Program in Applied and Computational Mathematics
Faculty
Director Director of Graduate Studies Executive Committee Associated Faculty
Yacine AïtSahalia, Economics Michael Aizenman, Physics, Mathematics William Bialek, Physics, LewisSigler Institute for Integrative Genomics David M. Blei, Computer Science Mark Braverman, Computer Science Carlos D. Brody, Molecular Biology, Princeton Neuroscience Institute Adam Burrows, Astrophysical Sciences Roberto Car, Chemistry Moses S. Charikar, Computer Science Bernard Chazelle, Computer Science Patrick Cheridito, Operations Research and Financial Engineering Mung Chiang, Electrical Engineering Erhan Çınlar, Operations Research and Financial Engineering Iain D. Couzin, Ecology and Environmental Biology David P. Dobkin, Computer Science Jianqing Fan, Operations Research and Financial Engineering Jason W. Fleischer, Electrical Engineering Christodoulos A. Floudas, Chemical and Biological Engineering Mikko P. Haataja, Mechanical and Aerospace Engineering
Gregory W. Hammett, Plasma Physics Lab, Astrophysical Sciences 
Associated Faculty (continued)
Isaac M. Held, Geosciences, Atmospheric and Oceanic Sciences Sergiu Klainerman, Mathematics Naomi Ehrich Leonard, Mechanical and Aerospace Engineering Simon A. Levin, Ecology and Evolutionary Biology Elliott H. Lieb, Mathematics, Physics Luigi Martinelli, Mechanical and Aerospace Engineering William A. Massey, Operations Research and Financial Engineering H. Vincent Poor, Electrical Engineering Warren B. Powell, Operations Research and Financial Engineering Frans Pretorius, Physics JeanHervé Prévost, Civil and Environmental Engineering Herschel A. Rabitz, Chemistry Peter J. Ramadge, Electrical Engineering Jennifer L. Rexford, Computer Science Clarence W. Rowley, Mechanical and Aerospace Engineering Szymon Rusinkiewicz, Computer Science
Robert E. Schapire, Computer Science José A. Scheinkman, Economics Frederik J. Simons, Geosciences Yakov G. Sinai, Mathematics Jaswinder P. Singh, Computer Science K. Ronnie Sircar, Operations Research and Financial Engineering Howard Stone, Mechanical and Aerospace Engineering John D. Storey, Molecular Biology, LewisSigler Institute for Integrative Genomics Sankaran Sundaresan, Chemical and Biological Engineering Salvatore Torquato, Chemistry Olga G. Troyanskaya, Computer Science, LewisSigler Institute for Integrative Genomics Geoffrey K. Vallis, Geosciences, Atmospheric and Oceanic Sciences Robert J. Vanderbei, Operations Research and Financial Engineering Ramon van Handel, Operations Research and Financial Engineering

Requirements
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.
It will be the student’s responsibility to choose three areas in which to be examined out of a list of six possibilities to be specified below. The student should chose their specific topics by the end of October. The director of graduate studies, in consultation with the student, will then appoint a set of advisers from among the faculty and associated faculty. The adviser in each topic will meet regularly with the student, monitor progress and assign additional reading material. Advisers are traditionally members of the Princeton University faculty within the department, but members of the faculty from other departments may serve as advisers with approval.They can be any member of the University faculty, but normally would be either program or associated faculty. The firstyear student should choose three topics from among the following six applied mathematics categories:
Asymptotics, analysis, numerical analysis and signal processing;Discrete mathematics, combinatorics, algorithms, computational geometry and graphics;
Mechanics and field theories (including computational physics/chemistry/biology);
Optimization (including linear and nonlinear programming and control theory);
Partial differential equations and ordinary differential equations (including dynamical systems); and
Stochastic modeling, probability, statistics and information theory.
Mechanics and field theories (including computational physics/chemistry/biology);
Optimization (including linear and nonlinear programming and control theory);
Partial differential equations and ordinary differential equations (including dynamical systems); and
Stochastic modeling, probability, statistics and information theory.
Additional topics may be considered with prior approval bythe director of graduate studies. Typically, students take regular or reading courses with their advisers in each of the three areas, completing the regular exams and course work for these courses.
