus on selected episodes in the history of mathematics from the seventeenth century to the present. Each participant will give at least one public presentation, which will be followed by discussion. Some weekly preparatory reading, often on the life and work of a prominent mathematician, will be required. Prerequisite: Mathematics 211 or concurrent registration with Mathematics 211 or consent of the instructor. 2 cr., MS, Offered in alternate years. Not offered in 2005-2006.
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— L. Chihara, E. Egge, M. Krusemeyer, G. Nelson
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 such as groups and rings and basic combinatorics. Prerequisite: Mathematics 232 or consent of the instructor. Students may not receive credit for both Computer Science 177 and Mathematics 236. 6 cr., MS, Fall,Winter,Spring—D. Haunsperger, G. Nelson, C. Roosenraad
MATH 241. Ordinary Differential Equations An introduction to the theory and methods of solution of ordinary differential equations. Prerequisites: Mathematics 232 or consent 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. Spring—S. Patterson
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 software to analyze real-life data. Prerequisites: Mathematics 215 (or equivalent) or 275. 6 cr., MS, Not offered in 2005-2006.
MATH 251. Chaotic Dynamics An exploration of the behavior of non-linear dynamical systems. Topics include one-dimensional dynamics, Feigenbaum's universality, Sarkovskii's Theorem, chaos, symbolic dynamics, fractals, structural stability, Smale's horseshoe map, strange attractors and bifurcation theory. Some point-set topology will be developed as needed. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, Offered in alternate years. Not offered in 2005-2006.
MATH 265. Probability Introduction to probability and its applications. Topics include: combinatorial analysis used in computing probabilities, random variables, independence, joint and conditional distributions, expectation, law of large numbers and properties of the most common probability distributions. Prerequisite: Mathematics 211. 6 cr., MS, Fall—R. Dobrow
MATH 275. Mathematical Statistics 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—L. Chihara
MATH 285. Topics in Probability and Statistics
Topic to be determined in 2006 based on students' and instructor's interests. Prerequisite: Mathematics 265 or consent of the instructor. 6 cr., MS, Spring—R. Dobrow
MATH 311. Topics in Numerical Analysis The design and comparison of effective methods for solving numerically, as on calculators or computers, problems arising in mathematics and areas of application. Topics chosen from the following: the numerical solution of algebraic, differential, and difference equations; integration; functional approximation; treatment of empirical data; computational linear algebra; error analysis. Prerequisite: Mathematics 232. No specific programming prerequisite. 6 cr., MS, Not offered in 2005-2006.
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 2005-2006.
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 2005-2006.
MATH 332. Advanced Linear Algebra
Vector spaces, linear operators, canonical forms, inner-product spaces. Emphasis on the interplay of theory and applications. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, Offered in alternate years. Fall—R. Dobrow
MATH 333. Combinatorial Theory Deciding on the existence of, finding, and/or counting arrangements, functions, and other desired structures involving finite sets. Some graph and network theory. Counting techniques include the inclusion-exclusion principle, generating functions, and recurrence relations. Existence criteria include the pigeonhole principle, Ramsey's theorem, and Hall's ("marriage") theorem. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, Offered in alternate years. Not offered in 2005-2006.
MATH 341. Partial Differential Equations 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—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. Riemannian geometry. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, Offered in alternate years. Not offered in 2005-2006.
