Mathematics

The Mathematics Department at Brown balances a lively interest in students and teaching with a distinguished research reputation. Our several strong research groups, Analysis, Algebraic Geometry, Geometry and Topology, and Number Theory, all have active weekly seminars that draw speakers ranging from the local to the international. We support 40 to 50 graduate students in a Ph.D. program whose graduates populate top mathematics departments and prominent positions in industry. Our joint graduate courses and seminars with the adjacent Division of Applied Mathematics add to the breadth of offerings available to our graduate students. The undergraduate program in mathematics at Brown is designed to prepare students for careers in the mathematical sciences and other careers requiring strong analytical skills, while engaging more ambitious students in creative projects that can culminate in a senior thesis.

For additional information, please visit the department's website: http://www.math.brown.edu/

Course usage information

MATH 0010A. First Year Seminar: A Taste of the Infinite.

The concept of infinity occurs in many disciplines - philosophy, mathematics, physics, religion, art, and so on. This class will focus on the mathematical aspects of infinity, surveying some of the ways that the infinite arises in mathematics. Topics will include: the sizes of infinity, rates of growth, computational complexity, construction of the real numbers, the notion of compactness, geometric spaces, transcendental numbers, and fractal sets. I will not assume any prior knowledge of mathematics beyond a good grounding in high school algebra and geometry. FYS

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MATH 0010B. Exploring the Fourth Dimension.

This interdisciplinary seminar explores all the mathematics students have seen or ever will see, concentrating on an engaging topic that begins with elementary geometry and branches out to literature, history, philosophy, and art as well as physics and other sciences. Guideposts to the fourth dimension include Salvador Dali's Corpus Hypercubicus, Edwin Abbott Abbott's Flatland, and Jeff Weeks' The Shape of Space. Students will investigate new mathematical topics such as combinatorics, regular polytopes, topology, and non-Euclidean geometry. Although students will use computers for visualization, no computer experience is required. There are no specific mathematical prerequisites except curiosity and a willingness to participate actively. Students considering concentrations in humanities, social sciences, and the arts are especially invited to this first-year seminar. Enrollment limited to 20 first year students. FYS

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MATH 0010C. From 'Flatland' to the Fourth Dimension.

No description available. Enrollment limited to 15 first year students. Instructor permssion required, after initial placement of students. FYS

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MATH 0050. Analytic Geometry and Calculus.

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. May not be taken for credit in addition to MATH 0070 or MATH 0090. S/NC only.

Fall MATH0050 S01 15809 TTh 9:00-10:20(11) (A. Landman)
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MATH 0060. Analytic Geometry and Calculus.

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.

Spr MATH0060 S01 24718 TTh 9:00-10:20(08) (A. Landman)
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MATH 0070. Calculus with Applications to Social Science.

A one-semester 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 0050 or MATH 0060 or MATH 0090. S/NC only.

Fall MATH0070 S01 15815 TTh 10:30-11:50(11) (A. Landman)
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MATH 0090. Introductory Calculus, Part I.

An intensive course in calculus of one variable including limits, differentiation, maxima and minima, the chain rule, rational functions, trigonometric functions, and exponential functions. Introduction to 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 the sciences or mathematics. May not be taken for credit in addition to MATH 0050 or MATH 0060 or MATH 0070. S/NC only.

Fall MATH0090 S01 15831 MWF 9:00-9:50(11) (A. Barron)
Fall MATH0090 S02 15832 MWF 10:00-10:50(11) (D. Katz)
Fall MATH0090 S03 15833 MWF 12:00-12:50(11) (X. Li)
Fall MATH0090 S04 15834 TTh 9:00-10:20(11) (Y. Wang)
Fall MATH0090 S05 15835 TTh 10:30-11:50(11) (K. Ascher)
Fall MATH0090 C01 15854 T 9:00-10:20 (S. Connolly)
Fall MATH0090 C02 15865 T 9:00-10:20 (G. Inchiostro)
Fall MATH0090 C03 15866 T 9:00-10:20 (A. Li)
Fall MATH0090 C04 15857 T 4:00-5:20 (A. Li)
Fall MATH0090 C05 15868 T 4:00-5:20 (S. Kakaroumpas)
Fall MATH0090 C06 15869 T 4:00-5:20 (G. Inchiostro)
Fall MATH0090 C07 15870 T 4:00-5:20 (P. Jiang)
Fall MATH0090 C08 15871 T 4:00-5:20 (S. Connolly)
Fall MATH0090 C09 15872 T 6:40-8:00PM (P. Jiang)
Fall MATH0090 C10 15863 T 6:40-8:00PM (S. Kakaroumpas)
Spr MATH0090 S01 24719 MWF 11:00-11:50(18) (D. Katz)
Spr MATH0090 S02 24720 MWF 2:00-2:50(18) 'To Be Arranged'
Spr MATH0090 C01 24721 T 9:00-10:20 'To Be Arranged'
Spr MATH0090 C02 24722 T 4:00-5:20 'To Be Arranged'
Spr MATH0090 C03 24723 T 4:00-5:20 'To Be Arranged'
Spr MATH0090 C04 24724 T 6:40-8:00PM 'To Be Arranged'
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MATH 0100. Introductory Calculus, Part II.

A continuation of the material of MATH 0090 including further development of integration, techniques of integration, and applications. Other topics include infinite series, power series, Taylor's formula, polar and parametric equations, and an introduction to differential equations. MATH 0090 or the equivalent are recommended for all students intending to concentrate in the sciences or mathematics.

