Applied Mathematics

Faculty of the School of Engineering and Applied Sciences Offering Instruction in Applied Mathematics

Donald G. M. Anderson, Gordon McKay Professor of Applied Mathematics
William H. Bossert, David B. Arnold, Jr. Professor of Science
Michael P. Brenner, Glover Professor of Applied Mathematics and Applied Physics (Director of Undergraduate Studies )
John W. Hutchinson, Abbott and James Lawrence Professor of Engineering and Gordon McKay Professor of Applied Mechanics
Kenneth Norman Kamrin, Lecturer on Applied Mathematics
L. Mahadevan, Lola England Professor of Applied Mathematics
Shreyas Mandre, Lecturer on Applied Mathematics
Scott A. Norris, Lecturer on Applied Mathematics
Vahid Tarokh, Perkins Professor of Applied Mathematics and Vinton Hayes Senior Research Fellow of Electrical Engineering
Salil P. Vadhan, Gordon McKay Professor of Computer Science and Applied Mathematics
Leslie G. Valiant, T. Jefferson Coolidge Professor of Computer Science and Applied Mathematics

Other Faculty Offering Instruction in Applied Mathematics

Roger W. Brockett, An Wang Professor of Electrical Engineering and Computer Science
Yiling Chen, Assistant Professor of Computer Science
Marie D. Dahleh, Senior Lecturer on Engineering Sciences
Evelyn Hu, Gordon McKay Professor of Applied Physics and of Electrical Engineering
Navin Khaneja, Gordon McKay Professor of Electrical Engineering
Zhiming Kuang, Assistant Professor of Climate Science
James R. Rice, Mallinckrodt Professor of Engineering Sciences and Geophysics
Eli Tziperman, Pamela and Vasco McCoy, Jr.Professor of Oceanography and Applied Physics
Patrick J. Wolfe, Associate Professor of Electrical Engineering on the Gordon McKay Endowment

The School of Engineering and Applied Sciences (www.seas.harvard.edu) offers undergraduate and graduate courses in Applied Mathematics, Applied Physics, Computer Science, Earth and Planetary Sciences, and Engineering Sciences. Recommended course programs at the undergraduate level may be obtained from the Academic Office, Pierce Hall 110. Engineering and Applied Sciences faculty also offer several courses in the section entitled Freshman Seminars, Extra-Departmental Courses, and House Seminars.

For information concerning the concentration in Applied Mathematics, please consult the Director of Undergraduate Studies or the Academic Office, School of Engineering and Applied Sciences, Pierce Hall 110. Many additional courses of interest to applied mathematicians can be found in the Computer Science, Engineering Sciences, Mathematics, and Statistics sections of the catalog.

Primarily for Undergraduates

Applied Mathematics 21a. Mathematical Methods in the Sciences
Catalog Number: 6395
Michael P. Brenner, Evelyn Hu, and L. Mahadevan
Half course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
Complex numbers. Multivariate calculus: partial differentiation, directional derivatives, techniques of integration and multiple integration. Vectors: dot and cross products, parameterized curves, line and surface integrals. Vector calculus: gradient, divergence and curl, Green’s, Stokes’ and Gauss’ theorems, including orthogonal curvilinear coordinates. Applications in electrical and mechanical engineering.
Note: May not be taken for credit by students who have passed Mathematics 21a. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.
Prerequisite: Mathematics 1b or equivalent.

Applied Mathematics 21b. Mathematical Methods in the Sciences
Catalog Number: 5074
Vahid Tarokh
Half course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
Linear algebra: matrices, determinants, eigenvalues, eigenvectors, Markov processes. Optimization and least-squares analysis. Ordinary differential equations. Infinite series and Fourier series. Orthogonality and completeness. Introduction to partial differential equations. Applications in electrical and mechanical engineering.
Note: May not be taken for credit by students who have passed Mathematics 21b. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.
Prerequisite: Applied Mathematics 21a or equivalent.

