Program in Applied and Computational Mathematics

Director

A. Robert Calderbank

Director of Graduate Studies

Paul D. Seymour

Executive Committee

A. Robert Calderbank, Electrical Engineering and Mathematics

René A. Carmona, Operations Research and Financial Engineering

Emily A. Carter, Mechanical and Aerospace Engineering

Ingrid C. Daubechies, Mathematics

Weinan E, Mathematics

Philip J. Holmes, Mechanical and Aerospace Engineering

Yannis G. Kevrekidis, Chemical Engineering

Paul D. Seymour, Mathematics

James M. Stone, Astrophysical Sciences

Sergio Verdú, Electrical Engineering

Associated Faculty

Yacine Ait-Sahalia, Economics

Michael Aizenman, Physics, Mathematics

William Bialek, Physics

Roberto Car, Chemistry

Bernard Chazelle, Computer Science

Mung Chiang, Electrical Engineering

Erhan Çinlar, Operations Research and Financial Engineering

F. Anthony Dahlen, Geosciences

Bradley W. Dickinson, Electrical Engineering

David P. Dobkin, Computer Science

Jianqing Fan, Operations Research and Financial Engineering

Christodoulos A. Floudas, Chemical Engineering

Isaac M. Held, Geosciences, Atmospheric and Oceanic Sciences

John J. Hopfield, Molecular Biology

Sergiu Klainerman, Mathematics

John A. Krommes, Astrophysical Sciences, Plasma Physics

Naomi Ehrich Leonard, Mechanical and Aerospace Engineering

Simon A. Levin, Ecology and Evolutionary Biology

Elliott H. Lieb, Mathematics and Physics

Maria P. Martin, Mechanical and Aerospace Engineering

Luigi Martinelli, Mechanical and Aerospace Engineering

William A. Massey, Operations Research and Financial Engineering

Jeremiah P. Ostriker, Astrophysical Sciences

H. Vincent Poor, Electrical Engineering

Jean-Hervé Prévost, Civil and Environmental Engineering

Herschel A. Rabitz, Chemistry

Peter J. Ramadge, Electrical Engineering

Clarence W. Rowley, Mechanical and Aerospace Engineering

Robert E. Schapire, Computer Science

José Scheinkman, Economics

Yakov G. Sinai, Mathematics

Burton H. Singer, Woodrow Wilson School

Jaswinder P. Singh, Computer Science

K. Ronnie Sircar, Operations Research and Financial Engineering

Sankaran Sundaresan, Chemical Engineering

Salvatore Torquato, Chemistry

Geoffrey K. Vallis, Geosciences, Atmospheric and Oceanic Sciences

Robert J. Vanderbei, Operations Research and Financial Engineering

 

The Program in Applied and Computational Mathematics offers a select group of highly qualified students the opportunity to obtain a thorough knowledge of branches of mathematics indispensable to science and engineering applications, including numerical analysis and other computational methods.

Before being admitted to a third year of study, students must sustain the general examination. The general examination, or generals, is designed as a sequence of interviews with assigned professors that takes place during the first year and covers three areas of applied mathematics. The generals culminate in a seminar on a research topic, usually delivered toward the end of the fourth term.

A student qualifies for the award of the Master of Arts (M.A.) degree by successfully completing all course work and passing the first portion of the general examination.

The doctoral dissertation may consist of a mathematical contribution to some field of science or engineering, or the development or analysis of mathematical or computational methods useful for, inspired by, or relevant to science or engineering.

Students may opt for a joint degree in materials and applied and computational mathematics. See the Princeton Institute for the Science and Technology of Materials (PRISM) entry for further detail.

Satisfactory completion of the requirements leads to the degree of Doctor of Philosophy in applied and computational mathematics.

Courses

APC 501 Mathematical Methods of Engineering Analysis I (see MAE 501)

APC 502 Mathematical Methods of Engineering Analysis II (see MAE 502)

APC 503 Analytical Techniques in Differential Equations (see AST 557)

APC 505 Numerical Methods in Computational Science

Staff

A basic graduate course in numerical analysis and scientific computing. The topics include methods for systems of linear and nonlinear equations, eigenvalue problems, interpolation, and quadrature. The principles and techniques of finite-difference, finite-element and finite-volume methods for differential equations are studied, and hierarchical methods and techniques for distributed computing are introduced.

APC 507 Basic Numerical Methods for Ordinary and Partial Differential Equations (also MAE 503)

Weinan E

Methods for molecular and continuum simulations of solids and fluids, molecular dynamics, and kinetic Monte Carlo methods are studied.