Fall MATH0100 S01 15836 MWF 11:00-11:50(04) (D. Katz)
Fall MATH0100 S02 15837 MWF 12:00-12:50(04) (Z. Wang)
Fall MATH0100 S03 15838 MWF 2:00-2:50(04) (Y. Xiao)
Fall MATH0100 S04 15839 MWF 9:00-9:50(04) (S. Fan)
Fall MATH0100 S05 15840 TTh 1:00-2:20(04) (D. Lowry)
Fall MATH0100 C01 15841 Th 9:00-10:20 (K. Lai)
Fall MATH0100 C02 15842 Th 9:00-10:20 (S. Asgarli)
Fall MATH0100 C03 15843 Th 9:00-10:20 (T. George)
Fall MATH0100 C04 15844 Th 4:00-5:20 (S. Asgarli)
Fall MATH0100 C05 15845 Th 4:00-5:20 (L. Li)
Fall MATH0100 C06 15846 Th 4:00-5:20 (T. George)
Fall MATH0100 C07 15847 Th 4:00-5:20 (S. Kim)
Fall MATH0100 C08 15848 Th 4:00-5:20 (K. Lai)
Fall MATH0100 C09 15849 Th 6:40-8:00PM (S. Kim)
Fall MATH0100 C10 15850 Th 6:40-8:00PM (L. Li)
Spr MATH0100 S01 24725 MWF 9:00-9:50(17) 'To Be Arranged'
Spr MATH0100 S02 24726 MWF 10:00-10:50(17) (D. Katz)
Spr MATH0100 S03 24727 TTh 9:00-10:20(17) 'To Be Arranged'
Spr MATH0100 S04 24728 TTh 10:30-11:50(17) 'To Be Arranged'
Spr MATH0100 S05 25876 MWF 11:00-11:50(17) 'To Be Arranged'
Spr MATH0100 C01 24729 Th 9:00-10:20 'To Be Arranged'
Spr MATH0100 C02 24730 Th 9:00-10:20 'To Be Arranged'
Spr MATH0100 C03 24731 Th 4:00-5:20 'To Be Arranged'
Spr MATH0100 C04 24732 Th 4:00-5:20 'To Be Arranged'
Spr MATH0100 C05 24733 Th 4:00-5:20 'To Be Arranged'
Spr MATH0100 C06 24734 Th 4:00-5:20 'To Be Arranged'
Spr MATH0100 C07 24735 Th 6:40-8:00PM 'To Be Arranged'
Spr MATH0100 C08 24736 Th 6:40-8:00PM 'To Be Arranged'
Spr MATH0100 C09 25877 Th 9:00-10:20 'To Be Arranged'
Spr MATH0100 C10 25878 Th 4:00-5:20 'To Be Arranged'
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MATH 0170. Advanced Placement Calculus.

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 examination scores or the results of tests given by the department during orientation week. May not be taken in addition to MATH 0100.

Fall MATH0170 S01 15875 MWF 9:00-9:50(04) (T. Silverman)
Fall MATH0170 S02 15877 TTh 1:00-2:20(04) (M. Nastasescu)
Fall MATH0170 S03 15878 TTh 2:30-3:50(04) (A. Weber)
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MATH 0180. Intermediate Calculus.

Three-dimensional analytic geometry. Differential and integral calculus for functions of two or three variables: partial derivatives, multiple integrals, line integrals, Green's Theorem, Stokes' Theorem. Prerequisite: MATH 0100, 0170, or 0190.

Fall MATH0180 S01 15882 MWF 12:00-12:50(04) (K. Widmayer)
Fall MATH0180 S02 15891 TTh 1:00-2:20(04) (Z. Fang)
Fall MATH0180 S03 15892 MWF 2:00-2:50(04) (K. Widmayer)
Fall MATH0180 C01 15893 T 12:00-12:50 (N. Malik)
Fall MATH0180 C02 15894 T 12:00-12:50 (Z. Ouyang)
Fall MATH0180 C03 15895 T 4:00-4:50 (N. Malik)
Fall MATH0180 C04 15896 T 4:00-4:50 (Z. Ouyang)
Spr MATH0180 S01 24737 MWF 9:00-9:50(15) 'To Be Arranged'
Spr MATH0180 S02 24738 MWF 10:00-10:50(15) (W. Lam)
Spr MATH0180 S03 24739 MWF 12:00-12:50(15) 'To Be Arranged'
Spr MATH0180 C01 24740 T 12:00-12:50 'To Be Arranged'
Spr MATH0180 C02 24741 Th 12:00-12:50 'To Be Arranged'
Spr MATH0180 C03 24742 Th 4:00-4:50 'To Be Arranged'
Spr MATH0180 C04 24743 Th 4:00-4:50 'To Be Arranged'
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MATH 0190. Advanced Placement Calculus (Physics/Engineering).

Covers roughly the same material and has the same prerequisites as MATH 0170, but 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. The extra hour is a weekly problem session.

Fall MATH0190 S01 15898 MWF 2:00-2:50(04) (S. Watson)
Fall MATH0190 S02 15905 TTh 2:30-3:50(04) (L. Walton)
Fall MATH0190 C01 15906 Th 12:00-12:50 (Z. Zhang)
Fall MATH0190 C02 15907 Th 12:00-12:50 'To Be Arranged'
Fall MATH0190 C03 15908 Th 4:00-4:50 (Z. Zhang)
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MATH 0200. Intermediate Calculus (Physics/Engineering).

