Course Descriptions
I. Primarily for Undergraduates 

0010  FirstYear Seminar  For freshmen only 
0020  What is Mathematics?  A broad overview of the subject, intended primarily for liberal arts students. What do mathematicians do, and why do they do it? We will examine the art of proving theorems, from both the philosophical and aesthetic points of view, using examples such as nonEuclidean geometries, prime numbers, abstract groups, and uncountable sets. Emphasis will be placed on appreciating the beauty and variety of mathematical ideas. The course will include a survey of important results and unsolved problems that motivate mathematical research. 
0030  Mathematics and Poetry (UC 3, English 38)  An interdisciplinary exploration into the creative process and use of imagination as they arise in the study of mathematics and poetry. The goal of the course is to guide each participant towards the experience of independent discovery, be if of a new insight into a math problem or an overlooked aspect of a poem. Students with and without backgrounds in either subject are welcome  no calculus will be required. No prerequisites. Enrollment limited to 0350. Written permission required. 
0040  Calculus and Its History (History of Mathematics 4)  In this course, students interested in learning why the calculus is justly described as one of the greatest achievements of the human spirit will find its concepts and techniques made more accessible by being placed in historical context. Beginning with the roots of calculus n the classical mathematics of antiquity, we will trace its development through the Middle Ages to the work of Newton and Leibniz and beyond. At each stage, we will examine the philosophical and practical challenges to existing mathematics that spurred this continuing development. While the course is aimed primarily at non science concentrators, it will also provide a thorough exposition of the basic techniques of calculus useful for further study of science and mathematics. 
0050, 0060  Analytic Geometry and Calculus  A slowerpaced introduction to calculus for students who require additional preparation for calculus. This sequence presents the same calculus topics as Mathematics 9, together with all the necessary precalculus topics. Students successfully completing this sequence will be prepared for Mathematics 0100. Placement in this course requires permission of the instructor. 
0070  Calculus with Applications to Social Sciences  A onesemester introduction to calculus recommended for students who wish to learn the basics of calculus for application to social sciences or for cultural appreciation as part of a broader education. Topics include functions, equations, graphs, exponentials and logarithms, and differentiation and integration; applications such as marginal analysis, growth and decay, optimization, and elementary differential equations. May not be taken for credit in addition to MA 0090 
0080  The Mathematical Way of Thinking  The course treats topics in geometry of four and higher dimensions, related to different parts of mathematics as well as interrelations with physical and biological sciences, literature, cognitive science, philosophy, and art. There are substantial writing assignments and final projects, involving mathematical and nonmathematical topics. There are no prerequisites. 
0090  Introductory Calculus  An intensive course in the calculus of one variable including limits; differentiation; maxima and minima, and the chain rule for polynomials, rational functions, trigonometric functions, and exponential functions. Introduction of integration with applications to area and volumes of revolution. Mathematics 0090 and 0100 or the equivalent are recommended for all students intending to concentrate in mathematics or the sciences. May not be taken in addition to 0050, 0060, or 0070; 0100 may not be taken in addition to MA 0170. 
0100  Introductory Calculus  A continuation of the material of Mathematics 9 including further development of integration, inverse trigonometric and logarithmic functions, techniques of integrations, and applications which include work and pressure. Other topics covered are infinite series, power series, Taylor's formula, polar coordinates, parametric equations, introduction to differential equations, and numerical methods. Mathematics 0090 and 0100 or the equivalent are recommended for all students intending to concentrate in mathematics or the sciences. 
0170  Advanced Placement Calculus  This course begins with a review of fundamentals of calculus, and includes infinite series, power series, paths, and differential equations of first and second order. Placement in this course is determined by the department on the basis of high school AP examinations scores or the results of tests given by the department during orientation week. May not be taken in addition to MA 0100. 
0180  Intermediate Calculus  Threedimensional analytic geometry. Differential and integral calculus of functions of two or three variables: partial derivatives, multiple integrals, Green's Theorem. Prerequisite: Mathematics 0100 or 0170. Mathematics 0180 may not be taken in addition to MA 0200 or MA 0350. 
0190  Advanced Placement Calculus (Physics/Engineering)  This course, which covers roughly the same material and has the same prerequisites as Mathematics 0170, is intended for students with a special interest in physics or engineering. The main topics are: calculus of vectors and paths in two and three dimensions; differential equations of the first and second order; and infinite series, including power series and Fourier series. 
0200  Intermediate Calculus (Physics/Engineering)  This course, which covers roughly the same material as Mathematics 0180, is intended for students with a special interest in physics or engineering. The main topics are: geometry of threedimensional space; partial derivatives; Lagrange multipliers; double, surface, and triple integrals; vector analysis; Stokes' theorem and the divergence theorem, with applications to electrostatics and fluid flow. Prerequisite: MA 0100, MA 0170, or MA 0190. 
0350  Honors Calculus  A third semester calculus course for students of greater aptitude and motivation. Topics covered include vector analysis, partial differentiation, multiple integration, line integrals, Green's theorem, Stokes' theorem, the divergence theorem, and additional material selected by the instructor. Advanced Placement. Written permission required. 
