 |
 |
|
|
|
|
Spring 2003 Course DescriptionsMATH 103: Mathematical Explorations
This course is for students who wish to experience how mathematical ideas naturally evolve. The homework consists of the students actively investigating mathematical ideas. The course emphasizes ideas and imagination as opposed to techniques and calculations. Topics vary depending on the instructor and are announced (www.math.cornell.edu) several weeks before the semester begins. Some assessment is done through writing assignments. Lecture 01: Winning Ways (James E. West) MATH 106: Calculus For The Life and Social Sciences
Course serves as an introduction to differential and integral calculus, partial derivatives, elementary differential equations. Examples from biology and the social sciences are used. MATH 111: Calculus I
Course topics include: functions and graphs, limits and continuity, differentiation and integration of algebraic, trigonometric, inverse trig, logarithmic, and exponential functions; applications of differentiation, including graphing, max-min problems, tangent line approximation, implicit differentiation, and applications to the sciences; the mean value theorem; and antiderivatives, definite and indefinite integrals, the fundamental theorem of calculus, substitution in integration, the area under a curve. Graphing calculators are used, and their pitfalls are discussed, as applicable to the above topics. MATH 111 can serve as a one-semester introduction to calculus or as part of a two-semester sequence in which it is followed by MATH 112 or 122. MATH 112: Calculus II
Course focus is on integration: applications, including volumes and arc length; techniques of integration, approximate integration with error estimates, improper integrals, differential equations (separation of variables, initial conditions, systems, some applications). Also covered are infinite sequences and series: definition and tests for convergence, power series, Taylor series with remainder, and parametric equations. MATH 122: Honors Calculus
Topics covered include: differentiation and integration of elementary transcendental functions, techniques of integration, applications, polar coordinates, infinite series, and complex numbers, as well as an introduction to proving theorems. The approach is more theoretical than in MATH 112. MATH 171: Statistical Theory and Application In The Real World
This introductory statistics course discusses techniques for analyzing data occurring in the real world and the mathematical and philosophical justification for these techniques. Topics include: population and sample distributions, central limit theorem, statistical theories of point estimation, confidence intervals, testing hypotheses, the linear model, and the least squares estimator. The course concludes with a discussion of tests and estimates for regression and analysis of variance (if time permits). The computer is used to demonstrate some aspects of the theory, such as sampling distributions and the Central Limit Theorem. In the lab portion of the course, students learn and use computer-based methods for implementing the statistical methodology presented in the lectures. (No previous familiarity with computers is presumed.) MATH 191: Calculus For Engineers
Course topics include: plane analytic geometry, differential and integral calculus, and applications. MATH 191 covers essentially the same topics as 190, but is designed for students with some previous successful experience with calculus. MATH 192: Calculus For Engineers
Course topics include: polar coordinates, infinite series, and power series. Also covered are: vectors and calculus of functions of several variables through double and triple integrals. MATH 213: Calculus III
Course topics include: vectors and vector-valued functions; multivariable and vector calculus including multiple and line integrals; first- and second-order differential equations with applications; systems of differential equations; and elementary partial differential equations. This course is designed for students who wish to master the basic techniques of calculus, but whose major will not require a substantial amount of mathematics. The course may emphasize different topics in the syllabus in different semesters. MATH 221: Linear Algebra and Differential Equations
Course covers linear algebra and differential equations. Topics include: vector algebra, linear transformations, matrices, and linear differential equations, as well as an introduction to proving theorems. This course is especially recommended for students who plan to major in mathematics or in a strongly related field. MATH 222: Multivariable Calculus
Course topics include: multivariable and vector differential and integral calculus, including multiple, line, and surface integrals. This course is especially recommended for students who plan to major in mathematics or in a strongly related field. MATH 224: Theoretical Linear Algebra and Calculus
Course topics include: vector fields; line integrals; differential forms and exterior derivative; work, flux, and density forms; integration of forms over parametrized domains; and Green's, Stoke's, and divergence theorems. MATH 231: Linear Algebra
Course topics include: vectors, matrices, and linear transformations, affine and Euclidean spaces, transformation of matrices, and eigenvalues. MATH 281: Deductive Logic (also PHIL 331)
For description, see PHIL 331. MATH 293: Engineering Mathematics
The conclusion of vector calculus, including line integrals, vector fields, Green's theorem, Stokes' theorem, and the divergence theorem; followed by an introduction to ordinary and partial differential equations, including Fourier series and boundary value problems. May include computer use in solving problems. MATH 294: Engineering Mathematics
