[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.


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