Mathematics

Professors Brooks and Haines (on leave, winter semester and Short Term); Associate Professors Ross, Rhodes, Chair, and Wong; Assistant Professors Shulman and Johann; Ms. Harder, Ms. Hemenway, and Ms. Cox

Mathematics today is a dynamic and ever-changing subject, and an important part of a liberal-arts education. Mathematical skills such as data analysis, problem solving, pattern recognition, statistics, and probability are increasingly vital to science, technology, and society itself. Entry-level courses introduce students to basic concepts and tools and hint at some of the power and beauty behind these fundamental results. Upper-level courses and senior thesis option provide majors with the opportunity to explore mathematical topics in greater depth and sophistication, and delight in the fascination of this "queen of the sciences."

During new-student orientation the Department conducts an information session on placement for all new students planning to study mathematics. Based on a student's academic background and skills, the Department recommends an appropriate starting course: Mathematics 103, 105, 106, 205, 206, or a more advanced course.

The major in mathematics consists of: 1) Mathematics 205, 206; 2) Mathematics s21, which should be taken during Short Term of the first year; 3) Mathematics 301, 309, and five elective mathematics or computer-science courses numbered 200 or higher; 4) a one-hour oral presentation; and 5) either a written comprehensive examination or a two-semester thesis (Mathematics 457-458). This option requires Departmental approval. Entering students may be exempted from any of the courses in 1) on the basis of work before entering college. Any mathematics or computer-science Short Term unit numbered 30 or above may be used as one of the electives in 3). One elective may also be replaced by a departmentally approved course from another department.

The mathematics major requirements accommodate a wide variety of interests and career goals. The courses provide broad training in undergraduate mathematics and computer science, preparing majors for graduate study, and for positions in government, industry, and the teaching profession.

While students must consult with their major advisors in designing appropriate courses of study, the following suggestions may be helpful: For majors considering a career in secondary education we suggest Mathematics 312, 314, 315, 341, and Computer Science 101 and 102. Students interested in operations research, business, or actuarial science should consider Mathematics 218, 239, 314, 315, 341, s32, and the courses in computer science. Students interested in applied mathematics in the physical and engineering sciences should consider Mathematics 218, 219, 308, 314, 315, 341, and the courses in computer science. Majors planning on graduate study in pure mathematics should particularly consider Mathematics 302, 308, 313, and 317. Mathematics majors may pursue individual research either through 360 (Independent Study) or 457-458 (Senior Thesis).

Students normally begin study of computer science with Computer Science 101. New students who have had the equivalent of 101 and would like to continue should consult with the Department.

The four core courses required for the secondary concentration in computer studies are Computer Science 101 and 102 and any two from Computer Science 201, 202, 301, or 302. Mathematics 218 and any of the computer-science courses or units not credited toward the core may be credited toward the three electives required for the concentration. The complete list of electives includes courses from other departments as well, and is designated annually by the Computing Services Committee.

Students interested in a career in computer science should consider not only computer-science courses, but also Mathematics 205, 218, 239, 314, and 315.

General Education. The quantitative requirement is satisfied by any of the mathematics or computer-science courses or units.

Courses
101. Working with Data. Techniques for analyzing data are described in ordinary English without emphasis on mathematical formulas. Graphical and descriptive techniques for summarizing data, design of experiments, sampling, analyzing relationships, statistical models, and hypothesis testing. Applications from everyday life: drug testing, legal discrimination cases, public-opinion polling, industrial quality control, and reliability analysis. Students are instructed in the use of the computer, which is used extensively throughout the course. Enrollment limited to 30. M. Harder.

103-104. Calculus with Algebra. Designed for students whose backgrounds in algebra are weak, or who have not taken any mathematics recently, the course covers in two semesters the material of Mathematics 105 together with a review of the necessary precalculus ideas. A student receives one course credit upon completion of the consecutive fall and winter semesters. Students may then enroll in Mathematics 106 if they wish to continue their study of calculus. Staff.

105. Calculus I. While the word "calculus" originally meant any method of calculating, it has come to refer more specifically to the fundamental ideas of differentiation and integration that were first developed in the seventeenth century. The subject's early development was intimately connected with understanding rates of change within the context of the physical sciences. Nonetheless, it has proved to be of wide applicability throughout the natural sciences and some social sciences, as well as crucial to the development of most modern technology. This course develops the key notions of derivatives and integrals and their interrelationship, as well as applications. An emphasis is placed on conceptual understanding and interpretation, as well as on calculational skills. Graphing calculators are used in the course for graphical and numerical explorations. Enrollment limited to 25 per section. B. Shulman, R. Sampson, S. Ross.