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: Senior standing and permission of the instructor. 6 cr., ND, Fall—C. Roosenraad
MATH 351. Functions of a Complex Variable Algebra and geometry of complex numbers, analytic functions, complex integration, series, residues, applications. Prerequisite: Mathematics 211. 6 credits cr., MS, Offered in alternate years. Winter—D. Appleyard
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 credits cr., MS, Offered in alternate years. Fall—L. Chihara
MATH 395. Topics in Graph Theory Possible topics include matchings in graphs, graph coloring, graceful and other labelings of graphs, Ramsey theory, perfect graphs, random graphs, eigenvalues of graphs, and the Tutte polynomial. Prerequisites: Mathematics 236 or permission of the instructor. 6 cr., MS, Spring—E. Egge
MATH 400. Integrative Exercise (Senior Examination) A three-hour written test on material from Mathematics 110 or 111, 121, 211, 232 and 236. Required of all senior majors 3 cr., S/NC, ND, Spring—Staff
MATH 400. Integrative Exercise (Senior Lecture) A mathematical talk on an assigned topic, presented by the registered senior mathematics major. Required of all senior majors. Prerequisite: Mathematics 236 and successful completion of three courses from among: Mathematics courses numbered above 236, Computer Science 227, Computer Science 237. 3 cr., S/NC, ND, Fall,Winter,Spring—Staff
Computer Science Courses (CS)
CS 107. Explorations in Computer Science
This course is designed for students who have never taken a computer science course. It provides an overview of computer technology and an introduction to the fundamental concepts and applications of computer science. We will explore both the technical aspects and the social, political, and ethical aspects of computers and computer technology. Students who have taken Computer Science 127 may not enroll in Computer Science 107. 6 cr., MS, Fall—A. Csizmar-Dalal
CS 117. Introduction to Computer Science This course introduces the fundamentals of computer science through problem solving. Students will learn the Java programming language by writing programs to solve interesting problems, such as procesing images. The design and implementation of more extensive projects will introduce students to computer science through such topics such as data representation, graphics, recursion, and object-oriented design. No previous programming experience is necessary. 6 cr., MS, Fall,Winter,Spring—Staff
CS 127. Data Structures Data structures form the foundation of computer science: they make algorithms more efficient and facilitate problem-solving on both small scales and large scales. In this course, we will explore these basic building blocks of computer science and develop computer programs based on these data structures. We will study stacks, queues, trees, linked lists, graphs and hash tables, as well as algorithms for recursion, searching, and sorting. Mathematical methods for analyzing the performance of algorithms will also be addressed. We will apply these concepts in a term-long software design project. Prerequisite: Computer Science 117 or consent of the instructor. 6 cr., MS, Fall,Winter,Spring—Staff
CS 177. Algorithms I A collection of topics useful in computer science topics: elements of logic and Boolean algebra; methods of proof; sets, relations, and functions; graphs, counting techniques; elementary complexity theory; and finite probability. Additional topics may be drawn from recurrence relations, finite-state machines, and linear algebra. Prerequisites: Mathematics 110 or 111; Computer Science 117 or concurrent registration in Computer Science 117. 6 cr., MS, Fall—D. Liben-Nowell
CS 207. Computer Organization and Architecture The design and organization of hardware and software. Topics include data representations, digital logic, micro-processor architecture, micro-programming, assembly languages, memory hierarchies, caches, RISC architectures and pipelining. Prerequisite: Computer Science 127 or consent of the instructor. 6 cr., MS, Fall—A. Csizmar Dalal
CS 217. Programming Languages
Are there other sorts of programming languages besides object-oriented ones? This course will survey a number of different programming paradigms in order to do a comparative analysis of features and design. Students will gain experience programming in a variety of programming languages, including functional and logical languages such as Scheme and Prolog. Furthermore, the course will examine such topics in programming language construction as syntax and semantics, mechanisms for parameter passing, typing, scoping, and control structures. Prerequisite: Computer Science 127 or consent of the instructor. 6 cr., MS, Spring—D. Musicant
CS 227. Algorithms II How to think of good solution methods for solving computational problems and how to find the best methods and prove that they are the best. We'll consider the design and analysis of algorithms: divide and conquer, dynamic programming, greedy method, backtracking, branch-and-bound, and inductive approaches; recurrence relations, applications, complexity theory, NP-completeness. Prerequisites: Computer Science 127; Computer Science 177 or Mathematics 236; Mathematics 121. 6 cr., MS, Winter—R. Nau
