The purpose of this document is to describe the graduate program in applied mathematics and computational science at Penn. See the AMCS home page for information about other aspects of the AMCS graduate group, including its colloquia, seminars and lecture series, and the research interests of the faculty.

ADMISSIONS AND APPLICATIONS

The Penn AMCS Graduate Program is an interdisciplinary program that offers training in advanced mathematics and its applications to problems in empirical science, medicine and engineering. We welcome applications from interested and dedicated students who have done substantial work in mathematics, and/or its applications of superior quality as undergraduates. Ideally, applicants for admission are expected to have a mathematical background that is comparable to an undergraduate mathematics major.  Applicants are encouraged to take courses, beyond calculus,  in mathematical analysis, linear algebra, probability,  and complex analysis. We are also happy to consider applications from students in applied fields with less complete mathematical preparation.  The latter students will be expected to bring their mathematical competency up to the required level in their first year of graduate study. Students will be expected to pass the our Masters Preliminary Exam, which covers the foundational mathematics expected of all students, by the end of the first year.  Mathematics course grades should mostly be A or A- (or the equivalent).

Applications must be submitted  electronically.  More information about the application procedure is available here.

All applicants must take the Graduate Record Examination (GRE): both the General test and the Advanced Mathematics Subject Test.

Applicants whose native language is not English must take the TOEFL (Test of English as a Foreign Language) or the TSE (Test of Spoken English). It is best to take both the TOEFL and the TSE, particularly in cases of applicants whose native language is unrelated to English. More information.

The GRE and English language exams should be taken early enough for the scores to be received at Penn in time to be considered by the Admissions Committee. Note that it can take a month or more after taking the exam before the score is reported, and that advanced registration is required for these tests.

Applicants are considered on the basis of their courses, grades, letters of recommendation, test scores (basic and advanced GRE, TOEFL, and TSE), and other supporting material. Letters of recommendation should preferably be from mathematics professors, or faculty from another scientific discipline,  who know the applicant well, and who can comment on the applicant's abilities, accomplishments, and potential for research in applied mathematics and computational science.

Applicants for the graduate program in AMCS are ordinarily admitted for the Fall semester each year, with the official deadline for applications December 15 of the previous calendar year. It is important to ensure that the application and all related materials, including letters of recommendation, transcripts, and test scores, arrive by the deadline (or at least by mid-January). It is therefore best to take the Advanced Mathematics GRE no later than November; those who take it in December should ensure that their scores arrive quickly. Applications that are late may be considered on the basis of available space and financial aid.

Applicants should indicate the final degree that they hope to be awarded from the Penn AMCS Graduate Program (either M.S., or Ph.D.). Students who intend to obtain an M.S. on the way to a Ph.D. at Penn should indicate that they intend to get a PhD.

FELLOWSHIPS AND FINANCIAL AID

Full-time students admitted to our Ph.D. program in the fall of 2008 and after are generally offered a financial support package for four years of study.  Support is provided by a combination of  fellowships and teaching/research assistantships, including a fellowship (with no teaching) in the first year and a teaching assistantship for the second year. This package, which covers tuition and fees and health insurance and also provides a stipend, continues for four years, for Ph.D. students in good standing.  As many students will write their dissertations in applied areas, it is expected that  their latter years of graduate study will be funded by outside grant support provided by their dissertation advisors.

Teaching assistants most often run recitations of calculus, meeting with up to four small groups of undergraduates once a week to discuss homework. Other possible TA assignments  include running problem sessions in more advanced courses such as advanced calculus or linear algebra, or running review sessions or grading for courses in which there are no problem sessions. Counting contact hours, office hours, grading, record keeping, etc., the TA responsibilities take up to twelve hours per week (and replace one of the graduate student's own courses). Before beginning a teaching assistantship, graduate students go through a several-day TA training program run by the Math Department. There is also mentoring for TA's during the semester.

Graduate students often teach courses over the summer, for an additional stipend. (Doing this is optional, but is good experience.) Unlike TA responsibilities during the academic year, grad students who teach during the summer have full responsibility for the course, giving the lectures and assigning homework and grades.

Students in the masters program are not normally offered financial aid. In some circumstances Master's students may  be considered for teaching assistantships, which provide only a stipend. These are awarded based on merit and availablity, and are decided on a semester-by-semester basis.

Correspondence concerning Admission and Financial Aid should be sent to the Graduate Group Chair.

