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There are several central mathematical
problems, or complexes of problems, that every
mathematician who is eager to acquire some broad
competence in the subject would like to understand, even
if he has no ambition to attack them all. That would be
out of the question! Those with the most intellectual and
aesthetic appeal to me are in number theory, classical
applied mathematics and mathematical physics. In spite of
forty years as a mathematician, I have difficulty
describing these problems, even to myself, in a simple,
cogent and concise manner that makes it clear what is
wanted and why. As a possible, but only partial, remedy I
thought I might undertake to explain them to a lay
audience. I shall try for a light touch
including, in particular, some historical background.
Nevertheless the lectures are to be about mathematics. In
the first set, there will be geometrical constructions,
simple algebraic equations, prime numbers, and perhaps an
occasional integral. Every attempt will be made to
explain the necessary notions clearly and simply, taking
very little for granted except the good will of the
audience.
Starting in the easiest place for me, I shall give,
during the academic year 1999/2000 about eight lectures
on pure mathematics and number theory with the motto beautiful
lofty things . Beginning with the Pythagorean theorem
and the geometric construction of the Pythagorean
pentagram, I shall discuss the algebraic analysis of
geometric constructions and especially the proof by Gauss
in 1796 of the possibility of constructing with ruler and
compass the regular heptadecagon. This was a very great
intellectual achievement of modern mathematics that can,
I believe, be understood by anyone without a great
aversion to high-school algebra. Then I will pass on to
Galois's notions of mathematical structure, Kummer's
ideal numbers, and perhaps even the relations between
ideal numbers and the zeta-function of Riemann. This
material will be a little more difficult, but I see no
reason that it cannot be communicated. It brings us to
the very threshold of current research.
Since this attempt is an experiment, the structure and
nature of the lectures will depend on the response of the
audience and on my success in revealing the fabric of
mathematics. If it works out, I would like to continue in
following years on classical fluid mechanics and
turbulence, with motto l'eau mêlée à la lumière ,
and then, with the somewhat trite motto s togo berega,
on the analytical problems suggested by renormalization
in statistical mechanics and quantum field theory.
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Duke University Mirror