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\markright{Statement of Purpose, Kevin M. Iga}
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Like everyone else who applies to graduate school in mathematics, I enjoy
doing and learning about math.  And like most, I'd like to do both for
the rest of my life.  Why?

When I was young, I discovered a path in a forest.  I was overjoyed.
A path no one knew about...but me.  How could a little boy {\em not}
be filled with wonder and excitement about where it might lead?
I followed it, and discovered more paths, and a stream, and some
rocks---a universe of beauty.  With time, it became familiar territory,
but by then I had found more paths.

When I learned about mathematics, it was a door to another universe.
And once I saw that, how could I not be filled with wonder and excitement
about what might be in it?

And throughout my schooling, guides (be they people or books) would
show me paths, and I would watch in wonder.  Sometimes a path would be
a major thoroughfare for science and engineering; sometimes, a narrow
ridge, awesome and beautiful to the mathematician willing to do some
climbing.  A number of times, I'd wander through the familiar ground
and find a new path, and my head would rush at the excitement.  And
many times, I would later find someone had been there before.  And I
would think, ``At least it was more exciting to discover it myself
than for someone else to show it to me.''  But every once in a while,
I'd find a path no one knew about.

Like my study of the problem: given a continuous real function $g(x)$,
find all continuous real functions $f(x)$ such that $f(f(x))=g(x)$.
I've solved this problem, by the way, for monotonic $g(x)$, and for a
certain class of non-monotonic $g(x)$ (I am currently writing up these
results and plan to submit them for publication soon).  I've also
investigated directions one can go from here, including changing the
space from the reals to the complex numbers, the circle, or n-space,
and changing the set of functions from continuous, to differentiable,
or analytic, and changing the number of times you compose $f$ with
itself, and so on.

When I sent out a query to the MAA, they couldn't find my particular
problem, but directed me to papers on similar topics.  When I found
those papers, I found nothing resembling my findings.  I then realized
I had truly done original mathematics.  And I was a little boy again,
wandering through the woods.

That's what I want to do.  That's why I want to learn and do mathematics.

But I don't only want to learn and do mathematics; I want to teach it,
too.  Throughout my undergraduate career, I've taught high school
students precalculus, calculus, elementary group theory, and
elementary mechanics.  I've also shared some of my more advanced
material with fellow undergraduates.  (In some way, teaching in a
classroom is not very different from sharing between friends).  In
these experiences, I've found I enjoy teaching.  My students enjoy
learning from me.  And most of all, they learned the material well, as
can be demonstrated in their subsequent performance in later classes.

The obvious road for me, someone who enjoys doing, learning, and
teaching mathematics, is the one for professorship.  Of course, that's
a long ways away.  But it is my goal.

On a different note, I'd like to tell you about how I approach
problems.  I'm a geometric thinker.  When trying to learn a subject, I
won't settle for an outright definition.  I'll try to think of ``what
it means,'' which might mean imagining some playing field where
functors follow shapes in left field and trace them into something
related in right field, or imagining a sprite-like group element act
on a set by physically moving the points around.  Some mathematical
objects (Stone \v{C}ech compactification comes to mind) simply do not
easily allow this sort of geometrization.  But I'll try hard.  To this
day, I imagine normal subgroups in terms of fundamental groups of
regular covering spaces!

Despite this, I think the greatest lesson MIT has taught me in mathematics is
that the other way of looking at mathematics, the formalism and the symbols,
is just as valuable as the intuition.  After all, in high school, the
relation between the formalism and the intuition was always obvious.  Past
high school, it's easy to get caught up in one without realizing you have lost
the other.  I've always tried to find the intuition in the formalism;
MIT has taught me to find the formalism behind the intuition.

Given my natural bent toward geometric intuition, I have naturally
been attracted to geometry, topology, and analysis.  This is not to
deny the applicability of algebra to such fields---I have no qualms
about using algebra in studying geometric objects---but it gives me a
secure sense that I'm exploring something that has meaning, and not
merely pushing symbols around.

Of the three, I've found the approach in topology clean, beautiful and
fascinating, as opposed to the unnatural deltas and epsilons, and
sequences in analysis, or the problems of needing to deal with
explicit coordinates in geometry.  Topology has also required my
patience, as many of the interesting problems in topology require a
great deal of background to even attempt.

In summary, I plan to learn, do, and teach mathematics as much as possible,
and graduate study is the next logical step in that direction.

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