The degree of Doctor of Philosophy in Applied Mathematics and Computational Science is an advanced degree designed for those who wish to pursue a career involving applied mathematics research. It is conferred in recognition of marked ability and high attainment in advanced applied and computational mathematics, including the successful completion of a significant original research project. The Ph.D. program is designed to guide students, year-by-year, toward becoming researchers in applied mathematicians. Typically the program takes four or five years to complete, including the dissertation (although it can be completed in less time, depending on the student).

There are several stages to the Ph.D. program. The first, which is centered on the course requirements and the Written Preliminary Exam, is designed to help the student acquire a broad background in applied mathematics and computational methods. The second stage includes the Ph.D. Oral Exam, and participation in seminars. Depth is one of the goals of this second stage, but the main objective at that point is to assist the student in choosing a field of specialization and in obtaining sufficient knowledge of this specialized field, including recent research developments. Ph.D. students also have an opportunity to earn a masters degree at this stage. The third and key stage of the Ph.D. program is the dissertation (or "Ph.D. thesis"), in which the student will make an original contribution to applied mathematics and computational science. The entire Ph.D. program is designed to help students move toward taking this significant step in creating new mathematics for applications or new applications of mathematics. Along the way, the student is required to acquire some teaching experience, this skill being essential for those entering an academic applied mathematics career. In addition, Applied Math students are required to satisfy a praxis requirement. This requirement is intended to provide students with a real world appreciation of the applied field in which they are working, which will inform their choice of research problems, and provide a gauge for the significance of mathematical results in their field. As statistics is the language of experimental data analysis, students in the AMCS PhD program are encouraged to satisfy a statistics proficiency requirement. Students whose research entails analysis of experimental data are required to demonstrate proficiency in statistics. This can be satisfied either through successful completion of a statistics course, at the level of STAT 541 or 512, or by passing an exam which will be offered as needed.

The program leading to this degree is described below, and may include work completed at the University of Pennsylvania for a Masters degree. (Up to eight courses taken at other universities, while a candidate for a graduate degree, can also be counted toward the Ph.D. requirements.)

**Detailed requirements:**

- Admission to candidacy: Admission to Ph.D. candidacy is achieved by passing the Ph.D. Oral Examination. Students must previously have passed the Written Preliminary Exam, and have taken at least six graduate courses approved by the chair of the graduate group (including those taken elsewhere).
- The Ph.D. Oral Examination: The purpose of the oral exam is to assess a student's readiness to transition into full-time research and eventually write his or her dissertation. It is something of a hybrid between the subject-oriented oral exam administered by the Math department and the thesis proposal used in many fields of science and engineering. As part of this process, the student will choose, with the advice of the Graduate Group Chair, a Thesis Committee to help supervise his/her dissertation research. A detailed description of the exam can be found on the Ph.D. Oral Exam web-page.
- Some important administrative details:
- This PhD Thesis committee is appointed by the Graduate Group Chair after consultation with the student. It consists of three or more faculty members, at least two of whom are full members of the graduate group.
- The PhD Oral Exam has two aspects:
- The student must prepare a written research proposal, outlining the problem they plan to pursue for their dissertation research. This proposal should be between 10 and 20 double-spaced pages. In any case, not to exceed 20 pages. The proposal should explain the mathematical significance of the proposed research.
- The student must prepare a syllabus of background material, which is needed to pursue research in their chosen field. The syllabus of background material should be prepared about six months before the exam, and must be approved by the Graduate Group Chair.

- The Graduate Group Chair notifies the AMCS faculty of the exam (date, time, place, committee members, reading lists). All affiliated faculty are to be invited, and are permitted to ask questions, however, only appointed members of the committee will vote on the candidate performance.
- If the student has elected to write a Masters Thesis related to the topic of their PhD dissertation research, then the oral defense of the Masters thesis may be combined with the Ph.D. Oral Exam.