FIRST YEAR CLASS SCHEDULE

All entering graduate students take the Masters Preliminary Examination before the Fall semester's classes begin. This is a five hour exam which covers essential undergraduate material in mathematics, including  linear algebra, advanced calculus, functions of a complex variable, and probability. It serves, in part, as a placement exam, to help determine each student's schedule of classes during the first year. Students are expected to exhibit a mastery of these topics at the undergraduate level.

After taking this exam, students meet individually with the Graduate Group Chair to design their first year program of courses. They also meet with their faculty advisors. (Incoming students are assigned advisors by the Graduate Group Chair, until such time as they choose advisors for themselves.) Most often, first-year students are advised to take the three main introductory Ph.D.-level courses in applied analysis, applied  algebra, and  probability and stochastic processes,  and a one semester course in computational science, plus an additional course of their choosing.  Those who demonstrate to the graduate group Chair that they already have a mastery of the material in these introductory courses can instead take more advanced courses. Students with a strong interest in computation may replace the second semester of stochastic processes with a course in computational science, for example CIS 502 (Analysis of Algorithms), or EAS 502 (Numerical Methods and Modeling). On the other hand, those whose performance on the Masters Prelim shows significant gaps in training may be advised to take one or more introductory masters-level courses during the first year in Algebra (Math 502, 503), or Analysis (Math 508, 509), and Probability.

Each year the AMCS offers a one semester no-credit course based on the lectures presented in the AMCS colloquium. The course meets once a week, before the colloquium talk. Each registered student is required to present a talk covering the background material for one of the lectures presented in the colloquium. All first  year students are required to take  this  course. 

One of the first year graduate courses, the "Proseminar" (MATH 500, 501), attempts to bridge the gap between undergraduate and graduate mathematics by treating the topics covered in the Masters Preliminary Exam more intensively than is typical in undergraduate courses, with an emphasis on problem solving and presenting proofs. Graduate students take this course during the first year of study here if they do not pass the Masters Preliminary Exam upon arrival.

The Masters Preliminary Exam is given for a second time at the end of the spring semester. Passing it by the end of the first year in the graduate program is a requirement for remaining in the graduate program. (The passing level is ordinarily higher for the Ph.D. program than for the masters.) The University's Qualifications Evaluation requirement is met by passing this exam and doing well in the first-year courses.

Year-by-year sequence for graduate students.

THE MASTERS DEGREE

The Masters of Science program in AMCS is intended for students who wish to go into professional careers in applied  mathematics, and who have a background corresponding to an undergraduate mathematics major or to an undergraduate major in a science or engineering topic, with a minor in mathematics. The masters degree is typically earned in approximately two years, though it is possible to earn it in less time.

Students in the masters program take at least eight graduate courses from the list of approved courses numbered 500 and above, including at least one each in applied algebra, applied analysis,  probability and statistics,  and numerical methods. The other courses can be in  pure mathematics, applied mathematics, and courses offered by other departments, which make essential usage of advanced mathematics. Every student's program of study must be approved by the graduate group chair. There is also a requirement to pass the Masters Preliminary Exam and to submit and defend a masters thesis.

The masters degree can also be earned by Penn Math Ph.D. students, on the way to their Ph.D. In addition, Penn Ph.D. students in allied fields of study can also apply to enter the masters program in applied mathematics and computational science, which can allow them to earn both degrees simultaneously.

Detailed requirements:

1. Admission to candidacy: Admission to Masters candidacy is achieved by passing the Masters Preliminary Exam by the end of the first year in the graduate program, and by satisfactory performance in the first year's courses.
2. Course requirements: A minimum of eight units of graduate courses at the University of Pennsylvania, numbered 500 and above, is required for the Masters degree. Among these courses, every student must normally take the  first year sequence in applied algebra, applied analysis, probability and statistics, and numerical methods.  The list of courses must be approved by the graduate group chair of the  AMCS graduate group.
   
3. Language requirement: There is no foreign language requirement for the Masters degree.
4. Masters thesis: Each student must write, under the supervision of a AMCS graduate group  faculty member, a satisfactory Masters thesis, of modest length, which is typically expository in nature, but may also be a research paper. The preparation of this thesis should involve the mastery of some area of applied mathematics or computational science beyond the curriculum of the courses that the student has taken, and should ordinarily take  one semester to complete.
5. The Masters General Examination: Following submission of the Masters thesis, the student will take a General Examination for the Masters degree. This exam consists of a presentation by the student of the content of the thesis, followed by questions from the faculty on the topic presented and related areas of applied mathematics.

6. Some important administrative details:
   • Administration of the Graduate Program: The Graduate Group Chair directs and oversees the administration of the graduate program, advises the graduate students in this program, and assesses their progress on a continuing basis. The Graduate Group Chair is aided in this by the the Curriculum Committee, and its  subcommittees.