Covers roughly the same material as MATH 0180, but 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. The extra hour is a weekly problem session. Recommended prerequisite: MATH 0100, 0170, or 0190.

Fall MATH0200 S01 15910 MWF 9:00-9:50(04) (X. Zhang)
Fall MATH0200 S02 15915 MWF 12:00-12:50(04) (A. Walker)
Fall MATH0200 S03 15916 TTh 1:00-2:20(04) (W. Lam)
Fall MATH0200 C01 15917 Th 12:00-12:50 (Y. Solomon)
Fall MATH0200 C02 15918 Th 6:00-6:50 'To Be Arranged'
Fall MATH0200 C03 15919 Th 4:00-4:50 (Y. Solomon)
Spr MATH0200 S01 24744 MWF 12:00-12:50(16) 'To Be Arranged'
Spr MATH0200 S02 24746 MWF 1:00-1:50(16) 'To Be Arranged'
Spr MATH0200 S03 24747 MWF 2:00-2:50(16) (B. Pausader)
Spr MATH0200 C01 24748 Th 12:00-12:50 'To Be Arranged'
Spr MATH0200 C02 24749 Th 12:00-12:50 'To Be Arranged'
Spr MATH0200 C03 24750 Th 4:00-4:50 'To Be Arranged'
Spr MATH0200 C04 24751 Th 4:00-4:50 'To Be Arranged'
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MATH 0350. Honors Calculus.

A third-semester calculus course for students of greater aptitude and motivation. Topics include vector analysis, multiple integration, partial differentiation, line integrals, Green's theorem, Stokes' theorem, the divergence theorem, and additional material selected by the instructor. Prerequisite: Advanced placement or written permission.

Fall MATH0350 S01 15921 MWF 10:00-10:50(04) (D. Abramovich)
Fall MATH0350 S02 15925 MWF 1:00-1:50(04) (B. Cole)
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MATH 0420. Introduction to Number Theory.

An overview of one of the most beautiful areas of mathematics. Ideal for any student who wants a taste of mathematics outside of, or in addition to, the calculus sequence. Topics include: prime numbers, congruences, quadratic reciprocity, sums of squares, Diophantine equations, and, as time permits, such topics as cryptography and continued fractions. No prerequisites.

Spr MATH0420 S01 24752 TTh 2:30-3:50(11) 'To Be Arranged'
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MATH 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. MATH 0520 or 0540 is a prerequisite for all 1000-level courses in Mathematics except MATH 1260 or 1610. Recommended prerequisite: MATH 0180, 0200, or 0350. May not be taken in addition to MATH 0540.

Fall MATH0520 S01 15927 MWF 11:00-11:50(11) (S. Lichtenbaum)
Fall MATH0520 S02 15928 MWF 2:00-2:50(11) (S. Fan)
Fall MATH0520 S03 15929 TTh 10:30-11:50(11) (M. Nastasescu)
Fall MATH0520 S04 17081 MWF 1:00-1:50(11) (B. Freidin)
Spr MATH0520 S01 24753 MWF 9:00-9:50(02) 'To Be Arranged'
Spr MATH0520 S02 24754 MWF 10:00-10:50(03) (J. Holmer)
Spr MATH0520 S03 24755 MWF 12:00-12:50(05) 'To Be Arranged'
Spr MATH0520 S04 24756 MWF 1:00-1:50(06) (S. Fan)
Spr MATH0520 S05 24757 MWF 11:00-11:50(04) 'To Be Arranged'
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MATH 0540. Honors Linear Algebra.

Linear algebra for students of greater aptitude and motivation, especially mathematics and science concentrators with a good mathematical preparation. Matrices, linear equations, determinants, and eigenvalues; vector spaces and linear transformations; inner products; Hermitian, orthogonal, and unitary matrices; and Jordan normal forms. Provides a more extensive treatment of the topics in MATH 0520. Recommended prerequisites: MATH 0180, 0200, or 0350.

Fall MATH0540 S01 15931 MWF 11:00-11:50(11) (B. Cole)
Fall MATH0540 S02 15932 MWF 1:00-1:50(11) (S. Treil)
Spr MATH0540 S01 24758 MWF 1:00-1:50(06) (B. Cole)
Spr MATH0540 S02 24759 TTh 1:00-2:20(10) (T. Aougab)
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MATH 0580. Mathematical Forms in Architecture.

This project will explore and advance innovative applications of mathematics to architecture using computational methods. Historically, architecture has been guided primarily by an intuitive creative process. In contrast to the end-results of intuitive design, many "optimal" forms--i.e. geometric shapes and configurations that satisfy extremal conditions--are unique because they are the result of systematic physical experiments or explicit mathematical study in addition to imaginative imput. Classic questions for which human intuition alone has been incapable of finding a solution include: What is the exact shape of the optimal arch?, or What is the shape of a child's slide that minimizes the time of travel? The use of computational methods to generate solutions to these problems will be made considerably simpler via optimization libraries in Mathematica. The application to architecture in this project will provide students a unique concrete backdrop to visualize solutions to these problems.

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MATH 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 and 3-space. Prerequisite: MATH 0180, 0200, or 0350. MATH 0520 or 0540 may be taken concurrently. Most students are advised to take MATH 1010 before MATH 1130.

