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 Masters 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 601/602), analysis (AMCS 608/609), probability and stochastic processes (AMCS 605/606), Algorithms CIS 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.)
Those students who do not pass the Masters 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 and stochastic processes (Math 546/547), instead of the corresponding 600-level courses. These students need to pass the masters 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 PhD 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.
Typically first year students spend 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.
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 take their Ph.D. Preliminary Exam ("oral exam"). The exam covers three topics, two different areas of applied mathematics, as well as third exam on material in the students applied field of study. Most often, students choose topics in which they have taken a course during their second year. (Exception: Students who took mostly 500-level courses in their first year may be permitted to defer their oral exam until their third year.) After students pass the exam, they take should finalize their choice of Ph.D. thesis advisor, and the research topic for their Ph.D. thesis. Typically their research topic involves one of the two areas of applied mathematics included on their oral exam.
Also during the second year, many Ph.D. students 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.)
As statistics is the language of experimental data analysis, students in the AMCS PhD 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.
Second-year students typically spend much of the following summer at Penn participating in research connected to their field of study. Some opporutnities 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.
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 years students are usually supported by research fellowships, which entail some responsibilities to their doctoral advisor, again with no teaching or grading responsibility.
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.