Below is a description of the typical sequence of events in the career of a Penn AMCS grad student.

**Ph.D. students**

### First year:

Fully-supported Ph.D. students ordinarily receive a fellowship in their first year, during which they have no teaching responsibilities and may take four courses per semester.

Upon arrival in the Ph.D. program, new students take the Written Preliminary Exam on essential aspects of undergraduate mathematics; this serves, in part, as a placement exam. Those who pass the exam ordinarily take the beginning Ph.D. level courses in algebra (AMCS 602/603), analysis (AMCS 608, and 609 ), probability and stochastic processes (STAT 530/531=MATH 546/547), and an elective. (Exception: Those who enter with a more advanced background have an opportunity to place out of the beginning courses, and instead to move on to more advanced courses. Students with well a defined research interest that would suggest different first year courses may petition the Graduate Group Chair to make such substitutions.)

Those students who do not pass the Written Preliminary Exam upon arrival will ordinarily take the Masters Proseminar (MATH 500/501), and possibly one or more of the masters level courses in algebra (Math 502/503), analysis (Math 509/509), and probability (Stat 430 or 510), instead of the corresponding higher-level courses. These students need to pass the Written Prelim by the end of their first year, to demonstrate their ability to move on to more advanced material.

There is no foreign language requirement for the Ph.D. in AMCS, but there is a teaching requirement. University regulations require students, whose native language is not English, to demonstrate their ability to communicate in English, before they may serve as teaching assistants. Those who cannot do so satisfactorily upon entry will take a special course in their first semester, designed to help them improve their English communication skills.

In addition to attending their courses, first year students are required to attend the AMCS Colloquium, where scientists from other universities speak on topics in applied mathematics and computational science of general interest. By the end of the first year, students are expected to have a clear idea of the field in which they would like to do their dissertation work, as it will inform the courses they take in the second year. As students in the AMCS Ph.D. program work on interdisciplinary problems, it is essential that a field of research is selected by the end of the first year. This assure that the student will have sufficient time to acquire a comprehensive understanding of both the applied field of study and the mathematics that underlies it.

Typically first year students spend the summer, following their first year, at Penn, and participate in a research project on an applied math or computational problem. Ideally the summer should be spent working in a "wet lab" (construed broadly), pursuing research related to the student's own research interest. Often these summer internships lead to the general field of research that the student will pursue in their dissertation work.

### Second year:

Second year students take more advanced courses of their choosing, in areas of mathematics and applied science connected to the field in which they plan to do their dissertation research. These courses enable students to attain a sufficient level of competence in the areas of study germaine to their Ph.D. thesis. (Exception: Those Ph.D. students who did not take the basic 600-level AMCS courses in their first year will need to do so in their second year.) As each student's program of study is, in a sense unique, it should be formulated in consultation with their advisor and must be approved by the graduate group chair.

During the second year, students begin to participate in specialized seminars in areas of their mathematical and applied science interests. Some of these seminars feature faculty from other universities speaking on their recent research. Others involve graduate students giving talks on research papers they have read. Second year students are required to attend the AMCS Colloquium.

Second year students typically serve as teaching assistants, with this activity taking the place of one course each semester. TA's often run recitations sections for calculus courses, in which they go over homework that a professor has assigned to undergraduates taking a first or second year calculus course. Some other TAs have different responsibilities, such as running problem sessions for a course in algebra or advanced calculus. Before beginning to serve as TA's, graduate students go through a several-day TA training program run by faculty and advanced graduate students in the Mathematics Department, and must demonstrate proficiency in spoken English. Depending upon their background, some AMCS students may serve as teaching assistants in other departments, for example, Physics, Biology, or Statistics.

In the spring semester of the second year, students will start to prepare for the Ph.D. Oral Exam. 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. The exam itself is usually taken by the end of the Fall semester of the third year. Before taking the exam, students should finalize their choice of Ph.D. thesis advisor, and the research topic for their Ph.D. thesis.

Also during the second year, Ph.D. students may choose to write a masters thesis, which is an expository paper of about 30 pages. This provides experience in learning mathematics and/or its applications on one's own, and presenting it in writing -- experience that is valuable later, when writing the Ph.D. thesis. Writing a masters thesis also enables students to obtain a masters degree on the way to the Ph.D. (The masters thesis is also a good idea for students who are uncertain whether to continue to pursue the Ph.D., or whether to graduate with a masters degree.) AMCS Ph.D. students can also obtain a masters degree if they have completed 10 approved graduate courses, with two on an advanced topic.

