Mathematics Course Descriptions
I. PRELIMINARY FOR UNDERGRADUATES
*Watch this short video describing the calculus and linear algebra courses.*
FirstYear Seminar 
For freshman 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 35. Written permission required. 
0040 
Calculus and Its History 
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 
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 0010. 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 Thining 
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 I 
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 II 
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 
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 0520 or MA 0540 is a prerequisite for all 100level courses in Mathematics except MA 1260. Prerequisite: MA 0100, MA 0170, or MA 0190. May not be taken in addition to MA 0540. 
0540 
Honors Linear Algebra 
This course provides a rigorous introduction to the theory of linear algebra. Topics covered include: matrices, linear equations, determinants, and eigenvalues; vector spaces and linear transformations; inner products; Hermitian, orthogonal, and unitary matrices; and Jordan normal forms. MATH 0540 provides a more theoretical treatment of the topics in MATH 0520, and students will have opportunities during the course to develop proofwriting skills. [Recommended prerequisites: MA 0100 or equivalent.] 
0750 
Introduction to Higher Mathematics 
This yearlong class is designed for incoming students who have an interest in learning serious mathematics, but is open to all undergraduates. In this course, which will be team taught by senior members of the mathematics faculty, students will be exposed to six fundamental areas of mathematics. Approximately 4 weeks will be devoted to each topic. The class will feature rigorous proofs, concrete interesting examples. This class is not mandatory for prospective math concentrators, but is strongly recommended. 
II. FOR UNDERGRADUATES AND GRADUATES
The standard requirements for all 100level mathematics courses except mathematics 1010 and 1260 are MA 0180, MA 0200, or MA 0350; and MA 0520 or MA 0540.
*Watch this short video describing the 1000level math courses.*
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 
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. 
1230 
Graph Theory 
Graph Theory, with an emphasis on combinatorics and applications in other areas of math. Topics include spanning trees, search algorithms, network flows, matching problems, coloring problems, planarity results, (and if time permits) an introduction to matroids. 
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 
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. 
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 
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. 
1910 
Race and Gender in the Scientific Community 
This course examines (1) disparities in representation in the scientific community, (2) issues facing different groups in the sciences, and (3) paths towards a more inclusive scientific environment. We will delve into the current statistics on racial and gender demographics in the sciences and explore their background through texts dealing with the history, philosophy, and sociology of science. We will also explore the specific problems faced by underrepresented and wellrepresented racial minorities, women, and LGBTQ community members. The course is readingintensive and discussionbased. 
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 
Algebraic Geometry 
Algebraic varieties over algebraically closed fields, affine and projective schemes, divisors, properties of morphisms, and sheaf cohomology. Further topics as chosen by the instructor. 
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 
Real Function Theory 
Point set topology, function spaces, Lebesgue measure and integration, Lp spaces, Hilbert spaces, Banach spaces, differentiability, and applications. 
2250 
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 
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 
Topology 
An introduction to algebraic topology. Topics include fundamental group, covering spaces, simplicial and singular homology, CW complexes, and an introduction to cohomology. 
2510 
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 
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 
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 2210 (or Applied Math 2110) concurrently. 
2710 
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 
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. 