106. Calculus II. A continuation of Calculus I. Further techniques of integration, both symbolic and numerical, are studied. The course then treats applications of integration to problems drawn from fields, such as physics, biology, chemistry, economics, and probability. Differential equations and their applications are also introduced, as well as approximation techniques, such as Taylor series. Graphing calculators are used in the course for graphical and numerical explorations. Prerequisite(s): Mathematics 105. Enrollment limited to 25 per section. J. Rhodes, P. Wong.

155. Mathematical Models in Biology. Mathematical models are increasingly important throughout the life sciences. This course provides an introduction to deterministic and stochastic models in biology, and to methods of fitting and testing them against data. Examples are chosen from a variety of biological and medical fields, such as ecology, molecular evolution, and infectious disease. Computers are used extensively for modeling and for analyzing data. This course is the same as Biology 155. Recommended background: Biology 101s. Enrollment limited to 30. Not open to students who have received credit for Biology 255. J. Rhodes.

205. Linear Algebra. Vectors and matrices are introduced as devices for the solution of systems of linear equations with many variables. Although these objects can be viewed simply as algebraic tools, they are better understood by applying geometric insight from two and three dimensions. This leads to an understanding of higher dimensional spaces and to the abstract concept of a vector space. Other topics include orthogonality, linear transformations, determinants, and eigenvectors. This course should be particularly useful to students majoring in any of the natural sciences or economics. Prerequisite: one 100-level mathematics course. Open to first-year students. D. Haines.

206. Multivariable Calculus. This course extends the ideas of Calculus I and II to deal with functions of more than one variable. Because of the multidimensional setting, essential use is made of the language of linear algebra. While calculations tend to make straightforward use of the techniques of single-variable calculus, more effort must be spent in developing a conceptual framework for understanding curves and surfaces in higher-dimensional spaces. Topics include partial derivatives, derivatives of vector-valued functions, vector fields, integration over regions in the plane and three space, and integration on curves and surfaces. This course should be particularly useful to students majoring in any of the natural sciences or economics. Prerequisite(s): Mathematics 106 and 205. Open to first-year students. J. Rhodes.

218. Numerical Analysis. This course studies the best ways to perform calculations that have already been developed in other mathematics courses. For instance, if a computer is to be used to approximate the value of an integral, one must understand both how quickly an algorithm can produce a result and how trustworthy that result is. While students will implement algorithms on computers, the focus of the course is the mathematics behind the algorithms. Topics may include interpolation techniques, approximation of functions, finding solutions of equations, differentiation and integration, solution of differential equations, Gaussian elimination and iterative solutions of linear systems, and eigenvalues and eigenvectors. Prerequisite(s): Mathematics 106 and 205 and Computer Science 101.
D. Haines.

219. Differential Equations. A differential equation is a relationship between a function and its derivatives. Many real world situations can be modeled using these relationships. This course is a blend of the mathematical theory behind differential equations and their applications. The emphasis is on first and second order linear equations. Topics include existence and uniqueness of solutions, power series solutions, numerical methods, and applications such as populations models and mechanical vibrations. Prerequisite(s): Mathematics 206. B. Shulman.

239. Linear Programming and Game Theory. Linear programming is an area of applied mathematics that grew out of the recognition that a wide variety of practical problems reduces to the purely mathematical task of maximizing or minimizing a linear function whose variables are restricted by a system of linear constraints. A closely related area is game theory, which provides a mathematical way of dealing with decision problems in a competitive environment, where conflict, risk, and uncertainty are often involved. The course focuses on the underlying theory, but applications to social, economic, and political problems abound. Topics include the simplex method for solving linear-programming problems and two-person zero-sum games, the duality theorem of linear programming, and the min-max theorem of game theory. Additional topics will be drawn from such areas as n-person game theory, network and transportation problems, and relations between price theory and linear programming. Computers are used regularly. This course is the same as Economics 239. Prerequisite(s): Computer Science 101 and Mathematics 205. Staff.

301. Real Analysis. An introduction to the foundations of mathematical analysis, this course presents a rigorous treatment of elementary concepts such as limits, continuity, differentiation, and integration. Elements of the topology of the real numbers will also be covered. Prerequisite(s): Mathematics 206 and s21. S. Ross.

308. Complex Analysis. This course extends the concepts of calculus to deal with functions whose variables and values are complex numbers. Instead of producing new complications, this leads to a theory that is not only more aesthetically pleasing, but is also more powerful. The course should be valuable not only to those interested in pure mathematics, but also to those who need additional computational tools for physics or engineering. Topics include the geometry of complex numbers, differentiation and integration, representation of functions by integrals and power series, and the calculus of residues. Prerequisite(s): Mathematics 206. Staff.