- Scheduling the Ph.D. Oral Exam: To take this exam, the student should have passed the Masters Preliminary Exam and be in good standing. The student also should have taken at least six approved graduate courses (including those taken elsewhere). The student discusses in advance the topics, syllabi and the composition of the oral exam committee with the Graduate Group Chair, whose written approval is needed. Normally, the exam is to be taken by the end of the first semester of the student's third academic year in the program. A one or two semester extension may be requested from the Graduate Group Chair. If the Ph.D. Oral Exam is not passed on the first try, it may be taken just once more, and this must occur before the end of the following semester. Passing the Ph.D. Oral Exam on the second try, at the latest, is a requirement for remaining in the Ph.D. program.
- Course Requirements: Twenty units of graduate courses, numbered 500 and above (or the equivalent), are required for the Ph.D. degree, including at least twelve courses taken at the University of Pennsylvania. The bulk of these courses should be drawn from the list of approved courses. (Independent study courses at Penn may be counted toward the twenty course requirement.) Among the courses, every student must take at least two semesters of graduate courses at the 600 level in each of applied algebra, and applied analysis, at least one semester of probability and stochastic processes, and one semester of computational science. In general, eight of the courses should be taken in AMCS itself or in the Mathematics department. To receive the Ph.D. degree a student must have at least a 3.0 cumulative grade point average. Sample programs of study in a range of fields can be found under the
menu bar. The Graduate Group Chair may, in exceptional cases, modify the requirement that at least 10 of the 20 graduate courses for the Ph.D. be in AMCS or mathematics. A maximum of 6 of these 20 courses may be reading courses (independent studies).*Academics* - Seminar Requirement: It is expected that all advanced graduate students will regularly attend and participate in at least one seminar series each semester. First year students must register for the AMCS colloquium course, and present at least one lecture. First and second year students are required to attend the AMCS colloquium talks, and all students are strongly encouraged to do so. Although one lecture is the formal minimum, it is expected that students will give a number of such talks to audiences of students and faculty. Guidance in the preparation of these lectures is provided by faculty members in the graduate group. The intention is for the student to gain experience in digesting and presenting advanced material and in fielding questions about it before an audience of scientists, as well as actively participating in research interactions and being a part of the community of graduate students in the AMCS graduate group.
- Praxis Requirement: What distinguishes Applied Mathematics, in part, from theoretical physics, or engineering, is the conceptual mathematical framework in which research is conducted. On the other hand, for mathematics to retain its relevance to an applied field, it is essential that the researcher be cognizant of the questions of importance, and the internal standards for progress within that field. For this reason we require each student to have a substantive experience in a laboratory conducting research on a problem of empirical science directly related to the students research. An internship is to be arranged in consultation with, and with the approval of the graduate group chair. This requirement should usually be completed by the beginning of the third year. Students with prior experience doing research in empirical science may, at the discretion of the graduate Chair, be excused from this requirement. This requirement must be satisfied no later than one year after admission to Ph.D. candidacy, generally by the end of the summer after the third year of study.
- Teaching requirement: In order to gain experience in classroom teaching, the student is required to perform satisfactorily as a teaching assistant or instructor for at least two semesters. Teaching for more than two semesters is encouraged, especially for those students who plan to teach after their Ph.D. Graduate students participate in a TA training program before they begin their teaching.
- The Dissertation: The dissertation, also known as the "Ph.D. thesis", is the heart of the Ph.D. program. It must be a substantial original investigation in a field of applied mathematics and computational science, done under the supervision of a faculty advisor.
- The Ph.D. Thesis Committee: This committee is appointed by the Graduate Group Chair after consultation with the student, at the time of the PhD Oral Exam. It consists of, at least, three faculty members, including the thesis advisor, and meets at least once a year with the student to discuss his or her progress and to offer guidance.
- Dissertation Examination: When the dissertation is complete, it must be defended in a Dissertation Exam, at which the student will be expected to give a short public exposition of the results of the thesis, and to satisfactorily answer questions about the thesis and related areas.
- The Final Dissertation Exam Committee: This committee is appointed by the Graduate Group Chair after consultation with the student, and consists of three or more faculty members, at least one of whom is a full member of the AMCS graduate group, and at least one of whom is from the area of specialization of the thesis. It will normally include the Thesis committee appointed at the time of the student's PhD Oral Exam. All AMCS faculty are to be invited to attend the Final Disseration Exam, however only those appointed by the Graduate Group Chair are voting members of the Exam Committee.
- The student must deliver a finished copy of the Ph.D. thesis to the graduate secretary at least two weeks before the Final Dissertation Exam, so that it will be available for reading by the AMCS faculty.
- Timing for the Final Dissertation Exam: This must be successfully completed no later than six years after entering the graduate program. It should occur no later than the end of the third academic year after admission to Ph.D. candidacy. In exceptional situations, the Graduate Group Chair may, after consultation with the Graduate Advising Committee, provide an extension.
- Satisfaction of Requirements: When not otherwise specified, this is determined by the Graduate Group Chair, in consultation with the Graduate Advising Committee and involved faculty members and (in the case of the teaching requirement) the Undergraduate Chair.