# Applied and Computational Mathematics

## Program Offerings

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, data science and many other areas. Students seeking to pursue an academic program with a strong focus on applied mathematics may major in mathematics with a course of study geared toward applications, or may major in the sciences or engineering and enroll in the certificate Program in Applied and Computational Mathematics (PACM).

Requirements for a major in the Department of Mathematics are a minimum of eight upper-level courses in mathematics or applied mathematics, including four basic courses on real analysis, complex analysis, algebra and geometry or topology. It is possible to design a course of undergraduate study aimed more specifically toward applications. Applied and computational mathematics faculty have developed core courses in applied mathematics where the emphasis is on computational methods and mathematical modeling. The latter is central to applied mathematics where it is not only necessary to acquire mathematical techniques and skills, but is also important to learn about the application domain.

The PACM certificate is designed for students from engineering and from the physical, biological and social sciences who wish 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 sciences and engineering. Students interested in the undergraduate certificate must contact the program's undergraduate representative on or before February 1 of their junior year to discuss their interests, and to lay out a plan for their course selection and research component.

### Goals for Student Learning

Mathematics is the language of science, and almost every discipline of science and engineering is ultimately founded on mathematics. At the same time, state-of-the-art computational methods are enabling the study of increasingly complex systems. The aim of the Program in Applied and Computational Mathematics certificate is to enable students from a broad range of disciplines to develop a stronger mathematical and computational foundation, and to promote dialogue between mathematics and its applications.

At the same time, the certificate aims to encourage students whose primary concentration is in pure mathematics to explore the impact of mathematics on science, engineering and technology.

A core tenet of the PACM certificate is the bilateral interaction between mathematical foundations and applications. A strong background in mathematics and computational methods provides students with powerful tools to address problems that arise in other disciplines, and may lead them to view these from a new perspective. Conversely, challenging mathematical and computational questions that arise in applications strongly motivate the study and development of mathematics. The PACM certificate promotes such interactions:

- By requiring all students to take both mathematical foundations and applications courses;
- By requiring all students to perform an independent research project that connects mathematics with its applications;
- By requiring all students to attend and present their independent work in the certificate seminar, where they are exposed to a broad range of applications of mathematics to different disciplines and have the opportunity to interact with students with applied mathematical interests from many different majors across campus.

At the same time, the flexible program is designed to allow each student to tailor their courses and independent work to their own interests, and to make mathematics as accessible as possible for students with diverse interests and backgrounds.

### Program of Study

The requirements for the certificate in applied and computational mathematics consist of:

- 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.
- Independent work consisting of a paper in one of the following formats: (a) a project that you are working on with a professor; or (b) a summer research project. However, the independent work may not be used to satisfy any requirements of your major or of any other minor or certificate. In particular, you may not use your junior paper or senior thesis to satisfy the independent work requirement for the certificate. A significant extension of the senior thesis or of a course project may however be used to satisfy the requirement, subject to approval of the PACM undergraduate representative.
- Students are required to participate during the spring semester of 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 a broad range of areas within applied mathematics.

The course requirement may be satisfied by a broad range of courses that place a particular emphasis on applied mathematics, which are offered by the mathematics department as well as the science, engineering and economics departments. The five required courses must be distributed between the following two areas, with at least two from each area:

- Mathematical foundations and techniques, including differential equations, real and complex analysis, discrete mathematics, probability, numerical methods, etc.
- Mathematical applications in diverse areas offered by the applied and computational mathematics program and by science, engineering and economics departments.

An extensive list of advanced undergraduate and some graduate courses that meet the certificate requirements can be found on the program website.

Courses that do not appear on this list may be approved by the PACM undergraduate representative. Specific programs should be tailored in consultation with the PACM undergraduate representative to meet the individual needs and interests of each student.

The independent work requirement is typically done under the supervision of a PACM core or affiliated faculty member, but external advisers are regularly accommodated. In the latter case, a second reader from PACM is asked to verify that the paper contains enough applied mathematics to satisfy the certificate requirements. In any case, plans for independent work must be approved by the undergraduate representative.

### Certificate of Proficiency

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

## Faculty

### Director

- Peter Constantin

### Director of Graduate Studies

- Maria Chudnovsky

### Director of Undergraduate Program

- Ramon van Handel

### Executive Committee

- Noga M. Alon, Mathematics
- René A. Carmona, Oper Res and Financial Eng
- Emily Ann Carter, Mechanical & Aerospace Eng
- Maria Chudnovsky, Mathematics
- Peter Constantin, Mathematics
- Paul Seymour, Mathematics
- Amit Singer, Mathematics
- Howard A. Stone, Mechanical & Aerospace Eng
- Romain Teyssier, Astrophysical Sciences
- Jeroen Tromp, Geosciences
- Ramon van Handel, Oper Res and Financial Eng

### Associated Faculty

- Ryan P. Adams, Computer Science
- Amir Ali Ahmadi, Oper Res and Financial Eng
- Michael Aizenman, Physics
- Yacine Aït-Sahalia, Economics
- William Bialek, Physics
- Mark Braverman, Computer Science
- Carlos D. Brody, Princeton Neuroscience Inst
- Adam S. Burrows, Astrophysical Sciences
- Roberto Car, Chemistry
- Bernard Chazelle, Computer Science
- Jianqing Fan, Oper Res and Financial Eng
- Jason W. Fleischer, Electrical & Comp Engineering
- Mikko P. Haataja, Mechanical & Aerospace Eng
- Gregory W. Hammett, PPPL Theory
- Isaac M. Held, Atmospheric & Oceanic Sciences
- Sergiu Klainerman, Mathematics
- Naomi E. Leonard, Mechanical & Aerospace Eng
- Simon A. Levin, Ecology & Evolutionary Biology
- Luigi Martinelli, Mechanical & Aerospace Eng
- William A. Massey, Oper Res and Financial Eng
- Assaf Naor, Mathematics
- H. Vincent Poor, Electrical & Comp Engineering
- Frans Pretorius, Physics
- Herschel A. Rabitz, Chemistry
- Peter J. Ramadge, Electrical & Comp Engineering
- Jennifer Rexford, Computer Science
- Clarence W. Rowley, Mechanical & Aerospace Eng
- Szymon M. Rusinkiewicz, Computer Science
- Mykhaylo Shkolnikov, Oper Res and Financial Eng
- Frederik J. Simons, Geosciences
- Jaswinder P. Singh, Computer Science
- Ronnie Sircar, Oper Res and Financial Eng
- Mete Soner, Oper Res and Financial Eng
- John D. Storey, Integrative Genomics
- Sankaran Sundaresan, Chemical and Biological Eng
- Robert E. Tarjan, Computer Science
- Corina E. Tarnita, Ecology & Evolutionary Biology
- Salvatore Torquato, Chemistry
- Olga G. Troyanskaya, Computer Science
- Robert J. Vanderbei, Oper Res and Financial Eng

### Professor

- Noga M. Alon
- Maria Chudnovsky
- Peter Constantin
- Amit Singer
- Romain Teyssier
- Jeroen Tromp

### Associate Professor

- Ramon van Handel

### Lecturer

- Henry F. Schreiner

### Visiting Professor

- Ehud Yariv

For a full list of faculty members and fellows please visit the department or program website.