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"I'm not motivated by classical problems so much as by structure, and
that's what really fascinated me about arithmetic algebraic geometry:
I took an instant liking to the structural aspects of it. That appealed
to me because I feel happiest when I'm doing algebra, but I like to
think geometrically.
"I think theory arises as a by-product of trying to solve problems.
I like solutions that have some surprise, and that is one of the really
nice things about arithmetic algebraic geometry. You have some fairly
concrete problems, and the beauty of these problems is that to get to
grips with them, you have to put in an enormous amount of technical
machinery and abstract structure, and that somehow justifies producing
this structure.
"In our dream world, we'd have an arithmetic cohomology theory that
associated to an algebraic variety certain infinite-dimensional spaces
with extra structure, which you could use to construct L-functions
to prove that they had nice analytic properties. I think this is a
dream of the way things might be, rather than a dream of how they are,
and it is such a long way from present knowledge."
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