0420  Introduction to Number Theory  An overview of one of the many beautiful areas of mathematics. It is ideal for any student who wants a taste of mathematics outside of, or in addition to, the calculus sequence. Topics to be covered include: prime numbers, congruences, quadratic reciprocity, sums of squares, Diophantine equations, and as time permits, such topics as cryptography and continued fractions. No prerequisites. 
0520  Linear Algebra  Vector spaces, linear transformations, matrices, systems of linear equations, bases, projections, rotations, determinants, and inner products. Applications may include differential equations, difference equations, least squares approximations, and models in economics and in biological and physical sciences. MA 0520 or MA 0540 is a prerequisite for all 100level courses in Mathematics except MA 1260. May not be taken in addition to MA 0540. 
0540  Honors Linear Algebra  Linear algebra for students of greater aptitude and motivation. Recommended for prospective mathematics concentrators, and science and engineering students who have a good mathematical preparation. Topics include: matrices, linear equations, determinants, characteristic polynomials, and eigenvalues; vector spaces and linear transformations; inner products; Hermitian, orthogonal, and unitary matrices; bilinear forms; elementary divisors and Jordan normal forms. Provides a deeper and more extensive treatment of the topics in MA 0520 and can be substituted for MA 0520 in fulfilling requirements. 
0750, 0760  Introduction to Higher Mathematics  This yearlong class will expose students to six fundamental areas of mathematics. It will be team taught by six members of the faculty. Fall topics will include logic, combinatorics, and analysis. Spring topics will include number theory, algebra, and geometry. Approximately 4 weeks will be devoted to each topic. The class will emphasize rigorous proofs and concrete interesting examples. The class will not be mandatory for prospective math concentrators, but it will be strongly recommended. 
II. For Undergraduates and Graduates 



1010  Analysis: Functions of One Variable  Completeness properties of the real number system, topology of the real line. Proof of basic theorems in calculus, infinite series. Topics selected from ordinary differential equations. Fourier series, Gamma functions, and the topology of Euclidean plane an 3Dspace. Prerequisite: MA 0180, MA 0200, or MA 0350. MA 0520 or MA 0540 may be taken concurrently. Most students are advised to take MA 1010 before MA 1130. 
1040  Fundamental Problems of Geometry  This class discusses geometry from a modern perspective. Topics include hyperbolic, projective, conformal, and affine geometry, and various theorems and structures built out of them. Prerequisite: MA 0520, MA 0540, or permission of the instructor. 
1060  Differential Geometry  The study of curves and surfaces in 2 and 3dimensional Euclidean space using the techniques of differential and integral calculus and linear algebra. Topics include curvature and torsion of curves, FrenetSerret frames, global properties of closed curves, intrinsic and extrinsic properties of surface, Gaussian curvature and mean curvatures, geodesics, minimal surfaces, and the GaussBonnet theorem. 
1110  Ordinary Differential Equations  Ordinary differential equations including existence and uniqueness theorems and the theory of linear systems. Topics may also include stability theory, the study of singularities, and boundary value problems. 
1120  Partial Differential Equations  The wave equation, the heat equation, Laplace's equation, and other classical equations of mathematical physics and their generalizations. Solutions in series of eigenfunctions, maximum principles, the method of characteristics, Green's functions, and discussion of wellposedness problems. 
1130, 1140  Functions of Several Variables  Calculus on manifolds. Differential forms, integration, Stokes' formula on manifolds, with applications to geometrical and physical problems, the topology of Euclidean spaces, compactness, connectivity, convexity, differentiability, and Lebesgue integration. It is recommended that a student take a 100level course in analysis before attempting MA 1130. 
1260  Complex Analysis  This subject is one of the cornerstones of mathematics. Complex differentiability, CauchyRiemann differential equations, contour integration, residue calculus, harmonic functions, and geometric properties of complex mappings. Prerequisite: MA 0180, MA 0200, or MA 0350. This course does not require MA 0520 or MA 0540. 
1270  Topics in Functional Analysis  Infinitedimensional vector spaces, with applications to some or all of the following topics: Fourier series and integrals, distributions, differential equations, integral equations, and calculus of variations. Prerequisite: at least one 100level course in Mathematics or Applied Mathematics or permission of the instructor. 
1410  Combinatorial Topology  Topology of Euclidean spaces, winding number and applications, knot theory, the fundamental group and covering spaces. Euler characteristic, simplicial complexes, the classification of twodimensional manifolds, vector fields, vector fields, the PoincareHopf theorem, and introduction to threedimensional topology. 
1530  Abstract Algebra  An introduction to the principles and concepts of modern abstract algebra. Topics will include groups, rings, and fields, with applications to number theory, the theory of equations, and geometry. MA 1530 is required of all students concentrating in mathematics. 
