Course Descriptions

Mathematics Course Descriptions

I. PRELIMINARY FOR UNDERGRADUATES

*Watch this short video describing the calculus and linear algebra courses.*

0010

First-Year 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 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

 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 0010. 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 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 non-mathematical 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

Three-dimensional 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 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 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 100-level 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 proof-writing skills. [Recommended prerequisites: MA 0100 or equivalent.]

0750
0760

Introduction to Higher Mathematics

This year-long 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 100-level 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 1000-level 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 3D-space. 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 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 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, Cauchy-Riemann 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 

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

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 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, p-adic 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. 

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 well-represented racial minorities, women, and LGBTQ community members. The course is reading-intensive and discussion-based.

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 

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
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 introduction to algebraic topology. Topics include fundamental group, covering spaces, simplicial and singular homology, CW complexes, and an introduction to cohomology. 

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 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 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
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.