[Editor: this story was originally submitted by Chris Woodward in April 2002.]

Item #3  Strategies for Graduate Students:  
A Perspective on Choosing an Advisor  

"Choosing an advisor"
Chris Woodward
ctw@carroll.rutgers.edu

I've gotten to the point now where I'm starting
to do more advising, but on the verge of forgetting,
to paraphrase G. C. Rota, the little sayings that I
used to repeat to myself in graduate school.  
For what it's worth, here are some of the sayings
I still remember.  I can't pretend that this
kind of advice is really going to be appropriate
for everyone - comments are welcome.

(1) "Avoid choosing an advisor"

Avoid getting a problem from anyone to work on
until you know something about the field.  Many
students finish the first and second year courses,
think of who their favorite teacher was and walk
into a prospective advisor's office knowing
nothing about the field.  The prospective advisor,
flattered for the interest, gives them a problem.  
A year later, after learning the background, they
realize that either (1) the problem is too hard
(2) they have no idea why the problem is interesting
(3) or they don't even like the field that much.  
It's embarrassing at this point to admit this -
it tends to make the student look bad from the
advisor's point of view.  If you put off asking for
a problem until you've read the advisor's papers,
gotten to know a little about what people in the
field are working on, then you can negotiate a
little about your problem, not to mention reject
problems that you know are way over your head.  
It doesn't look so bad if you reject a problem
say, after a week of thought.  If you want to be
a really perfect grad student, you should have read
(or at least own a copy) of all of your prospective
advisor's papers before bringing up the subject of
working together.  By the way, nothing prevents
you from trying to research in a field before you
commit to writing a Ph.D. with someone.

(2) "Avoid your advisor"

Obviously at the beginning you will need a lot of
guidance.  It's a good idea at the beginning to ask
if you can meet weekly with your prospective advisor.
Some will and some will not be willing to make
regular appointments.  At a certain point, though,
there is an expectation on the part of many advisors
that you will "go off into the woods" and solve your
problem, or at least, that when you show up when you
will have at least something small to tell them.  
That is, you should start responding to questions
about the paper you are reading by looking at other
papers, rather than by asking your advisor.  At
this point, it would be perfectly natural to start
meeting only once a month with your advisor.  This
is a good thing, since it means that you are learning
to research more independently. If your advisor seems
to be avoiding you, maybe this is because he/she is
trying to hint that you should be doing this.  
Basically, spending a lot of time talking to
someone who is much smarter than you is, in the end,
just demoralizing.  You need to build up your
confidence by figuring some things out for yourself,
using the literature.

(3) "Skip Class"

When you reach the "go off into the woods" stage,
you really need a lot of time and focus on your
particular subject. Classes on other subjects etc.
are really a distraction.  Most departments encourage
students to attend topics classes far too much,
in order to keep graduate enrollments up.

(4) "Avoid being a failed experiment"

Most mathematicians think like scientists, and this
applies to their students as well.  They have a number
of problems which they would like answers to, some of
which they could probably solve if they had time, and
some of which they have tried to solve but can't.  
Some advisors would be perfectly happy to have a
student spend five years on the second type of
problem and not come up with a solution.  That's
science, after all - not every experiment works out!
Try to figure out which type of problem the person is
suggesting, and pick the first kind.  Tip:  look for
an advisor who is very busy and just doesn't have
time to write up all the results he/she knows is
true. By the way, some problems will be considered
proprietary by the advisor - that is, if you decide
to switch advisors, you are not allowed to keep
working on the same problem.

(5) "Avoid abstraction".  The next two comments apply
to the type of problem to choose (in negotiation with
your advisor.) Very abstract problems are not a good
choice for most students. Instead, try to choose a
problem for which there are very concrete and
understandable examples, so that you learn the big
machinery through research.  On the other hand,
if you are Voevodsky, go for the Milnor conjecture
right away.

(6) "Be an explorer, not a problem solver".  
This is a quote I heard from the biologist E. O. Wilson.
I think it also applies to mathematics. For me,
it means:  most people in math are pretty smart.  
That means that if an approach to a problem has been
tried before by someone, there is very little chance
you are going to be able to add something new.  
Instead, try to find a problem/approach that no one
has tried before.  99% of results in mathematics are
easy results that the person got because no one else
had thought of trying it before.