309. Abstract Algebra I. An introduction to basic algebraic structures, many of which are introduced either in high-school algebra or in Mathematics 205. These include the integers and their arithmetic, modular arithmetic, rings, polynomial rings, ideals, quotient rings, fields, and groups. Prerequisite(s): Mathematics 205 and s21. P. Wong.

312. Foundations of Geometry. The study of the evolution of geometric concepts starting from the ancient Greeks (800 B.C.) and continuing to current topics. These topics are studied chronologically as a natural flow of ideas: conic sections from the Greek awareness of astronomy, continuing to Kepler and Newton; perspective in art and geometry; projective geometry including the Gnomic, Mercator, and Stereographic terrestrial maps; Euclidean and non-Euclidean geometries with their respective axiomatic structure; isometries; the inversion map in the plane and in three-space; curvature of curves and surfaces; graph theory including tilings (tesselations); fixed point theorems; space-time geometry. Geometers encountered are Euclid, Apollonius, Pappus, Descartes, Dürer, Kepler, Newton, Gauss, Riemann, A.W. Tucker, and others. R. Sampson.

313. Topology. A study of those geometric properties of space which are invariant under transformations. Properties include continuity, compactness, connectedness, and separability. Prerequisite(s): Mathematics 206 and s21. Staff.

314. Probability. Probability theory is the foundation on which statistical data analysis depends. This course together with its sequel, Mathematics 315, covers topics in mathematical statistics. Both courses are recommended for math majors with an interest in applied mathematics and for students in other disciplines, such as psychology and economics who wish to learn about some of the mathematical theory underlying the methodology used in their fields. Prerequisite(s): Mathematics 106. R. Brooks.

315. Statistics. The sequel to Mathematics 314. This course covers estimation theory and hypothesis testing. Prerequisite(s): Mathematics 314. M. Harder.

317. Differential Geometry. This course further develops the ideas of multivariable calculus to study the geometry of curves and surfaces. The concepts of the tangent space and orientation are used to understand the curvature of n dimensional surfaces. Geodesics on surfaces are introduced as analogs of straight lines in Euclidean space. Students with an interest in physics may find this course a useful introduction to some of the ideas behind mathematical formulations of general relativity. Prerequisite(s): Mathematics 206. Staff.

341. Mathematical Modeling. Often we are interested in analyzing complex situations (like the weather, a traffic flow pattern, or an ecological system) in order to predict qualitatively the effect of some action. The purpose of this course is to provide experience in the process of using mathematics to model real-life situations. The first half examines and critiques specific examples of the modeling process from various fields. During the second half each student creates, evaluates, refines, and presents a mathematical model from a field of his or her own choosing. Prerequisite(s): Mathematics 206. B. Shulman.

360. Independent Study. Independent study by an individual student with a single faculty member. Permission of the Department is required. Students are limited to one independent study per semester. Staff.

365. Special Topics. Content varies from semester to semester. Possible topics include chaotic dynamical systems, number theory, mathematical logic, representation theory of finite groups, measure theory, algebraic topology, combinatorics, and graph theory. Prerequisites vary with the topic covered but are usually Mathematics 301 and/or 309. Staff.

457-458. Senior Thesis. Prior to entrance into Mathematics 457, students must submit a proposal for the work they intend to undertake toward completion of a two-semester thesis. Open to all majors upon approval of the proposal. Required of candidates for honors. Students register for Mathematics 457 in the fall semester and Mathematics 458 in the winter semester. Staff.

Short Term Units

s21. Introduction to Abstraction. An intensive development of the important concepts and methods of abstract mathematics. Students work individually, in groups, and with the instructors to prove theorems and solve problems. Students meet for up to five hours daily to explore such topics as proof techniques, logic, set theory, equivalence relations, functions, and algebraic structures. The unit provides exposure to what it means to be a mathematician. Prerequisite; one semester of college mathematics. Required of all majors. J. Rhodes, Staff.

s32. Topics in Operations Research. An introduction to a selection of techniques that have proved useful in management decision-making: queuing theory, inventory theory, network theory (including PERT and CPM), statistical decision theory, computer modeling, and dynamic programming. Prerequisite(s): Mathematics 105 and a course in probability or statistics. Enrollment limited to 20. Written permission of the instructor is required. Staff.

s45. Seminar in Mathematics. The content varies. Recent topics have included Inverse Problems in the Mathematical Sciences and Introduction to Error Correcting Codes.

