Department of Mathematics
Course Descriptions
Course Descriptions
Preliminary For Undergraduates
Introductory Curriculum
Watch this short video describing the calculus and linear algebra courses.
For freshman only.
MATH 0050 and 0060 provide a slower-paced introduction to calculus for students who require additional preparation. Presents the same calculus topics as MATH 0090, together with a review of the necessary precalculus topics. Students successfully completing this sequence are prepared for MATH 0100. S/NC only.
A slower-paced introduction to calculus for students who require additional preparation. Presents the same calculus topics as MATH 0090, together with a review of the necessary precalculus topics. Students successfully completing this sequence are prepared for MATH 0100.
- Prerequisite: MATH 0050 or written permission. May not be taken for credit in addition to MATH 0070 or MATH 0090.
- S/NC only.
A survey of calculus 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 MATH 0060 or MATH 0090.
- S/NC only
The Math Teaching Fellows Program is a semester-long certificate program that provides UTAs with the skills and knowledge required to be an effective UTA in the Math Department. Admission into the program is by application only; participants will hold a UTA appointment in a Mathematics course approved by the Teaching Fellows Coordinators and must therefore be eligible for student employment at Brown during the term. Participants should have completed at least one of Math 0100, 0180, 0190, 0200, 0350, 0520, 0540.
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.
- MATH 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 MATH 0050, 0060, or 0070.
- S/NC only.
A continuation of the material of MATH 0090 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 0090 and 0100 or the equivalent are recommended for all students intending to concentrate in mathematics or the sciences.
- MATH 0100 may not be taken in addition to MATH 0170 or MATH 0190.
This course, which covers roughly the same material and has the same prerequisites as MATH 0100, 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 0100.
- MATH 0170 may not be taken in addition to MATH 0100 or MATH 0190.
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 0100, MATH 0170, or MATH 0190, or advanced placement or written permission.
- MATH 0180 may not be taken in addition to MATH 0200 or MATH 0350.
This course, which covers roughly the same material and has the same prerequisites as MATH 0100, 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 0190 may not be taken in addition to MATH 0100 or MATH 0170.
This course, which covers roughly the same material as MATH 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: MATH 0100, MATH 0170, or MATH 0190, or advanced placement or written permission.
- MATH 0200 may not be taken in addition to MATH 0180 or MATH 0350.
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 0350 covers the same material as MATH 0180, but with more emphasis on theory and on understanding proofs.
- Prerequisite: MATH 0100, MATH 0170, or MATH 0190, or advanced placement or written permission.
- MATH 0350 may not be taken in addition to MATH 0180 or MATH 0200.
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.
A first course in linear algebra designed to develop students' problem solving skills, mathematical writing skills, and gain facility with the applications and theory of linear algebra. Topics include: 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.
- MATH 0520 or MATH 0540 is a prerequisite for all 100-level courses in Mathematics except MATH 1460.
- Prerequisite: MATH 0100, MATH 0170, or MATH 0190.
- May not be taken in addition to MATH 0540.
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 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 prerequisite: MATH 0100, MATH 0170, or MATH 0190.
- MATH 0540 may not be taken in addition to MATH 0520.
This semester-long class will expose students to several fundamental areas of mathematics. It will be team taught by three members of the faculty. Topics, which will vary from year to year, will be chosen from logic and set theory, number theory, abstract algebra, combinatorics and graph theory, analysis, and geometry. Approximately 4 weeks will be devoted to each of the selected topics.
This year-long 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.
For Undergraduates and Graduates
The standard requirements for all 100-level mathematics courses except MATH 1010 and MATH 1460 are MATH 0180, MATH 0200, or MATH 0350; and MATH 0520 or MATH 0540.
Watch this short video describing the 1000-level math courses
Math 1001 is an introduction to proof-writing designed to prepare students for further exploration of rigorous mathematics. Students will be trained to identify and employ a variety of proof-techniques such as direct proof, proof by contradiction, proof by induction, and proof by cases, to name a few. Mathematical topics covered include samplings of set theory, logic, and number theory, with additional topics chosen at the instructors discretion if time permits.
- Recommended Prerequisites: MATH 0520, MATH 0540, MATH 0100, MATH 0180, MATH 0190, AMTH 0200, or MATH 0350. Students with little proof writing experience are encouraged to take MATH 1001 prior to taking, or concurrently with, other 1000-level Mathematics courses.
Due to limited space, students interested in Math 1000 must apply by filling in the following application: https://forms.gle/5t2kjPj8RkdwLG9Z8. All applications received by Friday, November 11th will be considered and a first round of decisions will be made on Monday, November 14th (the second to last day of pre-registration).
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: MATH 0180, MATH 0200, or MATH 0350.
- MATH 0520 or MATH 0540 may be taken concurrently.
- Most students are advised to take MATH 1010 before MATH 1630.
