Subject: Re: [HM] "theoretical tools" vs "problems"
From: Franz Lemmermeyer (lemmerm@mpim-bonn.mpg.de)
Date: Tue Jan 18 2000 - 05:34:59 EST
On Mon, 17 Jan 2000, Colin McLarty wrote:
> Aubin takes some quotes from the mathematician Christian Houzel,
> and agrees with them, and here I quote Aubin's own summary:
>
> While the previous period [1950s and 1960s] was one that
> had witnessed the development of powerful new theoretical
> tools of great generality, he [Houzel] noted, the 1970s
> were rather characterized by a tendency to revive an old
> interest in more concrete problems. (p.39)
>
> I wonder how to recognize this new trend in concrete examples. For
> example, in the 1950s and 1960s Serre and especially Grothendieck
> aimed their work very largely at one problem in number theory: the
> Weil conjectures, estimating the number of solutions to a set of
> polynomials in any finite field. Deligne finished the job, using
> Grothendieck's methods, in 1972. Are Serre and Grothendieck taken
> as pioneers in the new trend towards concrete problems?
I wouldn't think so - the main work here consisted in developing
cohomology theories that would fit the predictions of Weil. I see
this as a powerful tool of great generality. In a similar vein,
I would count, say, Iwasawa theory to the general toolkit even
though it can be used to answer concrete problems.
As far as number theory is concerned, a revival of interest
in concrete problems is not a total invention: the computer
made a lot of people think about algorithmic questions
(factorization, primality, class numbers ...); the great
building of conjectures put up in the first decades after
WWII (Birch-Swinnerton-Dyer, Taniyama-Shimura, main conjecture
of Iwasawa theory, Bloch-Kato - see Huisbergen's book)
asked for verification in special cases.
Although there is some truth in what Houzel is saying, I would
be cautious with statements of such generality. Aubin's
choice of words indicates a similar position.
franz
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