Program in Applied and Computational Mathematics
Director
A. Robert Calderbank
Undergraduate Representative
Weinan E
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
Amit Singer, Mathematics
James M. Stone, Astrophysical Sciences
Jeroen Tromp, Geosciences
Sergio Verdú, Electrical Engineering
Associated Faculty
Yacine Aït-Sahalia, Economics
Michael Aizenman, Physics, Mathematics
William Bialek, Physics, Lewis-Sigler Institute for Integrative Genomics
David M. Blei, Computer Science
Carlos D. Brody, Molecular Biology, Princeton Neuroscience Institute
Roberto Car, Chemistry
Moses S. Charikar, Computer Science
Bernard Chazelle, Computer Science
Patrick Cheridito, Operations Research and Financial Engineering
Mung Chiang, Electrical Engineering
Erhan Çinlar, Operations Research and Financial Engineering
Iain D. Couzin, Ecology and Environmental Biology
Bradley W. Dickinson, Electrical Engineering
David P. Dobkin, Computer Science
Jianqing Fan, Operations Research and Financial Engineering
Jason W. Fleischer, Electrical Engineering
Christodoulos A. Floudas, Chemical Engineering
Mikko P. Haataja, Mechanical and Aerospace Engineering
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 and Physics
Maria P. Martin-Aguirre, 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
Frans Pretorius, Physics
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 A. 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
John D. Storey, Molecular Biology, Lewis-Sigler Institute for Integrative Genomics
Sankaran Sundaresan, Chemical Engineering
Salvatore Torquato, Chemistry
Olga G. Troyanskaya, Computer Science, Lewis-Sigler Institute for Integrative Genomics
Geoffrey K. Vallis, Geosciences, Atmospheric and Oceanic Sciences
Robert J. Vanderbei, Operations Research and Financial Engineering
Ron Weiss, Electrical Engineering
Applied Mathematics at Princeton
There has never been a better time to be a mathematician. The combination of mathematics and computer modeling has transformed science and engineering and is changing the nature of research in the biological sciences. The requirements for the mathematics major are a minimum of eight upperclass courses in mathematics or applied mathematics including three basic courses on real analysis, complex analysis, and algebra. It is possible to design a course of undergraduate study aimed more strongly toward applications. Applied and computational mathematics/mathematics faculty have developed core courses in applied mathematics and several courses where the emphasis is mathematical modeling. The latter is central to applied mathematics where it is not only necessary to acquire mathematical techniques and skills, but also important to learn about the application domain.
Courses
APC 199 Math Alive (also MAT 199) — QR
An exploration of some of the mathematical ideas behind important modern applications, from banking and computing to listening to music. Intended for students who have not had college-level mathematics and are not planning to major in a mathematically based field. The course is organized in independent two-week modules focusing on particular applications, such as bar codes, CD players, population models, and space flight. The emphasis is on ideas and mathematical reasoning, not on sophisticated mathematical techniques. Two 90-minute classes, one computer laboratory. I. Daubechies
APC 307 Combinatorial Mathematics (see MAT 307)
APC 350 Introduction to Differential Equations (also CEE 350, MAT 350) — QR
An introduction to differential equations, covering both 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; and reaction-diffusion equations. Application of these equations and methods to finance and control. Prerequisites: MAT 102, 103, and 202. Two 90-minute lectures. W. E
APC 351 Topics in Mathematical Modeling (see MAT 351)
The Undergraduate Certificate
The certificate is designed for students from engineering and from the physical, biological, and social sciences who are looking to broaden their mathematical and computational skills. It is also an opportunity for mathematically oriented students to discover the challenges presented by applications from the natural sciences and engineering. Students interested in the undergraduate certificate contact the program’s undergraduate representative in the spring semester of their sophomore year to discuss their interests, and to lay out a plan for their course selection and research component.
Course of Study
The requirements for the undergraduate certificate in applied and computational mathematics (ACM) consist of:
1. A total of five courses normally 300 level or higher (requires letter grade; pass/D/fail not accepted), at least two of which are not included in the usual requirements for the candidates’ major concentration; and
2. Independent work consisting of a paper/course project/computational laboratory, possibly in the context of a course offered by ACM faculty or a senior thesis in the major department with a significant applied mathematics component (subject to approval of the ACM undergraduate representative). This independent work may not be used to satisfy the requirements of any other certificate.
Regardless of which option is selected in (2), students will also be required to participate during their junior and senior year in a not-for-credit colloquium offered by ACM. This will provide a forum for presentation and discussion of independent work among all certificate students and will introduce them to other areas of applied mathematics.
The five required courses may vary widely from department to department in order to include a broad spectrum of science and engineering students throughout the University. These courses should fit readily within the degree requirements of the respective departments of the engineering school or the economics, mathematics, physics, chemistry, molecular biology, and ecology and evolutionary biology, or other relevant departments, but will require a particular emphasis in applied mathematics.
The five required courses must be distributed between the following two areas, with at least two from each area:
1. Mathematical foundations and techniques, including differential equations, real and complex analysis, discrete mathematics, probability, and statistics, typically offered by the Department of Mathematics.