APC 514 Biological Dynamics (see MOL 514)

APC 515 Random Heterogeneous Materials (see MSE 515)

APC 518 Applied Stochastic Analysis and Methods (also ORF 518)

Weinan E

An introduction to stochastic models in the physical sciences, with an emphasis on numerical methods, asymptotics, and connection with partial differential equations. After a brief introduction to the basics of probability theory, the Markov process, and stochastic differential equations, the course concentrates on Fokker-Planck equations, invariant distributions, path integrals, and large deviation and rare events. Numerical methods for computing transition pathways and transition rates, and kinetic Monte Carlo methods are discussed. Prerequisite: elementary differential equations.

APC 539 Nonlinear Processes in Fluids and Plasmas (also AST 559)

John A. Krommes

A comprehensive introduction to the theory of nonlinear phenomena in fluids and plasmas, with an emphasis on turbulence and transport. Experimental phenomenology; fundamental equations, including Navier-Stokes, Vlasov, and gyrokinetic; numerical simulation techniques, including pseudo-spectral and particle-in-cell methods; coherent structures; transition to turbulence; statistical closures, including the wave kinetic equation and direct-interaction approximation; PDF methods and intermittency; and variational techniques are studied. Applications from neutral fluids, fusion plasmas, and astrophysics are studied as well.

APC 544 Topics in Computational Nonlinear Dynamics (see CHE 554)

APC 550 Introduction to Differential Equations

Weinan E

An introduction to differential equations for graduate students in science and engineering. Applications and fundamental theory; basic second-order differential equations (including the wave, heat, and Poisson equations); separation of variables and solution by Fourier series and Fourier integrals; boundary value problem and Green’s function; variational methods; normal mode analysis and perturbation methods; nonlinear first-order (Hamilton-Jacobi) equations and method of characteristics; reaction-diffusion equations; also, application of these equations and methods to, for example, finance and control. Necessary background material in ODEs is covered.

APC 551 Probability Theory (see ORF 551)

APC 571 Applied Dynamical Systems (see MAE 541)

APC 583 Wavelets: Applications of Wavelets in Mathematics and Other Fields (also MAT 593)

Ingrid C. Daubechies

Wavelet analysis, especially wavelet bases, is a functional analytic tool having applications in many fields. The course is aimed at building a bridge between mathematics and engineering, and should be accessible to students from either discipline, provided they have sufficient background in mathematical analysis. A graduate course open to undergraduates who have taken MAT 314 and 315. If in doubt about having the necessary background, the student should consult the instructor.

APC 584 Wavelets: Applications of Wavelets in Mathematics and Other Fields

Radu V. Balan

Course covers topics of wavelet and time-frequency analysis, with special emphasis on wavelet basis construction and filterbanks. It aims at building a bridge between the mathematics of harmonic analysis and its applications in engineering sciences. Two-thirds of the time is spent on theory, with the remaining one-third being devoted to applications.

APC 586 Topics in Discrete Mathematics: Discrete Math (see MAT 595)

APC 590 Computational Methods and Their Applications Across Disciplines

Staff

A survey of state-of-the-art computational methods useful across many scientific disciplines, emphasizing the practical application of methods such as computational fluid dynamics, molecular dynamics, Monte Carlo, multigrid, optimization, and statistical analysis (e.g., clustering, pattern matching, and classification). The course has a strong interdisciplinary flavor, and includes lectures, reading, and programming. Methods are introduced and discussed in the context of motivating problems in different disciplines, and key issues for high-performance, scalable computing are discussed.

APC 599 Summer Extramural Research Project

Ingrid Daubechies

A summer research project, designed in conjunction with the student’s advisor, APC, and an industrial, NGO, or government sponsor that will provide practical experience relevant to the student’s research area. Start date no earlier than June 1. A final paper and sponsor evaluation are required. Students considering applying for this course should review the recommended guidelines before consulting their adviser and director of graduate studies.