1540  Topics in Abstract Algebra  Galois theory together with selected topics in algebra. Examples of subjects which have been presented in the past include algebraic curves, group representations, and the advanced theory of equations. Prerequisite: MA 1530. May be repeated for credit. 
1560  Elementary Number Theory  Selected topics in number theory will be investigated. Unique factorization, prime numbers, modular arithmetic, quadratic number fields, finite fields, padic numbers, and related topics. Prerequisite: MA 1530 or written permission. 
1580  Cryptography  Topics include symmetric
ciphers, public key ciphers, complexity, digital signatures,
applications and protocols. Math 1530 will not be required for
the course. What is needed from abstract algebra and elementary
number theory will be covered. Prerequisite: MA 0520 or 0540. 
1610  Probability  Basic probability theory. ample spaces; random variables; normal, Poisson, and related distributions; expectation; correlation; and limit theorems. Applications in many fields (biology, physics, gambling, etc.). 
1620  Mathematical Statistics  Central limit theorem, point estimation, interval estimation, multivariate normal distributions, tests of hypotheses, and linear models. Prerequisite: MA 1610 or permission of the instructor. 
1810, 1820  Special Topics in Mathematics  Topics in special areas of mathematics not included in the regular course offerings. Offered from time to time when there is sufficient interest among qualified students. Contents and prerequisites vary. Written permission required. 
1970  Honors Conference  Collateral reading, individual conferences. 
III. Primarily for Graduates 

2010  Differential Geometry  Introduction to differential geometry (differentiable manifolds, differential forms, tensor fields, homogeneous spaces, fiber bundles, connections,and Riemannian geometry), followed by selected topics in the field. 
2050, 2060  Algebraic Geometry  Complex manifolds and algebraic varieties, sheaves and cohomology, vector bundles, Hodge theory, Kahler manifolds, vanishing theorems, the Kodaira embedding theorem, the RiemannRoch theorem, and introduction to deformation theory. 
2110  Introduction to Manifolds  Inverse function theorem, manifolds, bundles, Lie groups, flows and vector fields, tensors and differential forms, Sard's theorem and transversality, and further topics chosen by instructor. 
2210, 2220  Real Function Theory  Point set topology, function spaces, Lebesgue measure and integration, Lp spaces, Hilbert spaces, Banach spaces, differentiability, and applications. 
2250, 2260  Complex Function Theory  Introduction to the theory of analytic functions of one complex variable. Content varies somewhat from year to year, but always includes the study of power series, complex line integrals, analytic continuation, conformal mapping, and an introduction to Riemann surfaces. 
2370, 2380  Partial Differential Equations (Applied Mathematics 223, 224)  The theory of the classical partial differential equations as well as the method of characteristics and general first order theory. Basic analytic tools include the Fourier transform, the theory of distributions, Sobolev spaces, and techniques of harmonic and functional analysis. More general linear and nonlinear elliptic, hyperbolic, and parabolic equations and properties of their solutions, with examples drawn from physics, differential geometry, and the applied sciences. Generally, semester II of this course concentrates in depth on several special topics chosen by the instructor. 
2410, 2420  Topology  An introductory course with emphasis on the algebraic and differential topology of manifolds. Topics include simplicial and singular homology, de Rham cohomology, and Poincare duality. 
2510, 2520  Algebra  Basic properties of groups, rings, fields, and modules. Topics include: finite groups, representations of groups, rings with minimum condition, Galois theory, local rings, algebraic number theory, classical ideal theory, basic homological algebra, and elementary algebraic geometry. 
2530, 2540  Number Theory  Introduction to algebraic and analytic number theory. Topics covered during the first semester include number fields, rings of integers, primes and ramification theory, completions, adeles and ideles, and zeta functions. Content of the second semester will vary from year to year; possible topics include class field theory, arithmetic geometry, analytic number theory, and arithmetic Ktheory. Prerequisite: MA 2510. 
2630, 2640  Probability (Applied Mathematics 263, 264)  This course introduces probability spaces, random variables, expectation values, and conditional expectations. It develops the basic tools of probability theory, such fundamental results as the weak and strong laws of large numbers, and the central limit theorem. It continues with a study of stochastic processes, such as Markov chains, branching processes, martingales, Brownian motion, and stochastic integrals. Students without a previous course in measure and integration should take MA 2210 (or Applied Math 2110) concurrently. 
2710, 2720  Advanced Topics in Mathematics  Courses recently offered include: Advanced Differential Geometry, Algebraic Number Theory, Elliptic Curves and Complex Multiplication, Harmonic Analysis and Nonsmooth Domains, Dynamical Systems, Metaplectic Forms, Nonlinear Wave Equations, Operator Theory and Functional Analysis, Polynomial Approximation, Several Complex Variables, and Topology and Field Theory. May be repeated for credit. 
2910, 2920  Reading and Research  Independent research or course of study under the direction of a member of the faculty, which may include research for and preparation of a thesis. 
2990  Thesis Preparation  For graduate students who have met the tuition requirement and are paying the Registration Fee to continue active enrollment while preparing a thesis. No course credit. 