Linear algebra and its applications. Topics include matrices, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality and inner product spaces; applications include brief introductions to difference equations, Markov chains, and systems of linear ordinary differential equations. May include computer use in solving problems. MATH 311: Introduction to Analysis
Provides a transition from calculus to real analysis. Topics include: rigorous treatment of fundamental concepts in calculus: including limits and convergence of sequences and series, compact sets; continuity, uniform continuity and differentiability of functions. Emphasis will be placed upon understanding and constructing mathematical proofs. MATH 336: Applicable Algebra
An introduction to the concepts and methods of abstract algebra and number theory that are of interest in applications. Covers: basic theory of groups, rings and fields and their applications to such areas as public-key cryptography, error-correcting codes, parallel computing, and experimental designs. Also covers: elementary number theory, Euclidean algorithm, prime factorization, congruences, theorems of Fermat and Euler, elementary group theory, Chinese remainder theorem, factorization in the ring of polynomials, and classification of finite fields. Applications include the RSA cryptosystem and use of finite fields to construct error-correcting codes and Latin squares. MATH 356: Groups and Geometry
Groups were introduced in the nineteenth century as the sets of symmetries of algebraic or geometric objects. This viewpoint has become central in modern mathematics. This course studies the geometry of the plane and of patterns in the plane in terms of the group of symmetries ("isometries") of the plane. Prior knowledge of group theory is not a prerequisite. The purpose of the course is to prepare students for the 400-level courses in several ways. On one hand, the course offers experience in modern algebra and geometry (including the geometry of complex numbers). It presents some very beautiful and important topics and a sense of the unity of mathematics. On the other hand, special care is taken to initiate the student into the writing of proofs and the language of mathematics. Topics include: Symmetries, groups of transformations, subgroups and cosets. Homomorphisms and isomorphisms. Orbits and fixed points. Frieze groups, wallpaper groups ("2-dimensional crystallographic groups") and the associated tesselations of the Euclidean plane. MATH 401: Honors Seminar: Topics In Modern Mathematics
This course is a participatory seminar primarily aimed at introducing senior and junior mathematics majors to some of the challenging problems and areas of modern mathematics. The seminar helps students develop research and expository skills in mathematics, which is important for careers in any field that makes significant use of the mathematical sciences (i.e., pure or applied mathematics, physical or biological sciences, business and industry, medicine). The content varies from year to year. MATH 403: History of Mathematics
We will survey the development of mathematical ideas from antiquity to the present. The organizing theme for this semester will be the problems, ideas, and controversies that led to the development of the calculus and how these problems, ideas, and controversies are still influencing current mathematical research. As much as possible we will be reading from original sources in translation. There will be weekly writing assignments and occasional oral presentations that explore mathematical ideas from the past and their influences today. In addition, each student will complete a term project that will explore in depth an issue in the history of mathematics. As is appropriate for a senior-level mathematics course, the emphasis throughout will be on the mathematical ideas. MATH 408: Mathematics In Perspective
The purpose of this course is for students to step back and form an overview of the mathematics they have learned. The course is intended for junior and senior mathematics majors and other undergraduates with strong backgrounds in mathematics. MATH 414: Honors Introduction to Analysis II
The sequence MATH 413-414, designed for honors students, provides an introduction to the theory of functions of real variables, stressing a rigorous logical development of the subject rather than applications. Topics include: metric spaces, the real number system, continuous and differentiable functions, uniform convergence and approximation theorems, Fourier series, Riemann and Lebesgue integrals, calculus in several variables, and differential forms. MATH 418: Introduction To The Theory of Functions of One Complex Variable
A theoretical and rigorous introduction to complex variable theory. Topics include: complex numbers, differential and integral calculus for functions of a complex variable including Cauchy's theorem and the calculus of residues, elements of conformal mapping. Students interested in the applications of complex analysis should consider MATH 422. MATH 420: Differential Equations and Dynamical Systems
Course covers ordinary differential equations in one and higher dimensions: qualitative, analytic, and numerical methods. Emphasis is on differential equations as models and the implications of the theory for the behavior of the system being modeled and includes an introduction to bifurcations. MATH 422: Applied Complex Analysis
Course covers complex variables, Fourier transforms, Laplace transforms and applications to partial differential equations. Additional topics may include an introduction to generalized functions. MATH 424: Wavelets and Fourier Series