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.
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: MATH 0520, MATH 0540, or permission of the instructor.
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.
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.
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.
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.
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.
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.
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.
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: MATH 0180, MATH 0200, or MATH 0350.
- This course does not require MATH 0520 or MATH 0540.
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.
- MATH 1530 is required of all students concentrating in mathematics.
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: MATH 1530. May be repeated for credit.
Selected topics in number theory will be investigated. Unique factorization, prime numbers, modular arithmetic, arithmetic functions, quadratic reciprocity, finite fields, and related topics.
- Prerequisite: MATH 1530 or written permission.
1630: A rigorous introduction to real analysis, this course treats topics in point set topology, function spaces, differentiability of functions on Euclidean spaces, and Fourier series. Among the many topics and theorems we investigate in detail will be connectedness and compactness, the Arzela-Ascoli theorem, the inverse and implicit function theorems, and L^2 and pointwise convergence of Fourier series.
- It is recommended that a student take MATH 1010 before attempting MATH 1630
1640: A second course in real analysis, in this class we study measure theory and integration as well as Hilbert spaces. Among the many topics we study will be abstract measure and integration theory, Fourier transform, linear functionals and the Riesz representation theorem, compact operators, and the spectral theorem. The course may also include additional material of interest to the students and instructor.
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.
- Prerequisites: Abstract Algebra (Math 1530) and either Analysis: Functions of One Variable (Math 1010) or Real Analysis I (Math 1630)
Manifolds are a basic concept that unite ideas from linear algebra, multivariable calculus, geometry, and topology. Math and physics research require a familiarity with manifolds. Topics covered include the definition of a smooth manifold either as a smooth subset of $R^n$ or as an abstract manifold, coordinating charts, and transition functions. The course will cover concrete examples: e.g. surfaces and 3-manifolds. Other topics include tangent vectors, vector fields, tensors, exterior algebra, differential forms, change of coordinates, Stokes’ Theorem and the basics of De Rham cohomology.
- Prerequisites: Abstract Algebra (Math 1530) and either Analysis: Functions of One Variable (Math 1010) or Real Analysis I (Math 1630)
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.
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.
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.
Collateral reading, individual conferences.
Primarily for Graduates
Any undergraduate wishing to enroll in a 2000 level course should have completed all necessary prerequisites and must receive permission from the instructor.
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.
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: MATH 2510, MATH 2520, and undergraduates require permission from the instructor.
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: MATH 2510, MATH 2520, and undergraduates require permission from the instructor.
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: MATH 1060, MATH 1160, and preferably MATH 1710, and undergraduates require permission from the instructor.
Real numbers, outer measures, measures, Lebesgue measure, integrals of measurable functions, Holder and Minkowski inequalities, modes of onvergence, L^p spaces, product measures, Fubini's Theorem, signed measures, Radon-Nikodym theorem, dual space of L^p and of C, Hausdorff measure.
The basics of Hilbert space theory, including orthogonal projections, the Riesz representation theorem, and compact operators. The basics of Banach space theory, including the open mapping theorem, closed graph theorem, uniform boundedness principle, Hahn-Banach theorem, Riesz representation theorem (pertaining to the dual of C_0(X)), weak and weak-star topologies. Various additional topics, possibly including Fourier series, Fourier transform, ergodic theorems, distribution theory, and the spectral theory of linear operators.
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.
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.
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.
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.
An introduction to algebraic topology. Topics include fundamental group, covering spaces, simplicial and singular homology, CW complexes, and an introduction to cohomology.
- Undergraduate Prerequisites - MATH 1710, MATH 1530, and MATH 1010 and/or MATH 1630 and undergraduates require permission from the instructor.
A continuation of MATH 2410 - An introduction to algebraic topology. Topics include fundamental group, covering spaces, simplicial and singular homology, CW complexes, and an introduction to cohomology.
Undergraduate Prerequisites - MATH 1710, MATH 1530, and MATH 1010 and/or MATH 1630 and undergraduates require permission from the instructor.
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.
- Undergraduate Prerequisites
- MATH 2510 - MATH 1530 and MATH1540 and undergraduates require permission from the instructor.
- MATH 2520 - MATH 2510 and undergraduates require permission from the instructor.
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.
- Undergraduate Prerequisites
- MATH 2510 - MATH 1530 and MATH1540 and undergraduates require permission from the instructor.
- MATH 2520 - MATH 2510 and undergraduates require permission from the instructor.
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: MATH 2510.
- Undergraduate Prerequisites: MATH 2510 and undergraduates require permission from the instructor.
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: MATH 2510.
- Undergraduate Prerequisites: MATH 2510 and undergraduates require permission from the instructor.
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 MATH 2210 (or Applied Math 2110) concurrently.
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 MATH 2210 (or Applied Math 2110) concurrently.
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.
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.
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.
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.
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.