Pertinent Courses in Allied Departments

Astrophysical Sciences

551, 552 General Plasma Physics I, II

553 Plasma Waves and Instabilities

554 Irreversible Processes in Plasma

560 Computational Methods in Plasma Physics

Atmospheric and Oceanic Sciences

571 Introduction to Geophysical Fluid Dynamics

572 Atmospheric and Oceanic Wave Dynamics

573 Physical Oceanography

575 Numerical Prediction of the Atmosphere and Ocean

576 Current Topics in Dynamic Meteorology

577 Weather and Climate Dynamics

Chemical Engineering

442 Design, Synthesis, and Optimization of Chemical Processes

448 Introduction to Nonlinear Dynamics

501 Fluid Mechanics

521 Advanced Chemical Reactor Engineering

524 Introduction to Statistical Mechanics

527 Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications

528 Advanced Process Flowsheeting and Process Control

529 Hydrodynamic Stability

530 Systems Engineering

Chemistry

501 Introduction to Quantum Chemistry

502 Advanced Quantum Chemistry

Civil and Environmental Engineering

513 Introduction to Finite-Element Methods

521 Continuum Mechanics

522 Random Vibration Theory and Applications to Earthquake and Wind Engineering

532 Advanced Finite-Element Methods

558 Random Fields and Random Media

Computer Science

226 Algorithms and Data Structures

341 Discrete Mathematics

402 Artificial Intelligence

423 Theory of Algorithms

426 Computer Graphics

451 Computational Geometry

487 Theory of Computation

522 Computational Complexity

526 Advanced Computer Graphics

528 Data Structures and Graph Algorithms

551 Introduction to Computational Molecular Biology

557 Analysis and Visualization of Large-Scale Genomic Data Sets

593, 594 Advanced Topics in the Theory of Algorithms

597, 598 Advanced Topics in Computer Science

Ecology and Evolutionary Biology

324 Theoretical Ecology

519 Theoretical Ecology

Economics

511, 512 Advanced Economic Theory I, II

513 Advanced Econometrics: Time Series Models

514 Game Theory

519 Advanced Econometrics: Nonlinear Models

525, 526 Financial Economics I, II

Electrical Engineering

301 Circuits and Signal Processing

352 Physical Optics

391 The Wireless Revolution: Telecommunications for the 21st Century

454 Photonics and Light-Wave Communications

482 Digital Signal Processing

485 Signal Analysis and Communication Systems

520 Optimization and Optimal Control

521 Linear System Theory

523 Nonlinear System Theory

525 Random Processes in Information Systems

527 Selected Topics in Signal Processing

528 Information Theory

530 Theory of Detection and Estimation

532 Adaptive Systems

533 Multiuser Communication Theory

535 Machine Learning and Pattern Recognition

571 Digital Neurocomputing

Geosciences

557 Theoretical Geophysics

Materials Science and Engineering

504 Modeling and Simulation in Materials Science

515 Random Heterogeneous Materials

Mathematics

303 Ordinary Differential Equations

304 Introduction to Partial Differential Equations

305 Mathematical Programming

306 Introduction to Graph Theory

308 Theory of Games

311 Introduction to Modern Applied Mathematics

314 Introduction to Real Analysis

317 Complex Analysis with Applications

327 Introduction to Differential Geometry

328 Differential Geometry

330 Analysis I: Fourier Series and Partial Differential Equations

331 Analysis II: Complex Analysis

332 Analysis III: Integration Theory and Hilbert Space

333 Analysis IV: Special Topics in Analysis

390 Probability Theory

391 Random Processes

535, 536 Nonlinear Wave Equations

581, 582 Stochastic Processes

583, 584 Statistical Mechanics

585, 586 Mathematical Physics

591, 592 Applied Partial Differential Equations

Mechanical and Aerospace Engineering

305, 306 Mathematics in Engineering I, II

335 Fluid Dynamics

336 Viscous Flows

434 Modern Control

542 Advanced Dynamics

544 Aircraft Dynamics

545 Nonlinear Control

546 Optimal Control and Estimation

551 Fluid Mechanics

552 Viscous Flows and Boundary Layers

553 Turbulent Flow

554 Stability and Turbulence

555 Nonequilibrium Gas Dynamics

Molecular Biology

437 Computational Neurobiology and Computing Networks

457 Computational Aspects of Molecular Biology

Operations Research and Financial Engineering

307 Optimization

309 Probability and Stochastic Systems

311 Optimization Under Uncertainty

405 Regression and Applied Time Series

515 Asset Pricing II: Stochastic Calculus and Advanced Derivatives

522 Linear Optimization

523 Nonlinear Optimization

524 Statistical Theory and Methods

526 Stochastic Modeling

527 Stochastic Calculus and Finance

531 Computational Finance in C++

534 Financial Engineering

542 Controlled Markov Processes

547 Dynamic Programming

548 Large-Scale Optimization

549 Stochastic Programming

551 Probability Theory

553 Stochastic Differential Equations

554 Markov Processes

557, 558 Stochastic Analysis Seminar

Physics

403 Mathematical Methods of Physics

408 Modern Classical Dynamics

505, 506 Quantum Mechanics I, II

511 Thermodynamics, Kinetic Theory, and Statistical Mechanics

521 Introduction to Mathematical Physics

523 Introduction to Relativity

535 Condensed Matter/Many-Body Physics