Preliminary Exam
At the end of the first year, firstyear students will also take a preliminary exam, consisting of a joint interview by their three firstyear advisers. Each student should discuss with their firstyear advisers which of these courses are relevant for their areas.
In order to assess whether they have sufficient preparation, or whether it would be good to take a particular course, it is a good idea to obtain some typical homework or a final exam from a previous year. If the student fails the preliminary examination or a part thereof the first time, they make take it a second time.
General Examination
Before being admitted to a third year of study, students must pass 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.
Before being admitted to a third year of study, students must pass 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 who completes all departmental requirements (coursework, preliminary exams, with no incompletes) but fails the general examination may take it a second time. If the student fails the general examination a second time, then Ph.D. candidacy is automatically terminated.
Master of Arts
The Master of Arts degree is normally an incidental degree on the way to full Ph.D. candidacy, but may also be awarded to students who for various reasons leave the Ph.D. program. Students who have satisfactorily passed required coursework including the resolution of any incompletes and have passed the preliminary exam, may be awarded an M.A. degree. Students must complete the required “Advanced Degree Application form” upon learning the Department’s determination of their candidacy in order to receive the M.A.
The Master of Arts degree is normally an incidental degree on the way to full Ph.D. candidacy, but may also be awarded to students who for various reasons leave the Ph.D. program. Students who have satisfactorily passed required coursework including the resolution of any incompletes and have passed the preliminary exam, may be awarded an M.A. degree. Students must complete the required “Advanced Degree Application form” upon learning the Department’s determination of their candidacy in order to receive the M.A.
Doctoral Dissertation
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.
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.
Satisfactory completion of the requirements leads to the degree of Doctor of Philosophy in applied and computational mathematics.
Courses
APC 501/MAE 501 Mathematical Methods of Engineering Analysis I
Edgar Y. Choueiri
Edgar Y. Choueiri
Methods of mathematical analysis for the solution of problems in physics and engineering. Topics include an introduction to functional analysis, SturmLiouville theory, Green's functions for the solution of ordinary differential equations and Poisson's equation, and the calculus of variations.
APC 502/CBE 502 Mathematical Methods of Engineering Analysis II
Sankaran Sundaresan
APC 502/CBE 502 Mathematical Methods of Engineering Analysis II
Sankaran Sundaresan
Linear ordinary differential equations (systems of firstorder equations, method of Frobenius, twopoint boundaryvalue problems); spectrum and Green's function; matched asymptotic expansions; partial differential equations (classification, characteristics, uniqueness, separation of variables, transform methods, similarity); and Green's function for the Poisson, heat, and wave equations, with applications to selected problems in chemical, civil, and mechanical engineering.
APC 503/AST 557 Analytical Techniques in Differential Equations
Roscoe B. White
APC 503/AST 557 Analytical Techniques in Differential Equations
Roscoe B. White
Local analysis of solutions to linear and nonlinear differential and difference equations. Asymptotic methods, asymptotic analysis of integrals, perturbation theory, summation methods, boundary layer theory, WKB theory, and multiple scale theory. Prerequisite: MAE 306 or equivalent.
APC 506/MAE 502 Mathematical Methods of Engineering Analysis II
Philip J. Holmes
APC 506/MAE 502 Mathematical Methods of Engineering Analysis II
Philip J. Holmes
A complementary presentation of theory, analytical methods, and numerical methods. The objective is to impart a set of capabilities commonly used in the research areas represented in the Department. Standard computational packages will be made available in the courses, and assignments will be designed to use them. An extension of MAE 501.
APC 507/MAE 503 Basic Numerical Methods for Ordinary and Partial Differential Equations
Weinan E
APC 507/MAE 503 Basic Numerical Methods for Ordinary and Partial Differential Equations
Weinan E
Difference schemes for ordinary differential equations; analysis of simple difference schemes for model hyperbolic and parabolic problems; the linear advection condition; explicit and implicit schemes; difference and interpolation formulas on equal and unequal meshes with error estimates; Lagrange interpolation: Peano error estimates; least squares approximation: orthogonal polynomials' piecewise polynomial interpolation: splines; trigonometric interpolation and error estimate for spectral approximation; Chebyshev expansions; numerical quadrature; iterative solution of nonlinear equations; and inversion of sparse sets of equations.
