Notes for prospective PhD advisees
Since part of my professorial duties include directing PhD dissertations,
I have put together some general information that may be useful for
students considering working on a dissertation under my supervision.
I also recommend checking with my past and present students about
their experiences: Ruochuan Liu, Chris Davis, Liang Xiao, Jennifer
Balakrishnan, Fucheng Tan (joint with Mazur). Do beware though that
every student's experience is different, both because of the student and
because of the particular problems that the student opts to work on.
This page does not discuss particular research problems; I have a
separate page for this.
Preparation
What background do I expect from students working with me? Since
I am mostly interested in arithmetic algebraic geometry, essential tools
for me include algebraic number theory and algebraic geometry. There are
many books that cover the former at an appropriate level of detail
(including local fields, class field theory, and Galois cohomology); the
latter is conventionally studied first in Hartshorne's book (and then
further in EGA and SGA as needed).
Beyond those, the background required for the problems I'm interested in
typically depends on the problem itself, so I tend to prescribe
specific reading.
How to find a problem
Matching students with problems is a complicated and difficult art.
Some advisors take a completely hands-off approach; for instance,
Hendrik Lenstra claims that he can no more
suggest to his students what problems they should work on than who
they should marry.
In my experience, relatively few students manage this without any assistance,
so I tend to take a somewhat more interventionist approach. I tend to focus
on a small number of areas, which I discuss on my
page of questions of interest. I am happy
to discuss these questions to see whether one of these may lead to a suitable
problem.
On the other hand, I also strongly approach students to discuss with other
mathematicians about possibly interesting problems. I tend to know a little
bit about many topics (and am willing to learn more should the situation warrant),
so it is quite practical to work with me on a problem suggested by someone else.
Indeed, several of my students so far have done exactly this.
Meetings
It is the joint responsibility of advisor and student to meet often
enough to ensure that adequate progress is being made.
I am willing to have regular meetings, but I do not insist on this; I try
to keep myself available to meet on a more sporadic basis, as the need
warrants. (However, I do not recommend depending on me to remember that
a long enough time has passed since our last meeting that we need to meet again.)
Keep in mind that as a practicing mathematician, I travel quite frequently
to give seminars or attend conferences, which can interfere with my ability to
keep up a schedule of meetings. I keep an
online itinerary that can be used to determine
when I will be out of town.
Publishing
It is not absolutely necessary for students to publish their work before completing
their dissertation. (Here I include posting to a public web site, such as arXiv,
as a form of publishing.)
It is certainly reasonable to do so in many cases, and doing so
may help make your work known to the audience of people who will be looking at
your job applications later. However, the temptation should be resisted to
put material out prematurely; it is also an option to send preprints directly
to possibly interested individuals.
Rate of progress
Students at MIT are expected to make a reasonable attempt to graduate in four years,
and many of them do so. The others mostly finish in five years; getting funding
beyond the fifth year is usually quite difficult.
This means that students need to complete their course preparation, find an
advisor, and complete the qualifying examination by either the end of the first
year, or early in the second year. It is a good idea to start talking to
potential advisors immediately upon arriving, or possibly even sooner, to get
some sense of the expectations involved.
It is also an extremely good idea to make sure that one acquires adequate
teaching experience, since most job applications require a letter of reference
concerning teaching. Those with outside funding might be able to postpone
acquiring this experience until the year of graduation, but this is strongly
discouraged, as your second teaching experience is likely to turn out
better than the first.