Applied Mathematics

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

Donald G. M. Anderson, Gordon McKay Professor of Applied Mathematics (Director of Undergraduate Studies )
William H. Bossert, David B. Arnold, Jr. Professor of Science
Michael P. Brenner, Gordon McKay Professor of Applied Mathematics and Applied Physics
Roger W. Brockett, An Wang Professor of Electrical Engineering and Computer Science (on leave fall term)
Daniel S. Fisher, Professor of Physics and Professor of Applied Physics
Navin Khaneja, Assistant Professor of Electrical Engineering on the Gordon McKay Endowment
Dionisios Margetis, Lecturer on Applied Mathematics
Venkatesh Narayanamurti, John A. and Elizabeth S. Armstrong Professor of Engineering and Applied Sciences and Professor of Physics (Dean of the Division of Engineering and Applied Sciences)
James R. Rice, Mallinckrodt Professor of Engineering Sciences and Geophysics
Allan R. Robinson, Gordon McKay Research Professor of Geophysical Fluid Dynamics (on leave 2001-02)
Irvin C. Schick, Lecturer on Engineering Sciences
Rocco A. Servedio, Lecturer on Applied Mathematics
Howard A. Stone, Harvard College Professor and Gordon McKay Professor of Chemical Engineering and Applied Mechanics
Tai T. Wu, Gordon McKay Professor of Applied Physics and Professor of Physics (on leave spring term)

Other Faculty Offering Instruction in Applied Mathematics

Jeremy Bloxham, Harvard College Professor and Professor of Geophysics

The Division of Engineering and Applied Sciences offers undergraduate and graduate courses in Applied Mathematics, Applied Physics, Computer Science, Earth and Planetary Sciences, and Engineering Sciences. Recommended curricula may be obtained from the Academic Office, Pierce Hall 110a. Division faculty also offer several courses in the section entitled General Education Electives.

For information concerning the concentration in Applied Mathematics please consult the Director of Undergraduate Studies or the Academic Office, Division of Engineering and Applied Sciences, Pierce Hall 110a. 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
Jeremy Bloxham and Anthony Harkin
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
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.
Note: May not be taken for credit by students who have passed Mathematics 21a.
Prerequisite: Mathematics 1b or equivalent.

Applied Mathematics 21b. Mathematical Methods in the Sciences
Catalog Number: 5074
Jeremy Bloxham and Anthony A. Harkin
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
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.
Note: May not be taken for credit by students who have passed Mathematics 21b.
Prerequisite: Applied Mathematics 21a or equivalent.

*Applied Mathematics 91r. Supervised Reading and Research
Catalog Number: 7607
Donald G. M. Anderson
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 pass/fail 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 from the Academic Office, Pierce Hall 110a. Applicants 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 assign 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
Donald G. M. Anderson
Half course (fall term; repeated spring term). Hours to be arranged.
Provides an opportunity for students to engage in preparatory research and writing of a senior thesis. Graded on a satisfactory/unsatisfactory basis 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. Students should consult their advisers and concentration literature for further information and guidance.

For Undergraduates and Graduates

Applied Mathematics 105a. Complex and Fourier Analysis
Catalog Number: 7732
Dionisios Margetis
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
Functions of a complex variable: mapping, integration, branch cuts, series. Fourier series; Fourier and Laplace transforms; transforms applied to differential equations and data analysis; convolution and correlation; elementary probability theory.
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
Howard A. Stone
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
Irvin C. Schick
Half course (fall term). Tu., Th., 4:30–6. EXAM GROUP: 18
Introduction to abstract algebra and its applications. Topics covered include sets, subsets, and partitions; mappings, operations, and equivalence relations; groups, rings, and fields; counting and number systems; and polynomials. Though the primary aim of the course is to establish the mathematical formalism and conceptual apparatus necessary for some future mathematics and engineering courses, examples will be given from applications such as finite automata, encryption, computer coding, and modular arithmetic.

Applied Mathematics 107. Graph Theory and Combinatorics
Catalog Number: 6411
Rocco A. Servedio
Half course (spring term). Tu., Th., 10–11:30. 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
Donald G. M. Anderson
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Elementary numerical methods and their computer implementation: linear and nonlinear equations; interpolation, differentiation and quadrature; ordinary differential equation initial and boundary value problems.
Note: Expected to be given in 2002–03. Offered in alternate years.
Prerequisite: Applied Mathematics 21a and 21b, or Mathematics 21a and 21b. Ability to program in some high-level computer language like Fortran.

