Program in Applied and Computational Mathematics
Faculty
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Director Director of Graduate Studies Executive Committee Associated Faculty |
Associated Faculty (continued) |
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. This choice of topics should be achieved 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. (They can be any member of the University faculty, but normally would be either program or associated faculty.) The first-year 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.
Other topics as special exceptions might be possible, provided they are approved in advance by the 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. At the end of the first year, first-year students will also take a preliminary exam, consisting of a joint interview by their three first-year advisers. Each student should discuss with their first-year advisers which of these courses are relevant for their areas.
General Examination
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.
Master of Arts
A student qualifies for the award of the Master of Arts (M.A.) by successfully completing all course work and passing the first portion of the general examination.
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.
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
Applied and Computational Mathematics
APC 501/MAE 501 Mathematical Methods of Engineering Analysis I
Egemen Kolemen
Methods of mathematical analysis for the solution of problems in physics and engineering. Topics include an introduction to functional analysis, Sturm-Liouville 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
Morton D. Kostin
Linear ordinary differential equations (systems of first-order equations, method of Frobenius, two-point boundary-value 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
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
Edgar Y. Choueiri
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
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
This course derives how and why chemical bonds between atoms form, leading to the creation of molecules and condensed matter. State-of-the-art 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 multi-electron 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 reaction-diffusion equations in mathematical biology. Specific examples of Fisher-Kolmogorov-Petrovski-Piskunov equation and FitzHugh-Hagumo 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
No Description Available
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 gene-networks. 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
No Description Available
APC 520/MAT 540 Mathematical Analysis of Massive Data Sets
Amit Singer
Analysis of high dimensional data sets using spectral methods, how to perform tasks such as cluserting and classification, regression and out-of-sample extenion of empirical functions, semi-supervised 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
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
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
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; 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. Time-dependent 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
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 Probability Theory
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
Clarence W. Rowley
Phase-plane methods and single-degree-of-freedom 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
An introduction to topics of current interest in the dynamics of large-scale atmospheric flow. Possible topics include wave-mean flow interaction and nonacceleration theorems, critical levels, quasigeostrophic instabilities, topographically and thermally forced stationary waves, theories for stratospheric sudden warmings and the quasi-biennial oscillation of the equatorial stratosphere, and quasi-geostrophic turbulence.
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 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 will be spent on theory, with remaining one-third to be devoted to applications.
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
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.