Applied Mathematics 50 (formerly Applied Mathematics 50hf). Introduction to Applied Mathematics
Catalog Number: 9344
Marie D. Dahleh and Michael P. Brenner
Half course (spring term). M., W., 1–2:30. EXAM GROUP: 6, 7
Introduction to the problems and issues of applied mathematics. This will be accomplished both through the reading of papers that use mathematical arguments to have substantial impact on some field of human activity, as well as guest lecturers from around Harvard to discuss how mathematics is used in their field.

*Applied Mathematics 91r. Supervised Reading and Research
Catalog Number: 7607
Michael P. Brenner and Marie D. Dahleh
Half course (fall term; repeated spring term). Hours to be arranged.
An individual project of guided reading and research culminating in a substantial paper or other piece of work which can be meaningfully evaluated to assign a letter grade; may not be taken on a PA/FL basis. Students engaged in preparation of a senior thesis ordinarily should take Applied Mathematics 99r instead.
Note: May be taken as a half course in either term; normally may not be taken for more than two terms. Applications may be obtained at Pierce Hall 110. Students should consult their advisers and concentration literature for further information and guidance. Applications must be signed by the student, by the faculty member supervising the project (who will recommend the grade), and by the Director of Undergraduate Studies, who will sign the student’s study card once the project and its method of evaluation have been approved.

*Applied Mathematics 99r. Thesis Research
Catalog Number: 4648
Michael P. Brenner and Marie D. Dahleh
Half course (fall term; repeated spring term). Hours to be arranged.
Provides an opportunity for students to engage in preparatory research and the writing of a senior thesis. Graded on a SAT/UNS basis as recommended by the thesis supervisor. The thesis is evaluated by the supervisor and by two additional readers.
Note: May be taken as a half course in either term; normally may not be taken for more than two terms. The Director of Undergraduate Studies will sign the student’s study card once a faculty member has agreed in writing to supervise preparation of the thesis, and reaffirmed this agreement if the course is to be repeated. Applications may be obtained at Pierce Hall 110. Students should consult their advisers and concentration literature for further information and guidance.

For Undergraduates and Graduates

Applied Mathematics 101. Statistical Inference for Scientists and Engineers
Catalog Number: 3350
Patrick J. Wolfe
Half course (fall term). M., W., 2:30–4. EXAM GROUP: 7, 8
Introductory statistical methods for students in the applied sciences and engineering. Random variables and probability distributions; the concept of random sampling, including random samples, statistics, and sampling distributions; the Central Limit Theorem and its role in statistical inference; parameter estimation, including point estimation and maximum likelihood methods; confidence intervals; hypothesis testing; simple linear regression; and multiple linear regression. Introduction to more advanced techniques as time permits.
Note: May not be taken in addition to Engineering Sciences 101. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.
Prerequisite: Applied Mathematics 21b or Mathematics 21b or equivalent.

Applied Mathematics 105a. Complex and Fourier Analysis
Catalog Number: 7732
L. Mahadevan and Kenneth Norman Kamrin
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
Complex Analysis: complex numbers, functions, mapping, differentiation, integration, branch cuts, series expansions, residue theory. Fourier Analysis: Fourier series, Fourier and Laplace transforms, applications to differential equations and data analysis.
Note: Applied Mathematics 105a and 105b are independent courses, and may be taken in any order.
Prerequisite: Applied Mathematics 21a and 21b, or Mathematics 21a and 21b.

Applied Mathematics 105b. Ordinary and Partial Differential Equations
Catalog Number: 6316
Eli Tziperman and Scott A. Norris
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
Ordinary differential equations: power series solutions; special functions; eigenfunction expansions. Algebra and calculus of vectors, dyadics, and tensors. Elementary partial differential equations: separation of variables and series solutions; similarity solutions; comparison of elliptic, parabolic and hyperbolic systems. Asymptotics.
Note: Applied Mathematics 105a and 105b are independent courses, and may be taken in any order.
Prerequisite: Applied Mathematics 21a and 21b, or Mathematics 21a and 21b.