CS 237. Theory of Computation All about computing machines: abstract automata, especially finite state machines, push-down automata, and Turing machines. Formal languages, especially context-free languages. The relationship between automata and languages. Computability and solvability. Prerequisites: Computer Science 127; Computer Science 177 or 223 or Mathematics 236 or consent of the instructor. 6 cr., MS, Spring—D. Liben-Nowell
CS 257. Software Design Writing good software is difficult. In this course, we will study techniques, tools, and habits that greatly improve your chances of writing software well. Centering around several medium-sized programming projects, this course will investigate code construction techniques, debugging and profiling tools, testing methodologies, UML, principles of object-oriented design, design patterns, and user interface design. Prerequisite: Computer Science 127 or consent of the instructor. 6 cr., MS, Spring—A. Csizmar Dalal
CS 307. Operating Systems Introduction to the design and construction of operating systems. Sequential and concurrent processes, synchronization and mutual exclusion, memory management techniques, file systems design, security and protection systems, CPU scheduling, input/output device handling, and distributed operating systems. Prerequisite: Computer Science 207 or consent of the instructor. 6 cr., MS, Fall—J. Ondich
CS 317. Computer Graphics The raster graphics representation of 2- and 3- dimensional images. Topics include frame buffers, data structures for image storage, geometric transformations, hidden surface algorithms, raytracing, splines, and lighting models. Prerequisites: Computer Science 127, Mathematics 232 or consent of the instructor. 6 cr., MS, Spring—J. Goldfeather
CS 327. Artificial Intelligence
How can we design computer systems with behavior that seems "intelligent?" This course will examine a number of different approaches to this question, including intelligent agents (simulated with a robot), machine learning (including neural networks and genetic algorithms), and reasoning with uncertainty. We will also examine search methods, with an interest in computer game playing. The coursework is a mix of problem solving and computer programming based on the ideas that we discuss. Prerequisite: Computer Science 127 or consent of the instructor. 6 cr., MS, Offered in alternate years. Not offered in 2005-2006.
CS 337. Computer Networks
From the hotspots in coffee shops to the Internet in our homes, computer networking has increasingly pervaded our everyday lives. In this course, we'll study the technical details of computer networks, from local-area to wide-area networks, from the individual connections between machines to networked applications both new and old. Topics include the TCP/IP protocol stack, the OSI reference model, network architecture, protocols and their implementations, routing security, the structure of the Internet, DNS, and emerging applications such as VoIP and peer-to-peer networking. Prerequisite: Computer Science 207 or consent of the instructor. 6 cr., MS, Offered in alternate years. Winter—A. Csizmar Dalal
CS 347. Database Systems
Database systems are used in almost every aspect of computing, from storing data for websites to maintaining financial information for large corporations. Intrinsically, what is a database system and how does it work? This course takes a two-pronged approach to studying database systems. From a systems perspective, we will look at the low-level details of how a database system works internally, studying such topics as file organization, indexing, sorting techniques, and query optimization. From a theory perspective, we will examine the fundamental ideas behind database systems, such as normal forms and relational algebra. Prerequisite: Computer Science 127 or consent of the instructor. 6 credits cr., MS, Winter—D. Musicant
CS 377. Data Mining
How does Google understand what it is you're looking for? How does Amazon.com figure out what items you might want to buy? These questions and others are part of machine learning and data mining, two highly related fields at the crossroads of artificial intelligence, database systems, and statistics. Machine learning concerns itself with getting a computer to learn or discover patterns, whereas data mining focuses this task on large databases. Much of the material will be presented through primary source research papers, and the content will include techniques such as classification, clustering, association rules, web mining, collaborative filtering, and others. Prerequisite: Computer Science 127 or consent of the instructor. 6 cr., MS, Offered in alternate years. Not offered in 2005-2006.
CS 395. Robotics
This seminar will examine robot motion planning and other related topics. In addition to discussing these techniques, we will implement them on actual hardware. Prerequisites: Computer Science 127 or consent of instructor. 6 cr., MS, Fall—D. Musicant
CS 400. Integrative Exercise Senior Computer Science majors work in teams (typically four to seven students per team) on faculty-specified topics. Required of all senior majors. Prerequisite: Mathematics 121, Computer Science 117, 127, 177 (or Mathematics 236); three courses from among Computer Science 207, 217, 227, 237; one course from among Computer Science courses numbered 247 or above or Mathematics 311. 6 cr., S/NC, ND, Fall,Winter—A. Csizmar Dalal