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

Donald G. M. Anderson, Gordon McKay Professor of Applied Mathematics (on leave spring term)
William H. Bossert, David B. Arnold, Jr. Professor of Science (on leave spring term)
Michael P. Brenner, Glover Professor of Applied Mathematics and Applied Physics, Harvard College Professor, Area Dean for Applied Mathematics (on leave 2011-12)
Roger W. Brockett, An Wang Professor of Electrical Engineering and Computer Science (on leave 2011-12)
Yiling Chen, Assistant Professor of Computer Science (Director of Undergraduate Studies, fall term)
Marie D. Dahleh, Senior Lecturer on Engineering Sciences
Ozlem Ergun, Visiting Associate Professor of Applied Mathematics
Miranda C. Holmes, Lecturer on Applied Mathematics
Evelyn Hu, Gordon McKay Professor of Applied Physics and of Electrical Engineering, Area Dean for Electrical Engineering
Efthimios Kaxiras, John Hasbrouck Van Vleck Professor of Pure and Applied Physics
Navin Khaneja, Gordon McKay Professor of Electrical Engineering
David J. Knezevic, Lecturer on Computational Science
Margo S. Levine, Lecturer on Applied Mathematics
L. Mahadevan, Lola England de Valpine Professor of Applied Mathematics and of Organismic and Evolutionary Biology
Cherry Murray, John A. and Elizabeth S. Armstrong Professor of Engineering and Applied Sciences and Professor of Physics, Dean of the School of Engineering and Applied Sciences
Pavlos Protopapas, Lecturer on Computational Science
James R. Rice, Mallinckrodt Professor of Engineering Sciences and Geophysics
Mauricio Santillana, Lecturer on Applied Mathematics
Jenny Suckale, Lecturer on Applied Mathematics
Vahid Tarokh, Perkins Professor of Applied Mathematics and Vinton Hayes Senior Research Fellow of Electrical Engineering (on leave 2011-12)
Eli Tziperman, Pamela and Vasco McCoy, Jr.Professor of Oceanography and Applied Physics (Director of Undergraduate Studies, spring term) (on leave fall term)
Salil P. Vadhan, Vicky Joseph Professor of Computer Science and Applied Mathematics (on leave 2011-12)
Leslie G. Valiant, T. Jefferson Coolidge Professor of Computer Science and Applied Mathematics

Other Faculty Offering Instruction in Applied Mathematics

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. 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 Office of Student Affairs, 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
Evelyn Hu
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. 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
Margo S. Levine
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. 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 Mauricio Santillana
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
Marie D. Dahleh and Margo S. Levine
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
Marie D. Dahleh and Margo S. Levine
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
Miranda C. Holmes
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 104 (formerly Applied Mathematics 105a). Complex and Fourier Analysis
Catalog Number: 7732
Efthimios Kaxiras
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.
Prerequisite: Applied Mathematics 21a and 21b, or Mathematics 21a and 21b.

Applied Mathematics 105 (formerly Applied Mathematics 105b). Ordinary and Partial Differential Equations
Catalog Number: 6316
Eli Tziperman
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
Ordinary differential equations: power series solutions; special functions; eigenfunction expansions. Review of vector calculus. Elementary partial differential equations: separation of variables and series solutions; diffusion, wave and Laplace equations. Brief introduction to nonlinear dynamical systems and to numerical methods.
Prerequisite: Applied Mathematics 21a and 21b, or Mathematics 21a and 21b.

[Applied Mathematics 106. Applied Algebra]
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.
Note: Expected to be given in 2012–13.
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
Jenny Suckale
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), L. Mahadevan (spring term), and Brendan J. Meade (spring term)
Half course (fall term; repeated spring term). Fall: Section l: M., W., 1-2:30; Section ll: M., W., 11:30-1; Spring: M., W., 1-2:30. EXAM GROUP: 6, 7
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 biology, economics, engineering, physical and social sciences.
Prerequisite: Mathematics at least at the level of Applied Mathematics 21a, b but preferably at the level of Applied Mathematics 105 (formerly Applied Mathematics 105b). 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
Mauricio Santillana
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.
Prerequisite: Applied Mathematics 21b, or Mathematics 21b, or equivalent.

