[FOM] Dirichlet's theorem; boiling down proofs
Jacques Carette
carette at mcmaster.ca
Tue Aug 21 19:36:04 EDT 2007
Wayne Aitken wrote:
> On the other hand, I suspect one could find counter-examples to your
> thesis in the following areas, all already highly developed in the 19th
> century:
>
> * Elliptic and abelian integrals, and the associated theory of elliptic
> curves and jacobian varieties.
> * Projective geometry.
> * The differential geometry of Gauss and Riemann.
> * Galois theory.
> * Algebraic number theory.
> * Algebraic geometry (especially the Italian school).
> * Fourier analysis.
>
My undergraduate degree (in pure mathematics, completed a mere 17 years
ago) covered the last 5 topics in decent depth, and half of the first
topic, but no projective geometry. And in fact, the proofs were based
on exactly the tools you mention
> the best modern proofs use the
> (largely graduate level) mathematical machinery of the 20th century:
> measure theory, functional analysis, point set topology, more abstract
> forms of linear algebra, commutative algebra, homological algebra, and so
> on.
which were all covered (yes, even homological algebra) in those
undergraduate courses. And this was at the University of Waterloo, and
my colleagues who graduated from Berkeley, Harvard, Oxford and the Ecole
Polytechnique generally covered even more, although my colleagues from
less-renowned universities covered less.
I guess the only useful conclusion from this is: *which* undergraduate
education in mathematics are you using as your standard? The
``average'' american university, or what gets taught in the top 25 math
programs in the world? I suspect the answer will be /wildly/ different.
Jacques
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