As statistics is the language of experimental data analysis, students in the AMCS Ph.D. program are encouraged to attain a reasonable proficiency in ** statistics**. For students working on problems that make extensive usage of experimental data, this recommendation becomes a requirement. This can be accomplished by taking a statistics course, at the level of STAT 541, or STAT 512. Students should accomplish this requirement by the end of their second year. AMCS Ph.D. students are also strongly encouraged to develop a proficiency in computer programming, which can be facilitated by taking programming courses specifically designed for this purpose.

Second-year students typically spend much of the following summer at Penn participating in research connected to their field of study. Some opportunities also exist for graduate students to teach an undergraduate course in one of the two six-week summer sessions. (This is voluntary, and provides teaching experience and an additional stipend.) The same is true for third and fourth-year students

### Third year:

A third year student meets regularly with their his/her thesis advisor, who typically gives them research papers to read in order to help them prepare for the Ph.D. Oral Exam and write their Thesis proposal. Third year students typically take the Ph.D. Oral Exam by the end of the Fall semester. Students may also take more advanced courses, typically "topics" courses in areas of their interest that often go beyond what is available in textbooks, and may take independent study courses, in which they work with a faculty member one-on-one. In their third year, students participate actively in seminars in their area of mathematics and its applications. By the end of the third year, students generally have a well-defined thesis topic, and may have some preliminary research results. Students spend much of the following summer advancing those results.

Some students in their third year years are offered TAships or fellowships. Most third year students are supported by research fellowships, which may entail some responsibilities to their doctoral advisor.

It is recommended that each student fulfill the praxis requirement by the end of the summer after their third year. This provides a solid foundation in the practical aspects of their chosen field of research, which will inform the direction of their thesis research in their final years of graduate study.

### Fourth year:

Fourth-year students no longer officially take courses, but often choose to participate in advanced topics courses in their area of specialization in order to learn material that will be useful to them in their research. They continue to participate in seminars in their area of interest. Their main activity is working on their Ph.D. thesis, and during this year it is important for them to make significant research progress. Some students complete their Ph.D. at the end of the fourth year, though many require a fifth year to complete the degree. The summer after the fourth year is often a time when students find that their key research results come together in almost final form.

Fourth years students are usually supported by research fellowships, which entail some responsibilities to their doctoral advisor, again with no teaching or grading responsibility.

### Fifth year:

Additional financial support, in the form of a TAship or a research fellowship, is available for fifth-year students who have made significant progress toward the Ph.D. by the start of that year. During the fall of the fifth year, students work on strengthening their research results. Often they speak on this research in one of the seminars that they participate in. They also apply for jobs in the fall -- either academic or in industry, or both. In the spring they finish writing up their thesis, under the guidance of their thesis advisor. During that time they expect to hear back from places to which they have applied for jobs; and they may go on a series of job interviews, often giving talks about their research there. Later in the spring semester they present the key results of their Ph.D. thesis at an oral "defense"; and then they graduate at the end of the semester.

**Masters students**

The program for masters students begins in a similar way to the program for Ph.D. students, during the first year. One difference is that for masters students there is a lower passing grade on the Written Preliminary Exam. Masters students also have three chances to pass the Written Exam. Masters students are required to take eight one-semester courses and attend the AMCS colloquium. Often Masters students take two of the three two semester sequences in algebra, analysis, and the probability and stochastic processes (AMCS 602-3, AMCS 608-9, MATH 546-7), HOWEVER this is not a requirement. Depending upon the student's background, other programs of study may be arranged in consultation with the graduate group chair. Masters students may either write and defend a masters thesis of about 30 pages in length, or take two additional 1 credit courses of a more advanced character in a single topic. Only students writing a thesis take an oral exam. A Masters student will typically receive his/her degree by the end of the second year. In general, masters students do not receive financial support from the University or the AMCS graduate group, though they are often hired as "outside TAs" by the Math department on a course-by-course basis.

**Switching between degree programs**

Students in the AMCS Masters program may be permitted to switch to the Ph.D. program. If they do this, then the work that they have completed will be able to count toward the new program. This tyically requires the student to have Ph.D. passing grade on the written preliminary exam, an exemplary record in their course work, and a thesis advisor with financial resouces to fund their studies. In all cases this will require permission of the Graduate Group Chair.

Occasionally, graduate students in AMCS may wish to switch to a graduate program in another department, or vice versa. In these cases, the student speaks with the Graduate Chairs of both departments, to discuss whether such a switch is advisable. When such a switch occurs, credit for prior work *may* be given if appropriate. The awarding of such credit is at the disgression of the new graduate group Chair.