Applied Mathematics 106. Applied Algebra and Combinatorics
Catalog Number: 3871
Salil P. Vadhan
Half course (fall term). M., W., 2:30–4. EXAM GROUP: 7, 8
Introduction to abstract algebra and its applications. Sets, subsets, and partitions; mappings, operations, and equivalence relations; groups, rings, and fields, polynomials, encryption, computer coding, application of modular arithmetic, combinatorial designs, lattices, application of trellis representation of lattices, fast algorithms.
Prerequisite: Applied Mathematics 21a and 21b, or Mathematics 21a and 21b

Applied Mathematics 107. Graph Theory and Combinatorics
Catalog Number: 6411
Leslie G. Valiant
Half course (spring term). Tu., Th., 10-11:30, and a weekly section to be arranged. EXAM GROUP: 12, 13
Topics in combinatorial mathematics that find frequent application in computer science, engineering, and general applied mathematics. Specific topics taken from graph theory, enumeration techniques, optimization theory, combinatorial algorithms, and discrete probability.

Applied Mathematics 111. Introduction to Scientific Computing
Catalog Number: 7000
Zhiming Kuang
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Many complex physical problems defy simple analytical solutions or even accurate analytical approximations. Scientific computing can address certain of these problems successfully, providing unique insight. This course introduces some of the widely used techniques in scientific computing through examples chosen from physics, chemistry, and biology. The purpose of the course is to introduce methods that are useful in applications and research and to give the students hands-on experience with these methods.
Prerequisite: Applied Mathematics 21a and 21b, or Mathematics 21a and 21b, or permission of instructor.

Applied Mathematics 115. Mathematical Modeling
Catalog Number: 1768
William H. Bossert (fall term) and John W. Hutchinson (spring term)
Half course (fall term; repeated spring term). Fall: M., W., 1–2:30; Spring: Tu., Th., 11:30–1. EXAM GROUP: Fall: 6, 7; Spring: 13, 14
Abstracting the essential components and mechanisms from a natural system to produce a mathematical model, which can be analyzed with a variety of formal mathematical methods, is perhaps the most important, but least understood, task in applied mathematics. This course approaches a number of problems without the prejudice of trying to apply a particular method of solution. Topics drawn from mechanics, biology, economics and the behavioral sciences.
Prerequisite: Mathematics at least at the level of Applied Mathematics 21a,b. Additional skills in analysis, algebra, probability, statistics and computer programming will increase the value of the course to students.

Applied Mathematics 120. Applicable Linear Algebra
Catalog Number: 4378
Donald G. M. Anderson
Half course (fall term). Tu., Th., 2:30–4. EXAM GROUP: 16, 17
An algorithmic approach to topics in matrix theory which arise frequently in applied mathematics: linear equations, pseudoinverses, quadratic forms, eigenvalues and singular values, linear inequalities and optimization, linear differential and difference equations.
Note: Offered in alternate years.
Prerequisite: Applied Mathematics 21b, or Mathematics 21b, or equivalent.

Applied Mathematics 121. Introduction to Optimization: Models and Methods
Catalog Number: 3187 Enrollment: Limited to 40.
Yiling Chen
Half course (spring term). M., W., 1–2:30. EXAM GROUP: 6, 7
Introduction to basic mathematical ideas and computational methods for solving deterministic and stochastic optimization problems. Topics covered: linear programming, integer programming, branch-and-bound, branch-and-cut, Markov chains, Markov decision processes, queuing theory. Emphasis on modeling. Examples from business, society, engineering, sports, e-commerce. Exercises in AMPL, complemented by Maple or Matlab.
Note: May not be taken in addition to Engineering Sciences 102.
Prerequisite: Applied Mathematics 21b or Mathematics 21b (linear algebra) and some knowledge of probability and statistics at the level of Statistics 110 or Applied Mathematics 101 or permission of instructor.

