In
the afternoon, it’s time to meet the graduate students. I walk into class…and
count nine people there (all men). That’s good, since only six are enrolled in
the course. My graduate student is among them, sitting in the front row. I
recognize a couple of faces from last year’s graduate course; also good. This
is a Ph.D-level course (second-year graduate), so nine is a very decent number.
The
focus of the course is a “hot topic” in my field. Ten years ago, a program to
solve a famous problem in one area of mathematics using techniques from another
area was brought to fruition in unexpected and spectacular fashion, by a
brilliant individual working alone (the kind of thing that only happens in
math, I think.) The techniques involved in proving that theorem are very
powerful, not fully developed, and still have the potential to prove
interesting results in both fields; it’s the kind of topic where a beginner can
still find a good problem and make a contribution. “Regardless of what your main field of
interest is, having a paper in this area would be a good thing”, I tell them.
On
the other hand…since the topic bridges two areas, having some knowledge of both
at the beginning graduate level would be highly desirable, right? So I ask for
a show of hands. How many have taken a basic course in area A? Three hands
shoot up. A fourth student says `I took it in the Physics Department, does that
count?’ Now, that could take us far afield; but I just smile and say, `sure’. How
many have a basic course in area B behind them? Four hands (different ones).
Okay, so my first mission here is not to scare them away. When I was a student,
if something was new and difficult it was impossible so scare me off; mastering
new things is what it’s all about, even if it’s hard work. But by now I know
from experience that not everyone is like that.
I tell them “well, you’ll have to do some independent reading of
background material.” And I promise I won’t get too technical regarding area B.
In
fact, the requirement for a grade is minimal: I want them to give a talk on the
topic of the course—either present a result found in one of several monographs
in this area, or one from a recent paper. On the other hand (I tell them) the
more ambitious students should be thinking in terms of finding a good research
problem in this area by the end of the year. “You’re graduate students, and
what do graduate students do? They ask questions” (hint, hint). “Some of them
will be good questions” (they smile). “And, since you have no experience,
they’re not likely to be questions that would occur to me, or to someone
already working in the area”. That’s
their edge: being able to ask new questions, out of sheer ignorance.
I
start slowly, with a survey of the main results I would like to focus on during
the course, including simple examples one can do “with bare hands” (that is, on
the back of an envelope). I get one
good, creative question (from the guy who took it in Physics). “No, I don’t
think this has been done”, is the answer. Then I move on to a very simple
result that can be proved in the same spirit as more difficult theorems, and
where the techniques appear in their most basic form. This is standard “area A”
material. Unfortunately area A can get technical very quickly, and in the back
of my mind is the fear of discouraging those who are just planning to sit in to
get informed. I could do the “big ideas” thing, and never get technical; but
that would be cheating them of any chance to work in this area. The right balance
between ``technical” and “big ideas” (in lecture) is not a trivial one to find.
When
the lecture is over, I chat with my student for a few minutes. He tells me he
has a new job, as an instructor at a local community college. That’s good, but
it means he won’t have time to take any graduate courses for a grade. He is at
the thesis-writing stage, anyway (Master’s degree). He tells me he is “just
about finished” with writing up the derivation I asked him to do.
No comments:
Post a Comment