Applied Mathematics 121. Introduction to Optimization: Models and Methods
Catalog Number: 3187 Enrollment: Limited to 60.
Ozlem Ergun
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. 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
Margo S. Levine
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
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.
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

[MCB 198. Advanced Mathematical Techniques for Modern Biology ]

Primarily for Graduates

Applied Mathematics 201. Physical Mathematics I
Catalog Number: 3241 L. Mahadevan
Half course (fall term). M., W., 1–2:30. EXAM GROUP: 6, 7
Introduction to methods for developing accurate approximate solutions for problems in the sciences that cannot be solved exactly, and integration with numerical methods and solutions. Topics include: approximate solution of integrals, algebraic equations, nonlinear ordinary differential equations and their stochastic counterparts, and partial differential equations. Introduction to "sophisticated" uses of MATLAB. Prerequisite: Applied Mathematics 104 (formerly Applied Mathematics 105a), Applied Mathematics 105 (formerly Applied Mathematics 105b) or equivalent.

[Applied Mathematics 202. Physical Mathematics II]
Catalog Number: 6559
Instructor to be determined
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 104 (formerly Applied Mathematics 105a) and Applied Mathematics 105 (formerly Applied Mathematics 105b) or equivalent.

Applied Mathematics 205. Advanced Scientific Computing: Numerical Methods
Catalog Number: 1370
David Knezevic
Half course (fall term). M., W., F., at 10. EXAM GROUP: 3
An examination of the mathematical foundations of a range of well-established numerical algorithms, exploring their use through practical examples drawn from a range of scientific and engineering disciplines. Emphasizes theory and numerical analysis to elucidate the concepts that underpin each algorithm. There will be a significant programming component. Students will be expected to implement in Matlab a range of numerical methods through individual and group-based project work to get hands-on experience with modern scientific computing.
Prerequisite: Familiarity with linear algebra and calculus; basic programming knowledge at the Computer Science 50 level.

[Applied Mathematics 206. Advanced Applied Algebra]
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: Expected to be given in 2012–13. Meets with Applied Mathematics 106. Students enrolled in Applied Mathematics 206 will be assigned additional readings.

Applied Mathematics 207 (formerly Applied Mathematics 205b). Advanced Scientific Computing: Stochastic Optimization Methods
Catalog Number: 78757
Pavlos Protopapas
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Develops skills for computational research with focus on stochastic approaches, emphasizing implementation and examples. Stochastic methods make it feasible to tackle very diverse problems when the solution space is too large to explore systematically, or when microscopic rules are known, but not the macroscopic behavior of a complex system. Methods will be illustrated with examples from a wide variety of fields, ranging from simulating the immune system to strategies for investing in financial markets.
Prerequisite: Basic knowledge of a computer programming language (such as C or/and Python).

[Applied Mathematics 210. Elementary Functional Analysis]
Catalog Number: 2781
Instructor to be determined
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 2012–13. Offered in alternate years.
Prerequisite: Applied Mathematics 104 (formerly Applied Mathematics 105a) and Applied Mathematics 105 (formerly Applied Mathematics 105b) or equivalent.; and Applied Mathematics 120 or Mathematics 121, or equivalent.

[Applied Mathematics 211. Introduction to Numerical Mathematics]
Catalog Number: 1894
Instructor to be determined.
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 given in 2012–13.
Prerequisite: Applied Mathematics 105a and 105b; Applied Mathematics 111 or 120 would be helpful, but not required.

[*Applied Mathematics 215. Fundamentals of Biological Signal Processing]
Catalog Number: 23661 Enrollment: Limited to 20.
Instructor to be determined
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
The course will introduce Bayesian analysis, maximum entropy principles, hidden markov models and pattern theory. These concepts will be used to understand information processing in biology. The relevant biological background will be covered in depth.
Prerequisite: A strong background in Calculus, Linear Algebra, Fourier Analysis, complex analysis at the advanced undergraduate level and an introductory knowledge of probability theory is required. Knowledge of Statistical Mechanics and comfort with programming will be useful.