2. Mathematical applications, including signal processing, control theory, optimization, mathematical economics, typically offered by the economics, science, and engineering departments.
Specific choices must be approved by the ACM undergraduate representative.
The paper/course project/computational laboratories can be done as part of a course offered by applied and computational mathematics faculty or associated faculty on a wide range of topics of current interest in applied mathematics. Such courses vary from year to year and are designated to satisfy automatically the independent work requirement. Four courses developed and staffed by applied and computational mathematics faculty and offered regularly are the following:
CHE 448/MAT 448 Introduction to Nonlinear Dynamics
MAE 541/APC 571 Applied Dynamical Systems
MAT 594/APC 584 Wavelets: Applications of Wavelets in Mathematics and Other Fields
MAT 595/APC 586 Topics in Discrete Mathematics: Discrete Math
Any other course that students might use to satisfy the independent work requirement must have prior approval from the applied and computational mathematics undergraduate representative. Students may satisfy the independent work requirement outside of a course after consultation with and approval by the undergraduate representative. If the senior thesis option is selected, attempts will be made to coordinate it with departmental requirements.
Relevant Advanced Courses
Below is a list of representative advanced undergraduate and some graduate courses that meet the certificate requirements. This list is primarily illustrative and is by no means complete. Specific programs should be tailored by the program undergraduate representative in consultation with the student to meet individual and/or departmental needs.
Sample mathematical foundations courses
APC 350/CEE 350 Introduction to Differential Equations
APC 503/AST 557 Analytical Techniques in Differential Equations
MAE 305/MAT 301 Mathematics in Engineering I
MAE 306/MAT 302 Mathematics in Engineering II
MAE 501/APC 501 Mathematical Methods of Engineering Analysis I
MAE 502/APC 502 Mathematical Methods of Engineering Analysis II
MAE 503/APC 507 Basic Numerical Methods for Ordinary and Partial Differential Equations
MAT 303 Ordinary Differential Equations
MAT 304 Introduction to Partial Differential Equations
MAT 305 Mathematical Programming
MAT 306/COS 342 Introduction to Graph Theory
MAT 308/ECO 318 Theory of Games
MAT 312 Mathematical Logic
MAT 314 Introduction to Real Analysis
MAT 315 Real Analysis
MAT 317 Complex Analysis with Applications
MAT 323 Algebra
MAT 324 Topics in Algebra
MAT 325 Topology
MAT 326 Algebraic Topology
MAT 327 Introduction to Differential Geometry
MAT 328 Differential Geometry
MAT 330 Analysis I: Fourier Series and Partial Differential Equations
MAT 331 Analysis II: Complex Analysis
MAT 332 Analysis III: Integration Theory and Hilbert Space
MAT 333 Analysis IV: Special Topics in Analysis
MAT 390 Probability Theory
MAT 391 Random Processes
MAT 451, 452 Advanced Topics in Analysis
ORF 309/MAT 309 Probability and Stochastic Systems
PHY 403/MAT 407 Mathematical Methods of Physics
Sample mathematical applications courses
AOS 573 Physical Oceanography
APC 514/MOL 514/EEB 514 Biological Dynamics
CHE 448/MAT 448 Introduction to Nonlinear Dynamics
CHM 305 The Quantum World
CEE 361/MAE 325 Structural Analysis and Introduction to Finite Element Methods
CEE 532 Advanced Finite Element Methods
COS 423 Theory of Algorithms
COS 451 Computational Geometry
COS 487 Theory of Computation
ECO 312 Econometrics: A Mathematical Approach
ECO 317 The Economics of Uncertainty
ECO 414 Introduction to Economic Dynamics
ECO 481 Topics in Economics
ECO 513 Advanced Econometrics: Time Series Models
EEB 324 Theoretical Ecology
ELE 382 Distributed Algorithms and Optimization Methods for Engineering Applications
ELE 482 Digital Signal Processing
ELE 485 Signal Analysis and Communication Systems
ELE 488 Image Processing
ELE 521/MAE 547 Linear Systems Theory
ELE 523/MAE 548 Nonlinear System Theory
ELE 528 Information Theory
ELE 530 Theory of Detection and Estimation
GEO 424/CEE 424 Introductory Seismology and Oil Exploration
GEO 425/MAE 425 Introduction to Physical Oceanography
GEO 557 Theoretical Geophysics
MAE 335 Fluid Dynamics
MAE 336 Viscous Flows
MAE 433 Automatic Control Systems
MAE 434 Modern Control
MAE 541/APC 571 Applied Dynamical Systems
MAE 542 Advanced Dynamics
MAE 545 Nonlinear Control
MAE 546 Optimal Control and Estimation
MSE 515/APC 515 Random Heterogenous Materials
ORF 335/ECO 364 Introduction to Financial Engineering
ORF 405 Regression and Applied Time Series
PHY 304 Advanced Electromagnetism
PHY 305 Introduction to the Quantum Theory
PHY 408 Modern Classical Dynamics
Certificate of Proficiency
Students who fulfill all the requirements will receive a certificate upon graduation.