   • Timing: The program leading to the Masters degree should ordinarily be completed within two years of full time study, and in any case must be completed within three years of full time study. Part time study, when allowed, will have a corresponding timetable set in each case by the Graduate Group Chair.

   • Course schedules: Each semester, the student's course schedule must be approved in advance by the Graduate Group Chair and the student's advisor. Students are reminded that, while the course requirements for the masters degree include at least a semester of graduate courses in each of algebra, analysis, probability and statistics, and numerical methods, the course requirements for the Ph.D. include a full year in most of these subjects at a possibly higher level, and may want to take this into account when planning their course schedules.

   • Course requirement: Approval of the Graduate Group Chair is required for courses taken outside the list of approved courses to be counted toward the course requirement for the Masters Degree in AMCS.

   • The Masters General Examination Committee is appointed by the Graduate Group Chair after consultation with the student. It will consist of two or more faculty members, at least one of whom must be tenured, and one of whom will ordinarily be the supervising faculty member. The Graduate Group Chair will notify the entire mathematics faculty of the exam; all faculty are explicitly invited to attend, and those who do are full voting members of the Exam Committee.

   • If the Masters General Examination 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 Masters General Examination on the second try, at the latest, is a requirement for remaining in the graduate program.

   • Satisfactory performance in the graduate program is determined by the Graduate Group Chair, with the aid of the Graduate Advising Committee and (for first year courses) the First Year Course Committee. Satisfactory performance, which includes maintaining a B average every academic year and meeting requirements in a timely manner, is a requirement for remaining in the graduate program.
   • Financial aid: Students enrolled in a terminal masters program will not ordinarily be offered financial aid.
     

THE Ph.D. DEGREE

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 masters preliminary exam, is designed to help the student acquire a broad background in applied mathematics and computational methods. The second stage includes the oral Ph.D. preliminary exam on three topics of the student's choosing, 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:

1. Admission to candidacy: Admission to Ph.D. candidacy is achieved by passing the Ph.D. Preliminary Examination. Students must previously have passed the Masters Preliminary Exam, and have taken at least six graduate  courses approved by the chair of the graduate group (including those taken elsewhere). 
   

2. The Ph.D. Preliminary Examination: This is an oral exam in two distinct subfields of applied mathematics and computational science,  along with one in the applied field in which the student intends to specialize. The student should demonstrate both depth and breadth of knowledge, as well as real mastery of basic material and the ability to apply it to specific examples. Typically, the content of each of the two parts of this exam will be somewhat more than that of a one-semester graduate course at or above the 610 level (though sometimes the material in an advanced 500-level course may be appropriate).
3. 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.  We provide suggested courses of study for students in the four concentration areas: imaging science, inverse problems, mathematical biology, materials science.
   
4. 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.
5. 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.
6. 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.
7. 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.

8. The Ph.D. Advisory Committee: This committee is appointed by the Graduate Group Chair after consultation with the student, when the student begins to work on a thesis. It consists of 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.
9. 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.
10. Some important administrative details:
    The administrative details listed earlier in connection with the Masters degree program apply to the Ph.D. program as well. In addition, we have the following:
    • Combining the Masters General Examination with the Ph.D. Preliminary Examination: The oral defense of the Masters thesis may be used as the examination on one of the two subjects of the Ph.D. Preliminary Exam, provided that it occurs simultaneously with the other part of the Ph.D. exam, and that the requisite depth and breadth of knowledge are demonstrated.

    • The Ph.D. Preliminary Exam Committee: This 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 syllabi for the exam are agreed upon by the committee and the student, with the approval of the Graduate Group Chair. The Graduate Group Chair notifies the AMCS faculty of the exam (date, time, place, committee members, reading lists). All faculty are to be explicitly invited, and are invited to ask questions, however, only appointed members of the committee will vote on the candidate performance.

    • Scheduling the Ph.D. Preliminary 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. The exam is to be taken by the end of the student's second academic year in the program. A one or two semester extension may be requested from the Graduate Group Chair in the case of students who took mostly 500-level courses in their first year.

    • If the Ph.D. Preliminary 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. Preliminary Exam on the second try, at the latest, is a requirement for remaining in the Ph.D. program.

    • Course Requirements: 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).

    • Timing for satisfying the Praxis 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.

    • The 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. All AMCS faculty are to be explicitly invited to attend the exam, however only those appointed by the graduate group Chair are full 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 Dissertation Exam, so that it will be available for reading by the AMCS faculty.

    • Timing for the 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