Spr MATH1010 S01 24760 MWF 11:00-11:50(04) (K. Widmayer)
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MATH 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.

Spr MATH1040 S01 24761 TTh 9:00-10:20(08) (T. Aougab)
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MATH 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 surfaces, Gaussian curvature and mean curvature, geodesics, minimal surfaces, and the Gauss-Bonnet theorem.

Fall MATH1060 S01 15944 TTh 9:00-10:20(08) (G. Daskalopoulos)
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MATH 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.

Fall MATH1110 S01 15945 MWF 1:00-1:50(06) (Y. Wu)
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MATH 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. Prerequisites: MATH 0520 or MATH 0540, or instructor permission.

Spr MATH1120 S01 24762 MWF 2:00-2:50(07) (Y. Wu)
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MATH 1130. Functions of Several Variables.

A course on calculus on manifolds. Included are differential forms, integration, and 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 1000-level course in analysis (MATH 1010 or MATH 1260) before attempting MATH 1130.

Fall MATH1130 S01 15946 MWF 10:00-10:50(14) (S. Treil)
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MATH 1140. Functions Of Several Variables.

See Functions Of Several Variables (MATH 1130) for course description. Prerequisite: MATH 1130 or instructor permission.

Spr MATH1140 S01 24763 TTh 2:30-3:50(11) (N. Kapouleas)
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MATH 1230. Graph Theory.

This course covers important material about graph theory, such as spanning trees, network flow problems, matching problems, coloring problems, planarity, Cayley graphs, spectral theory on graphs, and Ramsey Theory. The emphasis will be on a combination of theory and algorithms. Depending on the instructor, connections to such fields as combinatorics, geometry, or computer science might be emphasized. Prerequisite: MATH 0180, 0200 or 0350 and MATH 0520 or 0540 are recommended. Enrollment limited to 40.

Spr MATH1230 S01 24764 TTh 1:00-2:20(10) (R. Kenyon)
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MATH 1260. Complex Analysis.

Examines one of the cornerstones of mathematics. Complex differentiability, Cauchy-Riemann differential equations, contour integration, residue calculus, harmonic functions, geometric properties of complex mappings. Prerequisite: MATH 0180, 0200, or 0350. This course does not require MATH 0520 or 0540.

Fall MATH1260 S01 15947 TTh 9:00-10:20(08) (R. Kenyon)
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MATH 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, calculus of variations. Prerequisite: At least one 1000-level course in Mathematics or Applied Mathematics, or permission of the instructor.

Fall MATH1270 S01 15948 MWF 2:00-2:50(07) (Y. Wu)
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MATH 1410. Combinatorial Topology.

Topology of Euclidean spaces, winding number and applications, knot theory, fundamental group and covering spaces. Euler characteristic, simplicial complexes, classification of two-dimensional manifolds, vector fields, the Poincaré-Hopf theorem, and introduction to three-dimensional topology. Prerequisites: MATH 0520 or MATH 0540, or instructor permission.

Fall MATH1410 S01 15950 TTh 10:30-11:50(13) (G. Daskalopoulos)
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MATH 1530. Abstract Algebra.

An introduction to the principles and concepts of modern abstract algebra. Topics include groups, rings, and fields; applications to number theory, the theory of equations, and geometry. MATH 1530 is required of all students concentrating in mathematics.

Fall MATH1530 S01 15951 TTh 2:30-3:50(03) (T. Goodwillie)
Spr MATH1530 S01 24765 TTh 2:30-3:50(11) (M. Chan)
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MATH 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: MATH 1530.

Spr MATH1540 S01 24766 TTh 10:30-11:50(09) (R. Kenyon)
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MATH 1560. Number Theory.

A basic introduction to the theory of numbers. Unique factorization, prime numbers, modular arithmetic, quadratic reciprocity, quadratic number fields, finite fields, Diophantine equations, and additional topics. Prerequisite: MATH 1530 or written permission.

Spr MATH1560 S01 24767 TTh 1:00-2:20(10) (M. Rosen)
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MATH 1580. Cryptography.

The main focus is on public key cryptography. Topics include symmetric ciphers, public key ciphers, complexity, digital signatures, applications and protocols. MATH 1530 is not required for this course. What is needed from abstract algebra and elementary number theory will be covered. Prerequisite: MATH 0520 or MATH 0540.

Fall MATH1580 S01 15952 MWF 10:00-10:50(14) (N. Pflueger)
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MATH 1610. Probability.

Basic probability theory. Sample spaces; random variables; normal, Poisson, and related distributions; expectation; correlation; and limit theorems. Applications in various fields (biology, physics, gambling, etc.). Prerequisites: MATH 0180, 0200 or 0350.

Fall MATH1610 S01 15953 MWF 11:00-11:50(02) (J. Holmer)
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MATH 1620. Mathematical Statistics.

Central limit theorem, point estimation, interval estimation, multivariate normal distributions, tests of hypotheses, and linear models. Prerequisite: MATH 1610 or written permission.

Spr MATH1620 S01 24768 TTh 10:30-11:50(10) 'To Be Arranged'
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MATH 1810A. Applied Algebraic Topology.

Topology is a powerful tool for identifying, describing, and characterizing the essential features of functions and spaces. In the recent years some of these methods have been adapted to study the shape of data collected from a range of different fields, including graphics and visualization, computational biology, etc. This course is an introduction to the basic concepts and topological structures behind these developments, focusing on persistent homology and mapper. Projects will involve using these methods to analyze and describe the shape of concrete data sets.

