From Frank Quinn's THE NATURE OF CONTEMPORARY CORE MATHEMATICS: "Mathematics has occasional fads, but for the most part it is a long-term solitary
activity.
In consequence the community lacks the customs evolved in physics to deal with
the aftermathematics of fads. If mathematicians desert an area no one comes in
afterwards to clean up.
Lack of large-scale cleanup mechanisms makes mathematical areas vulnerable
to quality control problems. There are a number of once-hot areas
that did not get cleaned up and will be hard to unravel when the developers
are not available. Funding agencies might watch for this and sponsor
physics-style review and consolidation activity when it happens."
Can you give examples of such once-hot areas in need of consolidation ?
Best Answer
Since Quinn's article is a long opinion piece which he says is 90% complete and welcomes comments, it seems entirely appropriate to contact him for clarification on this point. He would probably be happy to tell you more.
One example that springs immediately to my mind is the classification of finite simple groups. This was, by a safe margin, the largest scale collaborative activity in the history of mathematics, taking place over a decade or so. The accounts I have read describe Aschbacher, Thompson and (especially) Gorenstein as acting like army generals overseeing a war: they had the most insight into the global structure of the argument and they used it to apportion and subcontract various pieces of the proof. So far as I can think of at the moment, it is much more usual for a visionary mathematician (e.g. Langlands, Thurston, Hamilton) to lay out a program which other mathematicians are then inspired to work on as they see fit than to have this kind of explicit top-down organization.
The rest of the story is well-known: in the early 80's Aschbacher, Thompson and Gorenstein were photographed on an aircraft carrier in front of a victory banner (figuratively speaking of course) and all the other group theorists shouted hurrah and cleared out. But certain key parts of the argument had never been published in any form, as a small number of mathematicians (e.g. Serre) spent the next 20 years reminding the community. It seems fair to say that the finite group theorists cleared out a little too early. I don't really know why or exactly what motivated the recent moderate resurgence of interest in the classification, including the 2004 (!) publication of a two-volume work completing the quasi-thin case (a mere 1300 additional pages were required). In the last few years it seems that there has been "the right amount" of tidying up these massive argument by those involved in the "second generation" and "third generation" classification efforts.
See
http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
and the references therein for more details. Especially highly recommended is Aschbacher's 2004 Notices article
http://www.ams.org/notices/200407/fea-aschbacher.pdf
which, in addition to being gracefully written and informative, is admirably forthright.