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One way to help get through quals: The quals are,
on the surface, in some common justifications, to be
more sure the student can do the dissertation research.
Okay. Well, there's another way to be "more sure", really, more reliable than any
quals can ever be: Have the student do the dissertation
research independently before the quals. Now
in this case, just what
are the quals for? Or,
for an engineering Ph.D., a guy writes a good
dissertation in applied probability
with careful attention to the tricky
subject of measurable selection,
and want to hold him back
due to some qual with some tricky
issue about Feller I
probability? Or, there's a qual in optimization,
in part on the details of the Kuhn-Tucker
conditions,
but the student has already done original,
clearly publishable work, e.g., in JOTA, in optimization
and, in particular, the Kuhn-Tucker conditions? E.g., for problems in functional form, are the
Zangwill and Kuhn-Tucker constraint qualifications
independent? Along the way, given a closed
subset of R^n (usual topology), is there a function f: R^n --> R
positive on the closed set, 0 otherwise, and
infinitely differentiable (not quite the
same as the Whitney extension theorem)?
Is the Mandelbrot set closed and, thus, an
example? What about a sample path of
Brownian motion? So an infinitely
differentiable function can have
a bizarre level set? What does this say about
a question, without an answer, in the
famous paper in mathematical economics
by Arrow, Hurwicz, and Uzawa? And in this
case, the qual in optimization serves just
what purpose? The quals might be to see if the student
is prepared to take advanced grad courses,
but he's already done that and used
some of the best content in his Ph.D. research?
Now what are
the quals for? The quals can start to obscure a basic point
about the three things important in high end
academics, research, research, and research,
that is, the publishable kind. If a guy is
doing well in research, want to hold him
back because of what? Can there be such students? Yup. I have
an existence proof. Besides, quals can be awash in politics. The usual criteria for publication are
that the
work be "new, correct, and significant".
At some good research universities,
there is no coursework requirement for a Ph.D.,
and the dissertation is supposed to be
"an original contribution to knowledge
worthy of publication" or some such.
So, for a student, do some research
and publish it. As I recall, at one time the math department
at Princeton said that the grad courses
were introductions to research by
experts in their fields, that no courses
were given for preparation for the qualifying
exams, that students were expected to prepare
for the quals by independent study,
and students were expected to have some
research underway in their first year.
Good. I'd done a lot of
independent study before I
went to grad school. I got accepted as a grad student at the
Princeton math department but went elsewhere
(where my wife was still in grad school),
brought my own research problem with me
to grad school, and did my dissertation
research independently in my first summer
building on one of the courses in my
first year. Then the quals were for WTF? |