Fall MATH1810A S01 15955 TTh 2:30-3:50(03) (S. Kalisnik Verovsek)
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MATH 1810B. A Second Course in Linear Algebra.

We'll study various aspects of multilinear algebra, including tensors, differential forms and homological algebra, with emphasis on coordinate-free constructions and universal properties.

No background in abstract algebra, differential/algebraic geometry or manifold theory will be assumed. The only pre-requisites are a well-understood first course in linear algebra, a desire to do more with it, and a willingness to think abstractly.

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MATH 1820A. Introduction to Lie Algebras.

Lie groups and Lie algebras are important, because they are the symmetries of structures such as quadratic forms, differential systems and smooth manifolds. The prototype of a Lie algebra is the space of 3-vectors together with their cross product, which is closely related to the Lie group of rotations. We will see how this basic example generalizes, mostly in the context of matrices. We'll examine special types of Lie algebras, such as nilpotent, solvable and semi-simple, study root systems and their diagrams, explore some representation theory, and end with the classification of the simple Lie algebras. Prerequisite: MATH 1530.

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MATH 1820B. Special Topic: Combinatorics.

This course is an introduction to combinatorics, surveying some of its basic concepts, techniques and results. No special background is assumed beyond a liking for mathematics. Any needed set theory, group theory or proof technique will be developed in the course. Likely topics include counting, generating functions, max-min theorems, partitions, Polya theory, Ramsey theory, chromatic polynomials, and perhaps others depending on students' interests. Mandatory S/NC.

Spr MATH1820B S01 25710 F 3:00-5:30(15) (A. Landman)
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MATH 1970. Honors Conference.

Collateral reading, individual conferences. Section numbers vary by instructor. Please check Banner for the correct section number and CRN to use when registering for this course.

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MATH 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.

Fall MATH2010 S01 15956 TTh 10:30-11:50(13) (N. Kapouleas)
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MATH 2050. Algebraic Geometry.

Complex manifolds and algebraic varieties, sheaves and cohomology, vector bundles, Hodge theory, Kähler manifolds, vanishing theorems, the Kodaira embedding theorem, the Riemann-Roch theorem, and introduction to deformation theory.

Fall MATH2050 S01 15957 MWF 1:00-1:50(06) (N. Pflueger)
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MATH 2060. Algebraic Geometry.

See Algebraic Geometry (MATH 2050) for course description.

Spr MATH2060 S01 24769 MWF 11:00-11:50(04) (N. Pflueger)
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MATH 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.

Spr MATH2110 S01 24770 TTh 1:00-2:20 (N. Kapouleas)
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MATH 2210. Real Function Theory.

Point set topology, Lebesgue measure and integration, Lp spaces, Hilbert space, Banach spaces, differentiability, and applications.

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MATH 2220. Real Function Theory.

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.

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MATH 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.

Fall MATH2250 S01 15958 TTh 2:30-3:50(03) (J. Kahn)
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MATH 2260. Complex Function Theory.

See Complex Function Theory (MATH 2250) for course description.

Spr MATH2260 S01 24771 TTh 9:00-10:20(08) (J. Kahn)
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MATH 2370. Partial Differential Equations.

The theory of the classical partial differential equations; the method of characteristics and general first order theory. 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. Semester II concentrates on special topics chosen by the instructor.

Fall MATH2370 S01 15959 MWF 10:00-10:50(14) (W. Strauss)
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MATH 2380. Partial Differential Equations.

The theory of the classical partial differential equations; the method of characteristics and general first order theory. 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. Semester II of this course concentrates on special topics chosen by the instructor.

Spr MATH2380 S01 26121 MWF 11:00-11:50(04) (B. Pausader)
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MATH 2410. Topology.

An introductory course with emphasis on the algebraic and differential topology of manifolds. Topics include simplicial and singular homology, de Rham cohomology, and Poincaré duality.

Fall MATH2410 S01 15960 TTh 9:00-10:20(08) (T. Goodwillie)
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MATH 2420. Topology.

See Topology (MATH 2410) for course description.

Spr MATH2420 S01 24772 TTh 2:30-3:50(11) (T. Goodwillie)
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MATH 2450. Exchange Scholar Program.

Fall MATH2450 S01 14757 Arranged 'To Be Arranged'
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MATH 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.

Fall MATH2510 S01 15961 MWF 2:00-2:50(07) (S. Lichtenbaum)
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MATH 2520. Algebra.

See Algebra (MATH 2510) for course description.

Spr MATH2520 S01 24773 MWF 2:00-2:50(07) (D. Abramovich)
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MATH 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 varies from year to year; possible topics include class field theory, arithmetic geometry, analytic number theory, and arithmetic K-theory. Prerequisite: MATH 2510.

Fall MATH2530 S01 15962 MWF 11:00-11:50(02) (J. Silverman)
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MATH 2540. Number Theory.

See Number Theory (MATH 2530) for course description.

Spr MATH2540 S01 24774 MWF 10:00-10:50(03) (J. Silverman)
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MATH 2630. Probability.

Introduces probability spaces, random variables, expectation values, and conditional expectations. Develops the basic tools of probability theory, such fundamental results as the weak and strong laws of large numbers, and the central limit theorem. Continues with a study of stochastic processes, such as Markov chains, branching processes, martingales, Brownian motion, and stochastic integrals. Students without a previous course in measure theory should take MATH 2210 (or APMA 2110) concurrently.

