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Course Descriptions

I. Primarily for Undergraduates

0010 First-Year 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 non-Euclidean 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 35. 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 slower-paced 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 pre-calculus topics. Students successfully completing this sequence will be prepared for Mathematics 10. Placement in this course requires permission of the instructor.
0070 Calculus with Applications to Social Sciences  A one-semester 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 9
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 non-mathematical 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 9 and 10 or the equivalent are recommended for all students intending to concentrate in mathematics or the sciences. May not be taken in addition to 5, 6, or 7; 10 may not be taken in addition to MA 17.
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 9 and 10 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 10.
0180 Intermediate Calculus  Three-dimensional analytic geometry. Differential and integral calculus of functions of two or three variables: partial derivatives, multiple integrals, Green's Theorem. Prerequisite: Mathematics 10 or 17. Mathematics 18 may not be taken in addition to MA 20 or MA 35.
0190 Advanced Placement Calculus (Physics/Engineering)  This course, which covers roughly the same material and has the same prerequisites as Mathematics 17, 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 18, is intended for students with a special interest in physics or engineering. The main topics are: geometry of three-dimensional 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 10, MA 17, or MA 19.
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 This course will provide an overview of one of the most 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 52 or MA 54 is a prerequisite for all 100-level courses in Mathematics except MA 126. Prerequisite: MA 10, MA 17, or MA 19. May not be taken in addition to MA 54.
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 52 and can be substituted for MA 52 in fulfilling requirements. Prerequisite: MA 18, MA 20, or MA 35.
 

 II. For Undergraduates and Graduates

 

  The standard requirements for all 100-level mathematics courses except Mathematics 101 and 126 are MA 18, MA 20, or MA 35; and MA 52 or MA 54.
 
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 3D-space. Prerequisite: MA 18, MA 20, or MA 35. MA 52 or MA 54 may be taken concurrently. Most students are advised to take MA 101 before MA 113.
1040 Fundamental Problems of Geometry The topics covered are chosen from Euclidean, non-Euclidean, projective, and affine geometry. This course is highly recommended for all students who are considering teaching high school mathematics. Prerequisite: MA 52, MA 54, or permission of the instructor.
1060 Differential Geometry The study of curves and surfaces in 2- and 3-dimensional Euclidean space using the techniques of differential and integral calculus and linear algebra. Topics include curvature and torsion of curves, Frenet-Serret frames, global properties of closed curves, intrinsic and extrinsic properties of surface, Gaussian curvature and mean curvatures, geodesics, minimal surfaces, and the Gauss-Bonnet 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 well-posedness 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 100-level course in analysis before attempting MA 113.
1260 Complex Analysis This subject is one of the cornerstones of mathematics. Complex differentiability, Cauchy-Riemann differential equations, contour integration, residue calculus, harmonic functions, and geometric properties of complex mappings. Prerequisite: MA 18, MA 20, or MA 35. This course does not require MA 52 or MA 54.
1270 Topics in Functional Analysis Infinite-dimensional 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 100-level 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 two-dimensional manifolds, vector fields, vector fields, the Poincare-Hopf theorem, and introduction to three-dimensional 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 153 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 153. 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, p-adic numbers, and related topics. Prerequisite: MA 153 or written permission.
1580 Cryptography Topics include symmetric ciphers, public key ciphers, complexity, digital signatures, applications and protocols. Math 153 will not be required for the course. What is needed from abstract algebra and elementary number theory will be covered.
Prerequisite: MA 52 or 54.
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 161 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 Riemann-Roch 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 K-theory. Prerequisite: MA 251.
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 Maarkov chains, branching processes, martingales, Brownian motion, and stochastic integrals. Students without a previous course in measure and integration should take MA 221 (or Applied Math 211) 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 Non-smooth 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.