## Program in Applied and Computational Mathematics

#### Director

Peter Constantin

#### Executive Committee

Emmanuel A. Abbe, also Electrical Engineering

René A. Carmona, Operations Research and Financial Engineering

Emily A. Carter, also Mechanical and Aerospace Engineering

Maria Chudnovsky, also Mathematics

Peter Constantin, also Mathematics

Weinan E, also Mathematics

Yannis G. Kevrekidis, Chemical and Biological Engineering

Adam Marcus, also Mathematics

Paul D. Seymour, also Mathematics

Amit Singer, also Mathematics

Howard A. Stone, also Mechanical and Aerospace Engineering

James M. Stone, also Astrophysical Sciences

Jeroen Tromp, also Geosciences

Sergio Verdú, Electrical Engineering

#### Associated Faculty

Yacine Aït-Sahalia, Economics

Michael Aizenman, Physics, Mathematics

William Bialek, Physics, Lewis-Sigler Institute for Integrative Genomics

Mark Braverman, Computer Science

Carlos D. Brody, Molecular Biology, Princeton Neuroscience Institute

Adam S. Burrows, Astrophysical Sciences

Roberto Car, Chemistry

Bernard Chazelle, Computer Science

Patrick Cheridito, Operations Research and Financial Engineering

Mung Chiang, Electrical Engineering

David P. Dobkin, Computer Science

Jianqing Fan, Operations Research and Financial Engineering

Jason W. Fleischer, Electrical Engineering

Mikko P. Haataja, Mechanical and Aerospace Engineering

Gregory W. Hammett, Plasma Physics Lab, Astrophysical Sciences

Isaac M. Held, Geosciences, Atmospheric and Oceanic Sciences

Sergiu Klainerman, Mathematics

Naomi E. 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

Assaf Naor, Mathematics

H. Vincent Poor, Electrical Engineering

Warren B. Powell, Operations Research and Financial Engineering

Frans Pretorius, Physics

Jean-Hervé 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 M. Rusinkiewicz, Computer Science

Frederik J. Simons, Geosciences

Yakov G. Sinai, Mathematics

Jaswinder P. Singh, Computer Science

K. Ronnie Sircar, Operations Research and Financial Engineering

Howard A. Stone, Mechanical and Aerospace Engineering

John D. Storey, Lewis-Sigler Institute for Integrative Genomics

Sankaran Sundaresan, Chemical and Biological Engineering

Corina E. Tarnita, Ecology and Evolutionary Biology

Salvatore Torquato, Chemistry

Olga G. Troyanskaya, Computer Science, Lewis-Sigler Institute for Integrative Genomics

Ramon van Handel, Operations Research and Financial Engineering

Robert J. Vanderbei, Operations Research and Financial 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.

**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.

#### Program of Study

The requirements for the undergraduate certificate in applied and computational mathematics 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 candidate's major concentration; and

2. Independent work consisting of a paper in one of the following formats: (a) a course project/computational laboratory (possibly in the context of a course offered by Program in Applied and Computational Mathematics [PACM] faculty); (b) a project that you are working on with a professor; or (c) a summer research project that you are planning on undertaking. However, you may not use your junior paper or senior thesis to satisfy the independent work for the certificate program. Your paper should have a significant applied mathematics component (subject to approval of the PACM undergraduate representative). The independent work may not be used to satisfy the requirements of any other certificate. Students interested in the PACM certificate program must apply on or before December 31 of their junior year.

Regardless of which option is selected in (2), students will also be required to participate during their junior and senior years in a not-for-credit colloquium offered by PACM. 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, and optimization, mathematical economics, typically offered by the economics, science, and engineering departments.

Specific choices must be approved by the PACM 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. These courses should be taken in your junior year if you intend to use them as a paper for your independent work. Four courses developed and staffed by applied and computational mathematics faculty and offered regularly are the following:

CBE 448/MAT 481 Introduction to Nonlinear Dynamics

MAT323/APC 323 Topics in Mathematical Modeling

MAE 541/APC 571 Applied Dynamical Systems

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.

#### Certificate of Proficiency

Students who fulfill all requirements of the program will receive a certificate of proficiency in applied and computational mathematics upon graduation.

**Relevant Advanced Courses. **A list of representative advanced undergraduate and some graduate courses that meet the certificate requirements can be found on the program website. 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.

### Courses

APC 150 Introduction to Statistics QR

This course is an introduction to probability and statistical methods, and covers topics in probability, random variables, sampling, descriptive statistics, probability distributions, estimation and hypotheses testing, introduction to the regression model. The course emphasizes the practice, and students will learn how to perform data analysis using modern computational tools.
*
L. Martinelli*

APC 151 Problem Solving in Mathematics (also MAT 151) QR

This course is an introduction to mathematical modeling in physical and social sciences. Topics covered include modeling via simple first and second order differential equations, fitting experimental data, optimization and an introduction to modeling probabilistic events. One substantial goal of the course is to learn MATLAB through homework, weekly group projects and an individual final project. Equal emphasis will be put on practical implementations of the models through MATLAB scripts and on theoretical underpinnings of the models.
*
P. Holmes*

APC 192 An Integrated Introduction to Engineering, Mathematics, Physics (see EGR 192)

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.
* Staff*

APC 323 Topics in Mathematical Modeling (see MAT 323)

APC 350 Introduction to Differential Equations (also CEE 350/MAT 322) 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; reaction-diffusion equations. Application of these equations and methods to finance and control. Prerequisites: MAT 102, 103, and 202. Two 90-minute lectures.
*
P. Holmes*

APC 377 Combinatorial Mathematics (see MAT 377)

APC 441 Computational Geophysics (see GEO 441)

APC 486 Transmission and Compression of Information (see ELE 486)