[Applied Mathematics 147. Nonlinear Dynamical Systems]
Catalog Number: 7708
Instructor to be determined
Half course (spring term). Hours to be arranged.
An introduction to nonlinear dynamical phenomena, covering the behavior of systems described by ordinary differential equations. Topics include: stability; bifurcations; chaos; routes to chaos and universality; approximations by maps; strange attractors; fractals. Techniques for analyzing nonlinear systems are introduced with applications to physical, chemical, and biological systems such as forced oscillators, chaotic reactions, and population dynamics.
Note: Expected to be given in 2010–11.
Prerequisite: Mathematics 21a and 21b, or Applied Mathematics 21a and 21b.

Cross-listed Courses

[Earth and Planetary Sciences 100. The Missing Matlab Course: An Introduction to Programming and Data Analysis]
MCB 111. Mathematics in Biology

Primarily for Graduates

Applied Mathematics 201. Physical Mathematics I
Catalog Number: 3241
Michael P. Brenner
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Introduction to methods for developing accurate approximate solutions for problems in the physical sciences that cannot be solved exactly. Topics include: complex function theory, approximate solution of integrals, algebraic equations, nonlinear differential equations, and qualitative methods.
Prerequisite: Applied Mathematics 105a, b or equivalent.

Applied Mathematics 202. Physical Mathematics II
Catalog Number: 6559
L. Mahadevan
Half course (spring term). M., W., F., at 9. EXAM GROUP: 2
Theory and techniques for finding exact and approximate analytical solutions of partial differential equations with numerical evaluation: eigenfunction expansions, Green functions, variational calculus, transform techniques, perturbation methods, characteristics, line asymptotic methods and selected nonlinear PDE’s.
Prerequisite: Applied Mathematics 105a and 105b, or equivalent.

[Applied Mathematics 204. Geometrical Methods in the Physical and Engineering Sciences]
Catalog Number: 1763
Jene A. Golovchenko
Half course (spring term). Tu., Th., 11:30–1.
Introduction to geometrical concepts used to model physical phenomena. Coordinate and coordinate-free geometrical objects, fields, flows, calculus on manifolds, metrics, connections, integrability, symmetry and continuous group structures, gauge fields. Applications: mechanics and field theories.
Note: Expected to be given in 2010–11. Undergraduate courses in linear algebra, multivariable calculus, classical/analytical mechanics, and a field theory like electromagnetism, fluid mechanics or quantum mechanics are strongly recommended.

[Applied Mathematics 205. Practical Scientific Computing]
Catalog Number: 1370
Shreyas Mandre
Half course (fall term). Tu., Th., 1–2:30.
Computational methods at a sophisticated analytic level. Practical exercises emphasized. A wide range of topics from linear algebra to Fourier analysis will be covered.
Note: Expected to be given in 2010–11.
Prerequisite: Mathematics at the level of Applied Mathematics 105b. A previous course in computing is not required.

Applied Mathematics 206. Advanced Applied Algebra and Combinatorics
Catalog Number: 6018
Salil P. Vadhan
Half course (fall term). M., W., 2:30–4. EXAM GROUP: 7, 8
Sets, subsets, and partitions; mappings, operations, and equivalence relations; groups, rings, and fields, polynomials, encryption, computer coding, application of modular arithmetic, combinatorial designs, lattices, application of trellis representation of lattices, fast algorithms; selected readings.
Note: Meets with Applied Mathematics 106. Students enrolled in Applied Mathematics 206 will be assigned additional readings.

[Applied Mathematics 210. Elementary Functional Analysis]
Catalog Number: 2781
Donald G. M. Anderson
Half course (fall term). Tu., Th., 2:30–4.
An introduction to functional analysis and its applications: metric, Banach and Hilbert spaces; linear operators, spectral theory; differentiation and integration.
Note: Expected to be given in 2010–11. Offered in alternate years.
Prerequisite: Applied Mathematics 105a and 105b; and Applied Mathematics 120 or Mathematics 121, or equivalent.