s45B. Inverse Problems in the Mathematical Sciences. A myriad of important problems with applications in all the sciences are posed in a way that inverts a direct problem. Traditional undergraduate mathematics is dominated by these direct problems, but modern science and technology confront students with inverse problems (i.e., in remote sensing, medical imaging, non-destructive testing, environmental monitoring, seismic surveys). This unit treats the mathematical theory and practical applications of inverse problems with examples drawn from geology, biology, chemistry, and physics. Prerequisite(s): Mathematics 105 and 106. B. Shulman.

s45C. Introduction to Number Theory. Number Theory is one of the oldest branches of mathematics, and yet has extremely important applications in today's society. Many of its problems are so easy to state, yet are still unsolved; until just recently, Fermat's Last Theorem was such an example. We study properties of the natural numbers and ask questions about such topics as divisibility, prime numbers, solving certain forms of equations, and functions defined just on the natural numbers. Prerequisite(s): Mathematics s21. S. Ross.

s50. Individual Research. The Department permits registration for this unit only after the student submits a written proposal for a full-time research project to be completed during the Short Term and obtains the sponsorship of a member of the Department to direct the study and evaluate its results. Students are limited to one individual research unit. Staff.

Computer Science
101. Computer Science I. An introduction to computer science, with the major emphasis on the design, development, and testing of computer software. It introduces the student to a disciplined approach to problem-solving and system development in a modern programming environment using an object-based event-driven programming language. Students develop programs in Visual BASIC to run under the Windows operating system. The course is taught entirely in a hands-on laboratory setting. Students spend the last portion of the course on an individual or group project of their own choice. Enrollment limited to 16 per section. R. Brooks, P. Johann.

102. Computer Science II. A continuation of Computer Science I. The major emphasis of the course is on object-oriented software design and development using the C++ language. The object-oriented paradigm provides the context for studying additional topics such as data structures, software engineering, and large software systems. Students spend the last portion of the course on an individual or group project of their own choice. Computer Science 101 and 102 provide a foundation for further study in computer science. Prerequisite(s): Computer Science 101. Enrollment limited to 16 per section.
R. Brooks.

201. Principles of Computer Organization. Computer and processor architecture and organization including topics such as operating systems, memory organization, addressing modes, segmentation, input/output, control, synchronization, interrupts, multiprocessing, and multitasking. The course includes training in digital logic, machine language programming, and assembly language programming. Prerequisite(s): Computer Science 101. Open to first-year students. P. Johann.

202. Principles of Programming Languages. An introduction to the major concepts and paradigms of contemporary programming languages. Concepts covered include procedural abstraction, data abstraction, tail-recursion, binding and scope, assignment, and generic operators. Paradigms covered include imperative (e.g., Pascal and C), functional (e.g., LISP), object-oriented (e.g., Smalltalk), and logic (e.g., Prolog). Students write programs in SCHEME to illustrate the paradigms. Prerequisite(s): Computer Science 102. Open to first-year students. D. Haines.

301. Algorithms. The course covers specific algorithms (searching, sorting, merging, and network algorithms), related data structures, an introduction to complexity theory (O-notation, the classes P and NP, NP-complete problems, and intractable problems), and laboratory investigation of algorithm complexity and efficiency. Students gain extensive further computing experience, both in the programming of specific algorithms and in the empirical investigation of their efficiency. Prerequisite(s) or Corequisite(s): Computer Science 101 and 102. Open to first-year students. R. Brooks.

302. Theory of Computation. A course in the theoretical foundations of computer science. Topics include finite automata and regular languages, pushdown automata and context-free languages, Turing machines, computability and recursive functions, and complexity. Prerequisite(s): Computer Science 102. Staff.

360. Independent Study. Independent study by an individual student with a faculty member. Permission of the Department is required. Students are limited to one independent study per semester. Staff.

365. Special Topics. A seminar usually involving a major project. Recent topics have been: the mathematics and algorithms of computer graphics, in which students designed and built a computer-graphics system, and contemporary programming languages and their implementations, in which students explored new languages, in some cases using the Internet to obtain languages such as Oberon, Python, Haskell, and Dylan. Written permission of the instructor is required. Staff.

Short Term Units

s45. Seminar in Computer Science. The content varies. A recent topic was Cryptography and Data Security. Prerequisites vary with the topic covered. Staff.

s50. Individual Research. The Department permits registration for this unit only after the student submits a written proposal for a full-time research project to be completed during the Short Term and obtains the sponsorship of a member of the Department to direct the study and evaluate its results. Students are limited to one individual research unit. Staff.

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