Applied Mathematics 221. Advanced Optimization - (New Course) 
Catalog Number: 84323 
Ozlem Ergun 
Half course (fall term). M., W., 2:30–4.
Advanced techniques for modeling and solving large and difficult optimization problems as well as the core theory and geometry of linear inequalities, integer programming and combinatorial optimization. Topics covered: geometry and theory of linear programming, solving large scale optimization problems using column and constraint generation, network flows, computational complexity, basic integer programming models and algorithms, paths and trees, matchings, integrality of polyhedra, and matroids. Emphasis will be on developing an understanding of the core theory and solution methods. Exercises and the class project will involve developing and implementing optimization algorithms possibly using standard solvers such as AMPL.
Prerequisite: Applied Mathematics 21b or Mathematics 21b (linear algebra) and AM 121 or equivalent or permission of instructor. Comfort with programming.

[Applied Mathematics 272r. Kinetic Methods for Fluids: Theory and Applications]
Catalog Number: 27235
Instructor to be determined
Half course (spring term). W., 3-5, M., 7-9 pm.
Systematic introduction to kinetic methods for studying fluids, based on the lattice Boltzmann equation. Emphasizes theory, including discrete dynamics and symmetry, as well as hands-on programming of basic algorithms for fluid flow simulations, paying attention to understanding of the theoretical basis and connection to real fluid physics. The course lays the foundation for further research on the method extensions, particularly in complex fluids and micro/nano-fluidics and presents specific applications in various science and engineering problems.
Note: Expected to be given in 2012–13.
Prerequisite: Knowledge of basic classical physics, fluid dynamics, and numerical methods are desirable.

Applied Mathematics 274. Computational Fluid Dynamics - (New Course)
Catalog Number: 70261
David Knezevic
Half course (spring term). Tu., Th., 10–11:30.
A theoretical and practical introduction to the key tools in computational fluid dynamics. The course will examine a range of numerical algorithms relevant to fluids modeling, analyzing the stability, convergence and accuracy of each. Students will implement an extensive range of CFD algorithms. Topics include the hyperbolic partial differential equations and conservation laws, with a focus on numerical discretization via finite volume methods, followed by simulation of viscous incompressible fluids via the finite element method.
Prerequisite: A first course in scientific computing, e.g. Applied Mathematics 111 or 205, and knowledge of computer programming.

Applied Mathematics 275. Computational Design of Materials - (New Course)
Catalog Number: 18739
Zhenyu Zhang
Half course (spring term). M., W., 2:30–4. EXAM GROUP: 7, 8
This course will provide the background and an extensive set of examples showing how computational methods are applied to modern design of materials with desired functionality. The methods will span multiple length and time scales, including molecular dynamics simulations, first-principles approaches, stochastic methods for optimization and sampling, and continuum elasticity theory. Examples will include problems in electronic and photonic devices, materials for energy conversion, storage, and environmental protection, and those related to mechanical strength of materials.
Prerequisite: Undergraduate coursework in quantum mechanics, solid state physics, thermodynamics and statistical mechanics is recommended. Knowledge of physical chemistry and solid mechanics is 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. EXAM GROUP: 7, 8
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 2012–13.
Prerequisite: Undergraduate statistical mechanics or permission of the instructor.

Applied Mathematics 299r. Special Topics in Applied Mathematics
Catalog Number: 5798
Efthimios Kaxiras
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 Student Affairs Office, Pierce Hall 110.

Cross-listed Courses

Graduate Courses of Reading and Research

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

Catalog Number: 7333,6118
Donald G. M. Anderson 1061 (on leave spring term)

*Applied Mathematics 315,316. Stochastic Processes, Dynamical Systems, Applied Differential Geometry
Catalog Number: 2458,2459
Roger W. Brockett 3001 (on leave 2011-12)

*Applied Mathematics 317,318. Special Topics in Physical Mathematics
Catalog Number: 9160,2166
Michael P. Brenner 4101 (on leave 2011-12)
Note: (on leave 2011-12)

*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 (on leave spring term)

*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