Value of undergrad math research (was: math grad admissions)

From: hru...@xxxxxxxxxxxxxxxxxxxx (Herman Rubin)
I see no point of having an undergraduate do trivial research
when so little mathematics is known.

I believe you're attacking a straw man. In some obscure places in
math, it's possible for an "elementary" problem to amount to a
significant research project. I believe my undergraduate research
project in differential algebra was such a project, as was my later
self-project for generalized mis{e`}re Nim.

I'm not sure, but my undergraduate research project might have
trained my mind to tackle such long problems.

Prior to the undergraduate research project, my problem solutions
outside the regular curriculum consisted of:
- The five-sailors-on-island (coconuts and monkey) problem, which I
solved in about one hour from start to finish when I was in eighth
grade.
- The problems on the SCU high-school math contest, where we have
three hours to tackle six problems.
- "Elementary Problems" in the American Mathematical Monthly, where
I solved several problems and submitted solutions and got my name
mentionned as successful solver several times, and I estimate no
single problem I solved took more than an hour or so to solve.
Then suddenly the undergraduate research project had me spending
several summers working on a single huge problem, the first summer
basically learning the theory and methodology and reproducing a
known case (exponent 2), then the second summer solving the next
case in sequence (exponent 3), then the next summer generalizing
the generation of the generator to allow closed-expression (nested
summation and products and determinant of a matrix) of that
generator and writing software on the IBM 1620 for symbolic
mathematics of differential polynomials to avoid arithmetic
mistakes, and then upon seeing how things work, using Newton's
formulas for symmetric functions to get a proof of *all* cases
(exponents 4,5,6,...). The fourth summer was spent writing up those
first three summers of work for publication.

Meanwhile, each Nov/Dec there was the Putnam contest, a relatively
trivial (timewise) exercise in tackling problems that can be solved
in an hour or so each. The Putnam problems (more difficult than the
high-school problems and Monthly Elementary problems, although of
the same basic kind), and the undergraduate research project,
exercised different aspects of my problem-solving ability.

I can't be sure, but I suspect that although the experience on the
undergraduate research project didn't end up being used in a math
career, it *did* prepare my mind for later doing a major project on
generalized mis{e`}re Nim, including:
- the attention span to stick with a problem that took more than
three hours to solve, like months or years to get good results,
years or never to totally solve the problem, but at least a
good interim result within a few years worth publishing;
- the skills of breaking the problem into smaller pieces that can
be tackled in just a few days before moving onto the next more
general problem, which is a time-expanded version of what the
Putnam problems required;
- the skill of writing up formal proofs for publication, including
organizing the entire set of definitions and proof in an
understandable sequence.

Thus although I didn't do something really super-significant, like
solve the classification of all finite simple groups, or prove
Fermat's theorem on generalized triangle equation, or prove the
four-color conjecture, still I believe the undergraduate research
project did give me some of the useful skills towards tackling such
a problem.

The one crucial skill where the undergraduate research project
didn't test me, and consequently failed to discover my fatal
disability, was in memorizing large numbers of facts (definitions
and theorems) needed to do "REAL" research of the type you are
talking about, where you have to basically memorize everything that
everyone else has ever done in your field before you can put all
those pieces together to then advance the field one step further,
like the way Wiles combined several apparently unrelated fields to
finally solve the Fermat conjecture, and like the way Feit Thompson
et al put together all the Sylow theorems and other related matters
to solve the simple-group classification problem.

Too bad I didn't get interested in that Four Color conjecture,
because as it turns out it really didn't need combining "everything
known" in field(s), rather it just needed exhaustive case by case
breakdown aided by a computer database to keep track of all the
loose ends. I might have been able to solve that myself if I had
thought it was so easy that such an analysis would have gotten it
done.

Now that American Checkers has been proven a draw, using a simple
algorithm of backtracking to all possible situations with a small
number of pieces then minmax search from start of game to try to
reach these known end positions, maybe someday I'll use a similar
approach to find out who wins the variation (Cylendrical Checkers)
that I invented sometime around 1977 which is *never* a draw.
If I ever find anybody interested in such a line of research.
(For anyone curious: Board is 7 wide, from sidewall to sidewall, 10
forward around the cylinder. Starting position is analagous to
regular American Checkers, i.e. all but center pair of rows filled
in checkerboard pattern. No "kings", just keep moving (diagonally)
forward around the cylinder forever until somebody loses due to not
having any legal moves due to anihilation or blockage. Same rule
about forced jump, else opponent can "huff" or ignore or require
jump.)
.

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