Applied Mathematics 211. Introduction to Numerical Mathematics
Catalog Number: 1894
Donald G. M. Anderson
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Principles and techniques of numerical analysis, synthesis and computation: interpolation and approximation, numerical quadrature and differentiation, linear and nonlinear equations, optimization, differential and integral equations.
Prerequisite: Applied Mathematics 105a and 105b; Applied Mathematics 111 or 120 would be helpful, but not required.

[Applied Mathematics 212. Numerical Solution of Differential Equations]
Catalog Number: 6127
Donald G. M. Anderson
Half course (spring term). Tu., Th., 10–11:30.
The development, study and implementation of numerical methods for the approximate solution of ordinary and partial differential equation initial and boundary value problems, and related topics.
Note: Expected to be given in 2010–11. Offered in alternate years.
Prerequisite: Applied Mathematics 211, or equivalent; Applied Mathematics 202 or 210 would be helpful, but not required.

Applied Mathematics 213. Topics in Numerical Mathematics
Catalog Number: 1048
Donald G. M. Anderson
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Analytical and numerical methods for the approximate solution of integral equations.
Note: Offered in alternate years.
Prerequisite: Applied Mathematics 211, or equivalent; Applied Mathematics 202 or 210 would be helpful, but not required.

[Applied Mathematics 298r. Special Topics in Applied Mathematics: Self Assembly]
Catalog Number: 3882
Michael P. Brenner
Half course (spring term). M., W., 2:30–4.
This course will study the theoretical and mathematical basis for self assembly, focusing on what is required to make engineering-based self assembly a reality. Three parts: foundations, engineering solutions, and biological assembly.
Note: Expected to be given in 2010–11.
Prerequisite: Undergraduate statistical mechanics or permission of the instructor.

Applied Mathematics 299r. Special Topics in Applied Mathematics
Catalog Number: 5798
Michael P. Brenner (spring term)
Half course (fall term; repeated spring term). Hours to be arranged.
Supervision of experimental or theoretical research on acceptable applied mathematics problems and supervision of reading on topics not covered by regular courses of instruction.
Note: Open to graduate students and AB/SM candidates only. Students must arrange such work with a member of the School of Engineering and Applied Sciences. This course is graded and is ordinarily taken with the approval of the Committee on Higher Degrees. Applicants must file a project sheet before study cards are filed. Project sheets may be obtained from the Academic Office, Pierce Hall 110.

Cross-listed Courses

*Computer Science 365. SEAS Teaching Practicum
Catalog Number: 8195
John G. Morrisett 4853
Half course (spring term). W., 4–5:30. EXAM GROUP: 9
Gain effective skills for teaching applied sciences. Topics: presentation and communication, lesson planning, classroom practice, office hours and 1-on-1 interactions, feedback, assessment, and working with course staff. Seminar style with an emphasis on observation, practice, feedback, discussion, and reflection.

Graduate Courses of Reading and Research

Reading courses are odd-numbered; research courses are even-numbered.

*Applied Mathematics 311,312. Numerical Mathematics: Analysis, Synthesis and Computation
Catalog Number: 7333,6118
Donald G. M. Anderson 1061

*Applied Mathematics 315,316. Stochastic Processes, Dynamical Systems, Applied Differential Geometry
Catalog Number: 2458,2459
Roger W. Brockett 3001

*Applied Mathematics 317,318. Special Topics in Physical Mathematics
Catalog Number: 9160,2166
Michael P. Brenner 4101

*Applied Mathematics 319,320. Topics in Macroscopic Physics and Quantitative Biology
Catalog Number: 2084,4567
L. Mahadevan 4758

*Applied Mathematics 321,322. Biological Applications of Mathematics and Automatic Computers
Catalog Number: 7615,4243
William H. Bossert 1049

*Applied Mathematics 331,332. Theoretical Mechanics in the Earth and Engineering Sciences
Catalog Number: 0112,0251
James R. Rice 7270

*Applied Mathematics 341,342. Applied Probability and Statistical Inference, Classical and Quantum Information Theory
Catalog Number: 0970,6033
Navin Khaneja 4192