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 0090, together with all the necessary pre-calculus 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 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, exponential 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 non-mathematical topics. There are no prerequisites.

0090

Single Variable Calculus, Part 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

Single Variable Calculus, Part II

A continuation of the material of MATH 90 including further development of techniques of integration. Other topics covered are infinite series, power series, Taylor's formula, polar coordinates, parametric equations, introduction to differential equations, and numerical methods. MATH 90 and 100 or the equivalent are recommended for all students intending to concentrate in mathematics or the sciences. MATH 100 may not be taken in addition to MATH 170 or MATH 190.

0170

Single Variable Calculus, Part II (Accelerated)

This course, which covers roughly the same material and has the same prerequisites as MATH 100, covers integration techniques, sequences and series, parametric and polar curves, and differential equations of first and second order. Topics will generally include more depth and detail than in MATH 100. MATH 170 may not be taken in addition to MATH 100 or MATH 190.

0180

Multivariable Calculus

Three-dimensional analytic geometry. Differential and integral calculus of functions of two or three variables: partial derivatives, multiple integrals, Green's Theorem, Stokes' theorem, and the divergence theorem. Prerequisite: MATH 100, MATH 170, or MATH 190, or advanced placement or written permission. MATH 180 may not be taken in addition to MATH 200 or MATH 350.

0190

Single Variable Calculus, Part II 
(Physics/Engineering)

This course, which covers roughly the same material and has the same prerequisites as MATH 100, 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. MATH 190 may not be taken in addition to MATH 100 or MATH 170.

0200

Multivariable Calculus (Physics/Engineering)

This course, which covers roughly the same material as MATH 180, 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: MATH 100, MATH 170, or MATH 190, or advanced placement or written permission. MATH 200 may not be taken in addition to MATH 180 or MATH 350.

0350

Multivariable Calculus with Theory

This course provides a rigorous treatment of multivariable calculus. Topics covered include vector analysis, partial differentiation, multiple integration, line integrals, Green's theorem, Stokes' theorem, and the divergence theorem. MATH 350 covers the same material as MATH 180, but with more emphasis on theory and on understanding proofs. Prerequisite: MATH 100, MATH 170, or MATH 190, or advanced placement or written permission. MATH 350 may not be taken in addition to MATH 180 or MATH 200.

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

Linear Algebra with Theory

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 form. MATH 540 provides a more theoretical treatment of the topics in MATH 520, and students will have opportunities during the course to develop proof-writing skills. Recommended prerequisite: MATH 100, MATH 170, or MATH 190. MATH 540 may not be taken in addition to MATH 520.

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.

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, discussion of well-posedness problems. The method of characteristics, initial  and boundary conditions, separation of variables, solutions in series of eigenfunctions, Fourier series, maximum principles, Green’s identities and Green’s functions.

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, the Poincare-Hopf theorem, and introduction to three-dimensional topology. 

1530

Abstract Algebra 

A proof-based course that introduces 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. Previous proof-writing experience is not required. 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

Number Theory 

Selected topics in number theory will be investigated. Unique factorization, prime numbers, modular arithmetic, arithmetic functions, quadratic reciprocity, finite fields, and related topics. Prerequisite: MA 1530 or written permission. 

1580

Cryptography 

This course focuses on the mathematics underlying public key cryptosystems, digital signatures, and other topics in cryptography. A sampling of mathematical topics, such as groups, rings, fields, number theory, probability, complexity theory, elliptic curves, and lattices, will be introduced and applied to cryptography. No prior knowledge of these topics is assumed, nor is prior programming experience needed; any programming knowledge required will be covered in class.

1610

Probability 

Basic probability theory including random variables, distribution functions, independence, expectation, variance, and conditional expectation.  Classical examples of probability density and mass functions (binomial, geometric, normal, exponential) and their applications.  Stochastic processes including discrete and continuous time Poisson processes, Markov chains, and Brownian motion.  

1620

Mathematical Statistics 

Frequentist and Bayesian viewpoints and decision theory principles.  Concepts from probability, including the central limit theorem and multivariate normal distributions, and asymptotic estimates.  Inferences from independent, identically distributed sampling: point estimation, confidence intervals, and hypothesis testing.  Analysis of variance (ANOVA) and generalized linear models of regression.

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. 

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. *Undergraduate prerequisites: 2110 and undergraduates require permission from the instructor.

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. * Undergraduate prerequisites: 2510, 2520, and undergraduates require permission from 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. *Undergraduate prerequisites: 1060, 1140, and preferably 1410, and undergraduates require permission from the 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. * 2410 Undergraduate 1410, 1530, and 1010 and/or 1130 and undergraduates require permission from the instructor

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.  *2510 Undergraduate prerequisites:1530 and 1540 and undergraduates require permission from the instructor.  2520 undergraduate prerequisites: 2510 and undergraduates require permission from the instructor.

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.  *Undergraduate prerequisites: 2510 and undergraduates require permission from the instructor.

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.