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MATH 2640. Probability.

See MATH 2630 for course description.

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MATH 2710A. Probability, Quantum Field Theory, and Geometry.

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MATH 2710B. Solitary Waves.

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MATH 2710C. Gluing Constructions in Differential Geometry.

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MATH 2710D. Lie Groups and Lie Algebras.

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MATH 2710E. Arithmetic Groups.

Fall MATH2710E S01 15964 MWF 2:00-2:50(07) (J. Silverman)
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MATH 2710F. Stable Homotopy Theory.

No description available.

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MATH 2710G. Topics in Free Boundary Problems in Continuum Mechanics.

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MATH 2710H. Topics in Complex and p-adic Dynamics.

No description available.

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MATH 2710I. Topics in Effective Harmonic Analysis.

Graduate topics course in Harmonic Analysis.

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MATH 2710P. Harmonic Analysis on Polytopes and Cones.

Graduate Topics course in harmonic analysis.

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MATH 2710R. Problems of the Uncertainty Principle in Harmonic Analysis.

Graduate Topics course in Harmonic Analysis.

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MATH 2720A. Topics in Harmonic Analysis.

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MATH 2720B. Multiple Dirichlet Series.

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MATH 2720C. Curves on Surfaces and the Classification of Finitely Generated Kleinian Groups.

This is a course in the study of geometry and topology of hyperbolic 3- manifolds.

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MATH 2720D. Piecewise Isometric Maps.

This class will cover a variety of topics, all more or less related to dynamical systems that are defined by piecewise isometric maps. Topics may include:polygonal billiards and flat cone surfaces; outer billiards; interval exchange maps; The Gauss map and continued fractions; aperiodic tilings, such as the Penrose tiling; cut and paste theorems about polyhedra; and Ashwin's conjecture about piecewise rotation maps. A fairly large part of the class will be devoted to the explanation of the instructor's proof of the Moser-Neumann conjecture for outer billiards. For this part, the instructor will use his book on the subject. For other parts of the course, a variety of sources will be used.

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MATH 2720E. Advanced Topics in Mathematics.

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MATH 2720F. Topics in Geometric Analysis.

No description available.

Spr MATH2720F S01 24775 TTh 10:30-11:50 (G. Daskalopoulos)
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MATH 2720G. Introduction to Hodge Structures.

No description available.

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MATH 2720H. Discrete Groups, Ergodic Theory and Hyperbolic Geometry.

No description available.

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MATH 2720I. Automorphic Representations for GL(2).

Graduate topics course in automorphic representations for GL(2).

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MATH 2720N. Groups Acting on Trees.

This course will be an introduction to geometric group theory from the viewpoint of groups acting on trees. Some topics that may be covered included Bass-Serre theory, R-trees and the Rips machine and groups acting on quasi-trees.

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MATH 2970. Preliminary Exam Preparation.

No description available.

Fall MATH2970 S01 14758 Arranged 'To Be Arranged'
Spr MATH2970 S01 23849 Arranged 'To Be Arranged'
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MATH 2980. 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. Section numbers vary by instructor. Please check Banner for the correct section number and CRN to use when registering for this course.

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MATH 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.

Fall MATH2990 S01 14759 Arranged 'To Be Arranged'
Spr MATH2990 S01 23850 Arranged 'To Be Arranged'
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MATH XLIST. Courses of Interest to Graduate Students Majoring in Mathematics.

Fall 2016
The following courses may be taken for credit by graduate students majoring in Mathematics. Please check with the sponsoring department for times and locations.

Applied Mathematics
APMA 2110 Real Analysis
APMA 2630 Probability
Spring 2017
The following courses may be taken for credit by graduate students majoring in Mathematics. Please check with the sponsoring department for times and locations.

Applied Mathematics
APMA 2120 Hilbert Spaces and Their Applications
APMA 2640 Theory of Probability

Chair

Jeffrey F. Brock

Professor

Dan Abramovich
Professor of Mathematics

Thomas F. Banchoff
Professor Emeritus of Mathematics

Jeffrey F. Brock
Professor of Mathematics

Andrew Browder
Professor Emeritus of Mathematics

Brian J. Cole
Professor of Mathematics

Georgios D. Daskalopoulos
Professor of Mathematics

Thomas G. Goodwillie
Professor of Mathematics

Bruno Harris
Professor Emeritus of Mathematics

Brendan E. Hassett
Professor of Mathematics

Jeffrey Hoffstein
Professor of Mathematics

Jeremy A. Kahn
Professor of Mathematics

Nicolaos Kapouleas
Professor of Mathematics

Richard W. Kenyon
William R. Kenan, Jr. University Professor of Mathematics

Stephen Lichtenbaum
Roland George Dwight Richardson University Professor of Mathematics

Jonathan D. Lubin
Professor Emeritus of Mathematics and Engineering

Jill Catherine Pipher
Elisha Benjamin Andrews Professor of Mathematics

Michael I. Rosen
Professor Emeritus of Mathematics

Richard E. Schwartz
Chancellor's Professor

Joseph H. Silverman
Royce Family Professor of Teaching Excellence and Professor of Mathematics

Walter A. Strauss
L. Herbert Ballou University Professor of Mathematics and Applied Mathematics