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: Expected to be omitted in 2002–03. Offered in alternate years.
Prerequisite: Applied Mathematics 21b, or Mathematics 21b, or equivalent.

Applied Mathematics 147. Non-linear Dynamical Systems
Catalog Number: 7708
Daniel S. Fisher
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
An introduction to non-linear dynamical phenomena focussing on the behavior of systems described by ordinary differential equations such as oscillations, bifurcations and chaos. Approximations by maps, multiple-scale methods, and other techniques will be introduced. Some stochastic processes will also be studied. Applications will be made to physical and biological systems.
Prerequisite: Mathematics 21a and 21b, or Applied Mathematics 21a and 21b.

Primarily for Graduates

Applied Mathematics 201. Complex Function Theory with Applications
Catalog Number: 3241
Michael P. Brenner
Half course (fall term). M., W., F., at 9. EXAM GROUP: 2
First half: complex function theory: foundations, multiple-valued functions, analytic continuation, contour integration, conformal mapping. Second half: asymptotic methods: asymptotic expansions, method of steepest descent, approximate solutions of nonlinear differential equations, method of multiple scales.
Prerequisite: Applied Mathematics 105a, b or equivalent.

[Applied Mathematics 202. Partial Differential and Integral Equations]
Catalog Number: 6559
Donald G. M. Anderson
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 and integral equations, and related topics: eigenfunction expansions, Green functions, variational calculus, transform techniques, perturbation methods, characteristics.
Note: Expected to be given in 2002–03. Offered in alternate years.
Prerequisite: Applied Mathematics 105a and 105b, or equivalent.

Applied Mathematics 203. Topics in Applied Mathematics
Catalog Number: 6336
Tai T. Wu
Half course (fall term). M., W., F., at 9. EXAM GROUP: 2
Selected advanced mathematical methods. Topics vary from year to year. Examples are asymptotic analysis, WKB theory, multiscale analysis, solitons, singular integral equations, Wiener-Hopf theory.
Prerequisite: Applied Mathematics 201 or equivalent.

[Applied Mathematics 205. Practical Scientific Computing]
Catalog Number: 1370
William H. Bossert
Half course (fall term). Hours to be arranged.
Computational methods at a sophisticated analytic level. Practical exercises emphasized. Linear algebra; polynomial and rational function extrapolation; Chebyshev methods; special functions; nonlinear root finding; one- and multidimensional extremization; eigensystems; Fourier transform methods; linear and nonlinear model fitting; adaptive methods for differential equations; stochastic methods for integration and optimization of multidimensional functions.
Note: Expected to be given in 2002–03.
Prerequisite: Mathematics at the level of Applied Mathematics 105b. A previous course in computing is not required but ability to program in Fortran or C will be useful.

[Applied Mathematics 210. Elementary Functional Analysis]
Catalog Number: 2781
Donald G. M. Anderson
Half course (fall term). Tu., Th., 2:30–4. EXAM GROUP: 16, 17
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 2002–03. 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.
Note: Expected to be omitted in 2002–03. Offered in alternate years.
Prerequisite: Applied Mathematics 105a and 105b; Applied Mathematics 120, or equivalent, would be helpful. Ability to program in some high-level computer language like Fortran or C.

Applied Mathematics 212. Numerical Solution of Differential Equations
Catalog Number: 6127
Donald G. M. Anderson
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
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 omitted in 2002–03. Offered in alternate years.
Prerequisite: Applied Mathematics 211, or equivalent; Applied Mathematics 202 and 210 would be helpful, but are not required.

Applied Mathematics 299r. Special Topics in Applied Mathematics
Catalog Number: 5798
Venkatesh Narayanamurti
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 A.B./S.M. candidates only. Students must arrange such work with a member of the Division. This course is ordinarily taken with the approval of the Committee on Higher Degrees in certain cases when a letter grade is required. Applicants should file a project sheet before study cards are filed. Project sheets may be obtained from the Academic Office, Pierce Hall 110a.

Graduate Courses of Reading and Research

Reading courses are odd-numbered; research courses are even-numbered.
*Applied Mathematics 311,312. Numerical Mathematics and Scientific Computing
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 (on leave fall term)

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

*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 and Scientific Computation in Earth and Engineering Sciences
Catalog Number: 0112,0251
James R. Rice 7270

*Applied Mathematics 333,334. Dynamics of the Ocean; Interdisciplinary Modeling; Geophysical Fluid Dynamics
Catalog Number: 4947,4948
Allan R. Robinson 2133 (on leave 2001-02)

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