Both Fourier series and wavelets provide methods to represent or approximate general functions in terms of simple building blocks. Such representations have important consequences, both for pure mathematics and for applications. Fourier series use natural sinusoidal building blocks and may be used to help solve differential equations. Wavelets use artificial building blocks that have the advantage of localization in space. A full understanding of both topics requires a background involving Lebesgue integration theory and functional analysis. This course presents as much as possible on both topics without such formidable prerequisites. The emphasis is on clear statements of results and key ideas of proofs, working out examples, and applications. Related topics that may be included in the course: Fourier transforms, Heisenberg uncertainty principle, Shannon sampling theorem, and Poisson summation formula. MATH 428: Introduction To Partial Differential Equations
Topics selected from first-order quasilinear equations, classification of second-order equations, with emphasis on maximum principles, existence, uniqueness, stability, Fourier series methods, approximation methods. MATH 432: Introduction To Algebra
An introduction to various topics in abstract algebra, including: groups, rings, fields, factorization of polynomials and integers, congruences, and the structure of finitely generated abelian groups. Optional topics: modules over Euclidean domains, Sylow theorems. MATH 434: Honors Introduction To Algebra II
Math 433-434 is the honors version of MATH 431-432. MATH 433-434 is more theoretical and rigorous than 431-432 and includes additional material such as multilinear and exterior algebra. MATH 452: Classical Geometries
This is an introduction to hyperbolic and projective geometry º the classical geometries that developed as Euclidean geometry was better understood. For example, the historical problem of the independence of Euclid's fifth postulate is understood when the existence of the hyperbolic plane is realized. Straightedge (and compass) constructions and stereographic projection in Euclidean geometry can be understood within the structure of projective geometry. Topics in hyperbolic geometry include: models of the hyperbolic plane and relations to spherical geometry. Topics in projective geometry include: homogeneous coordinates and the classical theorems about conics and configurations of points and lines. Optional topics include: principles of perspective drawing, finite projective planes, orthogonal Latin squares, and the cross ratio. MATH 454: Introduction To Differential Geometry
Course covers differential geometry of curves and surfaces. Also covers curvature, geodesics, and differential forms. Serves as an introduction to n-dimensional Riemannian manifolds. This material provides some background for the study of general relativity; connections with the latter are indicated. MATH 472: Statistics
Statistics have proved to be an important research tool in nearly all of the physical, biological, and social sciences. This course will serve as an introduction to statistics for students who already have some background in calculus, linear algebra, and probability theory. Topics covered in the course will include parameter estimation, hypothesis testing, and linear regression. The course will emphasize both the mathematical theory of statistics as well as techniques for data analysis that are useful in solving scientific problems. MATH 481: Mathematical Logic (also PHIL 431)
Course covers: propositional and predicate logic; classical proof procedures; completeness and compactness; decidability and undecidability; the Godel incompleteness theorem; and elements of set theory. MATH 486: Applied Logic (also COM S 486)
Course covers: propositional and predicate logic; compactness and completeness by tableaux, natural deduction, and resolution. Other possible topics include: equational logic; Herbrand Universes and unification; rewrite rules and equational logic, Knuth-Bendix method and the congruence-closure algorithm and lambda-calculus reduction strategies; topics in Prolog, LISP, ML, or Nuprl; and applications to expert systems and program verification. MATH 490: Supervised Reading and Research
Supervised reading and research by arrangement with individual professors. Not for material currently available in regularly scheduled courses. MATH 507: Issues in Teaching and Learning Secondary Mathematics
Math 507 is a tour through contemporary issues in Mathematics Education. The course is appropriate for anyone who is going to teach mathematics, and those who will be teaching teachers of mathematics: all mathematics graduate students, senior undergraduate math majors, and math education MAT students. We will address such questions as: What does it mean to think mathematically? How can we empower a broader range of students to do mathematics? How can we help students to integrate mathematics within their world experience? What new activities and approaches are being developed and used in k-12 mathematics? We will look at mathematics education through the following lenses: Philosophy of Mathematics Education; contemporary approaches and materials; NCTM standards and frameworks; problem solving, reasoning, intuition and proof; exploring mathematics in a context; social issues/systemic change; appropriate technology (including Geometerís Sketchpad.) Each week we will explore some math activity or problem and read an article as a basis for discussion about a central/current issue. Exemplary activities from elementary/middle/high school topics as well as videos of classroom lessons will supplement our inquiry. Theory and practice combine to deepen our understanding of a whole view of teaching and learning mathematics, and enhance our practice of teaching. Participants will do observations in local school classes, write short papers on discussion topics, and occasionally present ideas and lead discussions. MATH 508: Mathematics For Secondary School Teachers
An examination of the principles underlying the content of the secondary school mathematics curriculum, including connections with the history of mathematics and current mathematics research. Graduate CoursesMATH 790: Supervised Reading and Research
Last modified: April 7, 2003 |