APC 509 Methods and Concepts in Electronic Structure Theory
Emily A. Carter
APC 509 Methods and Concepts in Electronic Structure Theory
Emily A. Carter
This course derives how and why chemical bonds between atoms form, leading to the creation of molecules and condensed matter. Stateoftheart electronic structure theory methods are discussed and compared in terms of strengths, weaknesses, and numerical implementations. Students will learn how to predict molecular structure, qualitative character of the electron distribution (hybridization), and relative chemical bond strengths directly from a simple multielectron wavefunction. Condensed matter electronic structure will be introduced via band theory, followed by an analysis of the pros and cons of modern density functional theory methods.
APC 511/EEB 511 Seminar in Mathematical Biology
Georgi S. Medvedev
Applications of reactiondiffusion equations in mathematical biology. Specific examples of FisherKolmogorovPetrovskiPiskunov equation and FitzHughHagumo system of equations. Discussion of modelling principles, as well as methods of mathematical analysis, the latter to include phase plane analysis for ordinary differential equations, maximum principle for parabolic PDE's, mthods for singularly perturbed systems, and matched asymnptotic expansions.
APC 513/ORF 517 Empirical Processes & Bootstrap Method
Dragan Radulovic
APC 513/ORF 517 Empirical Processes & Bootstrap Method
Dragan Radulovic
No Description Available
APC 514/MOL 514/EEB 514 Biological Dynamics
Curtis G. Callan
APC 514/MOL 514/EEB 514 Biological Dynamics
Curtis G. Callan
Introduction to the mathematical desciption of quantitative phenomena in living systems; Hodgkin Huxley equations of nerve membranes; the generation of spatial patterns in development, single cells, and colonies of cells; chemotaxis; the population dynamics of disease; dynamics activity of networks of neurons; intracellular chemical and genenetworks. Emphasis on formulation and experimental basis for the equations, and their relationship to significant biological issues.
APC 515/MSE 515 Random Heterogeneous Materials
Salvatore Torquato
APC 515/MSE 515 Random Heterogeneous Materials
Salvatore Torquato
Composites, porous media, foams, colloidal suspensions, geological media, polymer blends, and biological media are all examples of heterogeneous materials. Often the microstructure of such materials is random. The relationship between the macroscopic (transport, mechanical, electromagnetic, and chemical) properties and microstructure of random heterogeneous materials is formulated. Topics include statistical characterization of the microstructure via npoint distribution functions; percolation theory; fractal concepts; sphere packings; Monte Carlo simulation techniques; and image analysis of microstructures; homogenization theory; effective
APC 520/MAT 540 Mathematical Analysis of Massive Data Sets
Amit Singer
Amit Singer
Analysis of high dimensional data sets using spectral methods, how to perform tasks such as clustering and classification, regression and outofsample extenion of empirical functions, semisupervised learning, and independent component analysis (ICA). Applications to image and signal processing, structural biology, dynamical systems, text and document analysis, search and data mining, and finance, among others, will be covered.
APC 523/AST 523 Scientific Computation in Astrophysics
James M. Stone, Robert H. Lupton
APC 523/AST 523 Scientific Computation in Astrophysics
James M. Stone, Robert H. Lupton
A broad introduction to scientific computation using examples drawn from astrophysics. From computer science, practical topics including processor architecture, parallel systems, structured programming, and scientific visualization will be presented in tutorial style. Basic principles of numerical analysis, including sources of error, stability, and convergence of algorithms. The theory and implementation of techniques for linear and nonlinear systems of equations, ordinary and partial differential equations will be demonstrated with problems in stellar structure and evolution, stellar and galactic dynamics, and cosmology.
APC 524/MAE 506 Software Engineering for Scientific Computing
Clarence Rowley III, James Stone, Robert Lupton
APC 524/MAE 506 Software Engineering for Scientific Computing
Clarence Rowley III, James Stone, Robert Lupton
Course discusses elements of software engineering, geared towards beginning graduate students interested in scientific computing.