Sergei R. Treil
Professor of Mathematics

John Wermer
Professor Emeritus of Mathematics

Associate Professor

Justin A. Holmer
Associate Professor of Mathematics

Eva Kallin
Associate Professor Emerita of Mathematics

Alan Landman
Associate Professor of Mathematics

Benoit J. Pausader
Associate Professor of Mathematics

Visiting Associate Professor

Sinai Robins
Visiting Associate Professor of Mathematics

Assistant Professor

Tarik B. Aougab
Tamarkin Assistant Professor of Mathematics

Melody T. Chan
Assistant Professor of Mathematics

Sin Tsun Edward Fan
Tamarkin Assistant Professor of Mathematics

Wai Yeung Lam
Tamarkin Assistant Professor of Mathematics

Maria M. Nastasescu
Tamarkin Assitant Professor of Mathematics

Nathan K. Pflueger
Tamarkin Assistant Professor of Mathematics

Samuel S. Watson
Tamarkin Assistant Professor of Mathematics

Klaus M. Widmayer
Tamarkin Assistant Professor of Mathematics

Yilun Wu
Tamarkin Assistant Professor of Mathematics

Visiting Assistant Professor

Ulrica Wilson
Visiting Assistant Professor of Mathematics

Lecturer

Daniel J. Katz
Lecturer in Mathematics

Mathematics

Mathematics is a grouping of sciences, including geometry, algebra, and calculus, that study quantity, structure, space, and change.  Mathematics concentrators at Brown can explore these concepts through the department’s broad course offerings and flexible concentration requirements. The concentration leads to either the Bachelor of Arts or Bachelor of Science degree (the latter is strongly recommended for students interested in pursuing graduate study in mathematics or related fields). Concentrators begin their learning with multivariable calculus, linear algebra, and abstract algebra. Beyond these prerequisites, students take a variety of advanced topics on the 1000 and 2000 level based on their interests. Students also have the option of completing a thesis project.

Concentrators in mathematics should complete the prerequisites by the end of their sophomore year.   It is strongly recommended that students take MATH 1010 before taking MATH 1130.

Standard program for the A.B. degree

Prerequisites:
Multivariable calculus and linear algebra (choose one of the following sequences):2
Intermediate Calculus
and Linear Algebra
Intermediate Calculus
and Honors Linear Algebra
Intermediate Calculus (Physics/Engineering)
and Linear Algebra
Honors Calculus
and Honors Linear Algebra
Or the equivalent
Program:
MATH 1530Abstract Algebra1
Five other 1000- or 2000-level Mathematics courses 5
Total Credits8

Standard program for the Sc.B. degree

Prerequisites:
Multivariate calculus and linear algebra (choose one of the following sequences):2
Intermediate Calculus
and Linear Algebra
Intermediate Calculus
and Honors Linear Algebra
Intermediate Calculus (Physics/Engineering)
and Linear Algebra
Honors Calculus
and Honors Linear Algebra
Or the equivalent
Program:
MATH 1130
MATH 1140
Functions of Several Variables
and Functions Of Several Variables
2
MATH 1530Abstract Algebra1
MATH 1540Topics in Abstract Algebra1
or MATH 1560 Number Theory
Four other 1000- or 2000- level Mathematics courses.4
Four additional courses in mathematics, science, economics, or applied mathematics approved by the concentration advisor.4
Total Credits14

Honors

Honors degrees may be recommended for students who have exhibited high achievement in mathematics. Candidates must complete at least eight mathematics courses at the 1000 or 2000 level with sufficiently good grades and must write an honors thesis under the guidance of a faculty member. The honors thesis is usually written while the candidate is enrolled in MATH 1970. The candidate should consult with the concentration advisor for the precise grade requirements.

Those interested in graduate study in mathematics are encouraged to take:

Functions of Several Variables
Functions Of Several Variables
Complex Analysis
Combinatorial Topology
Topics in Abstract Algebra

Mathematics-Computer Science

Students may opt to pursue an interdisciplinary Bachelor of Science degree in Math-Computer Science, a concentration administered cooperatively between the mathematics and computer science departments. Course requirements include math- and systems-oriented computer science courses, as well as computational courses in applied math. Students must identify a series of electives that cohere around a common theme. As with other concentrations offered by the Computer Science department, students have the option to pursue the professional track of the ScB program in Mathematics-Computer Science.

Requirements for the Standard Track of the Sc.B. degree.

Prerequisites
Three semesters of Calculus to the level of MATH 0180, MATH 0200, or MATH 03503
MATH 0520Linear Algebra1
or MATH 0540 Honors Linear Algebra
Core Courses
MATH 1530Abstract Algebra1
Select one of the following series:2
Series A
Introduction to Object-Oriented Programming and Computer Science
and Introduction to Algorithms and Data Structures
Series B
Computer Science: An Integrated Introduction
and Computer Science: An Integrated Introduction
Series C
Accelerated Introduction to Computer Science
and an additional CS course not otherwise used to satisfy a concentration requirement; this course may be CSCI 0180, an intermediate-level CS course, or a 1000-level CS course
CSCI 0320Introduction to Software Engineering1
or CSCI 0330 Introduction to Computer Systems
CSCI 0220Introduction to Discrete Structures and Probability1
or CSCI 1010 Theory of Computation
Three 1000-level Mathematics courses3
Three advanced courses in Computer Science 13
Three additional courses different from any of the above chosen from Mathematics, Computer Science, Applied Mathematics, or related areas 23
A capstone course in Computer Science or Mathematics 31
Note: CSCI 1450 may be used either in place of CSCI 220 or 510 in the core courses or as an advanced course. CSCI 1450 was formerly known as CSCI 450: they are the same course and hence only one may be taken for credit. Applied Math 1650 may be used in place of CSCI 1450. However, concentration credit will be given for only one of Applied Math 1650 and CSCI 1450.
Total Credits19
1

These courses must be at the 1000-level or higher. The three courses must include a pair of courses with a coherent theme. A list of pre-approved pairs may be found at the approved-pairs web page. You are not restricted to the pairs on this list, but any pair not on the list must be approved by the director of undergraduate studies.

