The requirements for the undergraduate certificate in applied and computational mathematics are the following:

- 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 requirements for the candidate's major concentration.
- Independent work consisting of a paper on a research project with a significant applied mathematics component that is done under the supervision of a faculty advisor. The independent work cannot be used to qualify for anything else; for instance, you cannot use a junior paper or senior thesis for the independent work, without modification.
- 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.

## Coursework Requirement

The course requirement may be satisfied by a broad range of courses that place particular emphasis in applied mathematics, which are offered by the mathematics department and by 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 here. Courses that do not appear on this list may nonetheless be approved by the PACM undergraduate representative. Specific course choices should be tailored to meet the individual needs and interests of each student and must be approved by the PACM undergraduate representative, Prof. Ramon van Handel.

## Independent Work Requirement

The independent project can take several forms, depending on the interests and schedule of each student:

- Students may pursue an independent research project during the academic year under the supervision of a Princeton faculty member. Typically such a project is done on the side along with a regular academic program in either junior or senior year; the scope of such a project may be comparable to that of a final project in a course. Students who wish to pursue a larger project should discuss with the PACM undergraduate representative how that may be incorporated into the academic program.
- Students may pursue an independent research project over the summer, either at Princeton or off-campus at another institution.

There are three requirements for the independent work: (1) it must include a significant applied or computational mathematics component; (2) it cannot count toward any other requirement for your major or other certificates; (3) it must be independent work of the student under the supervision of their advisor(s), that is, the work cannot be a group project. The final paper that is submitted to satisfy the certificate requirement must be written by the student.

As the PACM independent work may not count toward any other requirement, your senior thesis or junior paper cannot satisfy the requirement without modification. It is, however, possible to use a prior research project as a basis for further work: for example, a separate paper extending the work in your senior thesis or a course project can satisfy the PACM independent work requirement. On the other hand, students may also take the opportunity to explore a completely different topic from prior independent work. Which route is taken depends entirely on the interests and schedule of each student.

Normally PACM independent projects are done under the supervision of a PACM core or affiliated faculty member. Initially, students should contact a faculty member, consult with them on the topic, and have them agree to be the advisor. After you choose your advisor, you must meet with them regularly. This will allow them to keep track of your progress and make sure that the paper has a sufficient amount of mathematics content to qualify for a certificate. A list of the research interests of PACM core and affiliated faculty may be found here.

In some cases, students may want to do a project with a faculty member who is not formally affiliated with PACM (either due to the nature of the subject matter or as the work is being done off-campus). In such cases, a PACM core or affiliated faculty member will be asked to serve as the second reader to verify that the paper contains enough applied mathematics to satisfy the certificate requirements.

Plans for independent work must be approved by the PACM undergraduate representative, Prof. Ramon van Handel.

## Certificate Seminar Requirement

The certificate seminar is held during 4-5 afternoon sessions (approximately 1.5 hours each) in the spring semester. During the seminar, every senior enrolled in the certificate is asked to give one presentation about their PACM independent work. All juniors and seniors are required to attend at least 75% of these presentations, in total over the junior and senior years. (Any sophomores who are enrolled in the certificate are cordially invited to participate as well.) Note that you will be asked to present your PACM independent work in your senior year, regardless of what year the independent work was performed in.