APC 539/AST 559 Nonlinear Processes in Fluids and Plasmas
Gregory W. Hammett
APC 539/AST 559 Nonlinear Processes in Fluids and Plasmas
Gregory W. Hammett
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 NavierStokes, Vlasov, and gyrokinetic; numerical simulation techniques, including pseudospectral and particleincell methods; coherent structures; transition to turbulence; statistical closures, including the wave kinetic equation and directinteraction approximation; PDF methods and intermittency; variational techiques. Applications from neutral fluids, fusion plasmas, and astrophysics.
APC 544/CBE 554 Topics in Computational Nonlinear Dynamics
Yannis G. Kevrekidis
The numerical solution of partial differential equations (finite element and spectral methods); computational linear algebra; direct and interactive solutions and continuation methods; and stability of the steady states and eigen problems. Timedependent solutions for large systems of ODEs; computation and stability analysis of limit cycles; Lyapunov exponents of chaotic solutions are explored. Vectorization and FORTRAN code optimization for supercomputers as well as elements of symbolic computation are studied.
APC 550 Introduction to Differential Equations
Weinan E
APC 550 Introduction to Differential Equations
Weinan E
Both applications and fundamental theory will be discussed. Topics include wave, heat and Poisson equations, separation of variables, solution by Fourier series and Fourier integrals, Green's function, variational methods, nonlinear first order equations and methods of characteristics, applications to control, finance and fluid flow. Background material in ODEs will be covered.
APC 551/ORF 551 Random Measures and Levy Processes
Erhan Çinlar
APC 551/ORF 551 Random Measures and Levy Processes
Erhan Çinlar
Introduction to probability theory, beginning with a review of measure and integration; various concepts of convergence, laws of large numbers, and central limits; martingale theory, filtrations, and stopping times; and Brownian motion, Lévy processes, and Poisson random measures.
APC 571/MAE 541 Applied Dynamical Systems
Philip J. Holmes
Phaseplane methods and singledegreeoffreedom nonlinear oscillators; invariant manifolds, local and global analysis, structural stability and bifurcation, center manifolds, and normal forms; averaging and perturbation methods, forced oscillations, homoclinic orbits, and chaos; and Melnikov's method, the Smale horseshoe, symbolic dynamics, and strange attractors. Offered in alternate years.
APC 576/AOS 576 Current Topics in Dynamic Meterology
Isaac M. Held
APC 576/AOS 576 Current Topics in Dynamic Meterology
Isaac M. Held
An introduction to topics of current interest in the dynamics of largescale atmospheric flow. Possible topics include wavemean flow interaction and nonacceleration theorems, critical levels, quasigeostrophic instabilities, topographically and thermally forced stationary waves, theories for stratospheric sudden warmings and the quasibiennial oscillation of the equatorial stratosphere, and quasigeostrophic turbulence.
APC 583/MAT 593 Wavelets
Ingrid C. Daubechies
APC 583/MAT 593 Wavelets
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. This is a graduate course open to undergraduate students who have taken Math 314, 315. If in doubt about having the necessary background, the student should consult the instructor before signing up.
APC 584/MAT 594 Wavelets
Radu V. Balan
Course covers topics of wavelet and timefrequency 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. Twothirds of the time will be spent on theory, with remaining onethird to be devoted to applications.
APC 585/MAT 585 Topics in Discrete Mathematics
Paul D. Seymour
APC 585/MAT 585 Topics in Discrete Mathematics
Paul D. Seymour
Various branches of discrete mathematics and combinatorial theory. Material treated is relevant to discrete optimization and algorithmic complexity but focuses most strongly on combinatorial theory. Students should have already taken a course in elementary graph theory.
APC 595/MAT 595 Topics in Discrete Mathematics
Paul D. Seymour
APC 595/MAT 595 Topics in Discrete Mathematics
Paul D. Seymour
Various branches of discrete mathematics and combinatorial theory. Material treated is relevant to dsicrete optimization and algorithmic complexity but focuses most strongly on combinatorial theory. Student should have already taken a course in elementary graph theory.
APC 599 Summer Extramural Research Project
Staff
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; end date no later than Labor Day. A final paper and sponsor evaluation is required.Staff