2

 These must be approved by a concentration advisor.

3

A one-semester course, normally taken in the student's last undergraduate year, in which the student (or group of students) use a significant portion of their undergraduate education, broadly interpreted, in studying some current topic in depth, to produce a culminating artifact such as a paper or software project.


Requirements for the Professional Track of the Sc.B. degree.

The requirements for the professional track include all those of the standard track, as well as the following:

Students must complete two two-to-four-month full-time professional experiences, doing work that is related to their concentration programs. Such work is normally done within an industrial organization, but may also be at a university under the supervision of a faculty member.

On completion of each professional experience, the student must write and upload to ASK a reflective essay about the experience addressing the following prompts, to be approved by the student's concentration advisor:

  • Which courses were put to use in your summer's work? Which topics, in particular, were important?
  • In retrospect, which courses should you have taken before embarking on your summer experience? What are the topics from these courses that would have helped you over the summer if you had been more familiar with them?
  • Are there topics you should have been familiar with in preparation for your summer experience, but are not taught at Brown? What are these topics?
  • What did you learn from the experience that probably could not have been picked up from course work?
  • Is the sort of work you did over the summer something you would like to continue doing once you graduate? Explain.
  • Would you recommend your summer experience to other Brown students? Explain.

Mathematics-Economics

Standard Mathematics-Economics Concentration 

Economics
ECON 1130Intermediate Microeconomics (Mathematical) 11
ECON 1210Intermediate Macroeconomics1
ECON 1630Econometrics I1
Two courses from the "mathematical-economics" group: 22
Welfare Economics and Social Choice Theory
Advanced Macroeconomics: Monetary, Fiscal, and Stabilization Policies
Market Design: Theory and Applications
Bargaining Theory and Applications
Econometrics II
Financial Econometrics
Investments II
Data, Statistics, Finance
Economics and Psychology
Behavioral Economics
Theory of Economic Growth
The Theory of General Equilibrium
Game Theory and Applications to Economics
One course from the "data methods" group: 21
Economics of Education I
Economics of Education: Research
Labor Economics
Health Economics
Urban Economics
Public Economics
Economic Development
The Economic Analysis of Institutions
Health, Hunger and the Household in Developing Countries
Applied Research Methods for Economists
Econometrics II
Financial Econometrics
Data, Statistics, Finance
Finance, Regulation, and the Economy: Research
Two additional 1000-level economics courses2
Mathematics
Calculus: MATH 0180 or higher1
Linnear Algebra - one of the following:1
Linear Algebra
Honors Linear Algebra
Probability Theory - one of the following:1
Probability
Mathematical Statistics
Statistical Inference I
Analysis - one of the following:1
Analysis: Functions of One Variable
Functions of Several Variables
Functions Of Several Variables
Differential Equations - one of the following:1
Ordinary Differential Equations
Partial Differential Equations
One additional course from the Probability, Analysis, and Differential Equations courses listed above1
Total Credits14
1

Or ECON 1110 with permission.

2

No course may be "double-counted" to satisfy both the mathematical-economics and data methods requirement.

Honors and Capstone Requirement:

Admission to candidacy for honors in the concentration is granted on the following basis: 3.7 GPA for Economics courses, and 3.5 GPA overall.  To graduate with honors, a student must write an honors thesis in senior year following the procedures specified by the concentration (see Economics Department website).  Beginning with the class of 2016, students not writing an honors thesis must complete an alternative senior capstone project and obtain approval of a faculty sponsor. 

Professional Track 

The requirements for the professional track include all those of the standard track, as well as the following:

Students must complete two two-to-four month full-time professional experiences, doing work that is relevant to their concentration programs.  Such work is normally done within an industrial organization, but may also be at a university under the supervision of a faculty member.

On completion of each professional experience, the student must write and upload to ASK a reflective essay about the experience addressing the following prompts, to be approved by the student's concentration advisor:

  • Which courses were put to use in your summer's work? Which topics, in particular, were important?
  • In retrospect, which courses should you have taken before embarking on your summer experience? What are the topics from these courses that would have helped you over the summer if you had been more familiar with them?
  • Are there topics you should have been familiar with in preparation for your summer experience, but are not taught at Brown? What are these topics?
  • What did you learn from the experience that probably could not have been picked up from course work?
  • Is the sort of work you did over the summer something you would like to continue doing once you graduate? Explain.
  • Would you recommend your summer experience to other Brown students? Explain.

Mathematics

The department of Mathematics offers a graduate program leading to the Doctor of Philosophy (Ph.D.) degree. Ph.D. students may also earn a transitional  A.M. or Sc.M. en route to the Ph.D.

For more information on admission and program requirements, please visit the following website:

http://www.brown.edu/academics/gradschool/programs/mathematics