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Statement of purpose

%I could tell you that I like to do math.  But you already knew that;
%if I didn't, I would not have gotten this far.  I could tell you that
%I'd like to learn more math.  But you already knew that; if I didn't,
%I wouldn't be applying to graduate school in math.  And you can probably
%guess I want to eventually do research in math, because

%I like to do math, and I like to learn math.  I would like to continue
%doing both.  There.  Now I can talk about how I might be different from
%other applicants.

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, and eventually, it became familiar territory.

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, a guide would show me a path, and in my
free time, I'd wander through the familiar ground and find new paths
and follow them.  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 find a new path, and my head would rush at the
excitement.  And many times, I would 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 ask around about a path I'd found and no
one knew about it.

>>>

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 group ``beast'' act
on a set by physically moving the points around.  Some mathematical objects
(Stone Cech 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 tended to find the intuition in the formalism;
MIT has taught me to find the formalism behind the intuition.


Puzzle solving
World discovering
Frontier thing
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