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 ?

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This "long-term solitary activity" is more collaborative than ever. This quotation seems to be an extract from a very long polemic, and unsuitable as an MO topic. – Robin Chapman May 24 '10 at 10:50
@RC: Eh, I've seen worse. The excerpt is from a polemic (which gives me hope that it will be interesting to read), but the question itself does not strike me as especially polemical. – Pete L. Clark May 24 '10 at 11:33
This probably should be community wiki since you are asking for a list of examples. – Grétar Amazeen May 24 '10 at 13:23
Can representation theory be regarded as an example? My impression from hearing experts talk about it is that there was a big push in maybe the 40's or 50's, driven in part by considerations from physics, to classify all unitary representations of this and that. This seems to have generated a massive, nearly impenetrable literature of which only a handful of experts have a big picture view. The pattern seems to be that algebro-geometric methods have just replaced all the analysis from back then. I would be interested if someone more informed could elaborate on or qualify this in an answer. – Paul Siegel May 25 '10 at 23:34
A digression re the apposite pun "aftermathematics": pace "Fermat's Last Tango", the English word "aftermath" has nothing to do with mathematics; the "math" at the end of "aftermath" is an archaic derivative of "mow" (presumably as in grow > growth, long > length, etc.). – Noam D. Elkies Aug 23 '11 at 14:47

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

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.

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I know Richard Lyons at Rutgers was (as of a couple of years ago) still working on cleaning things up and making it as human readable as possible. I don't know how much of that's done, though. – Charles Siegel May 24 '10 at 15:52
People were aware of the Aschbacher-Smith preprint on quasi-thin groups for several years before its publication, and I think it was more of a cause of the resurgence of interest than it was "included" in the resurgence. Also, the second-generation project was already underway long before this resurgence. – S. Carnahan May 24 '10 at 16:47
Some of the "clearing out" you mention is simply young finite group theorists' inability to get jobs once math departments decided the main problem in their field had been resolved. I think this is an effect that is less internal to the field than you suggest. – S. Carnahan May 24 '10 at 16:53
@Scott: you may well be right. I hope I was clear that I was offering no explanation whatsoever for the phenomenon. On the other hand, that the main problem in one's field has just been solved and there is seemingly relatively little left to do does seem like an unusual phenonemon, and thus somewhat internal to finite group theory, n'est-ce pas? Or can you think of other instances of this? – Pete L. Clark May 24 '10 at 20:12
I remember an (apocryphal?) story about a student coming to Richard Borcherds (Scott C's advisor) with a proposal for a thesis problem, and Richard saying "oh, all the interesting problems there are solved". The student then asked, "But what about <<what Scott C is doing>>?", and Richard demurred that that had been the last interesting problem. – Scott Morrison May 27 '10 at 6:20

In the seventies and eighties of the preceding century, existence and classification of vector bundles on projective space $\mathbb P^n$ were all the rage, with contributions from such luminaries as Artin, Atiyah, Hartshorne and Mumford among many others. I have the feeling that not much progress has been made since.

For example, as far as I know, Hartshorne's apparently naïve question "Does there exist an indecomposable algebraic vector bundle of rank 2 on $\mathbb P^n_k \; ?\:"$ is still open for all fields $k$ and all integers $n\geq 6$.

Update[Next day] My colleagues André Hirschowitz and Arnaud Beauville, who are well informed about these questions, have allowed me to report that they feel quite confident that Hartshorne's question is still unsolved.

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Although in this case, it is not clear to me, if poor documentation stopped the work in the area or if people are just afraid that the problem is too hard for them to solve. – Konrad Voelkel Jun 20 '10 at 9:52
And this question adresses directly the statement: mathoverflow.net/questions/13990/… – Martin Brandenburg Aug 23 '11 at 8:30
@Martin Brandenburg: thanks for the links, Martin. – Georges Elencwajg Aug 23 '11 at 9:57

In email, Frank Quinn mentions that Surgery theory from the 1970s and 80s has a mostly primary literature aimed at other experts, a lack of textbooks, and now has few new people working on it.

To me, this seems like a similar problem to properly documenting a computer program as you go along so that others (and your future self) can understand it, otherwise coming back to it can require the same or greater effort to go through it, as was required to create it in the first place, but the temptation is to skimp on that, and just plough ahead.

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Novikov has written about the "Period of Decay" in the 70s and 80s as well here arxiv.org/abs/math-ph/0004012 – j.c. May 25 '10 at 20:56
+1 for the computer program analogy. This is a problem in software engineering that took decades for people to realise how important it is (i.e. a good ratio of documentation per code). How to do this the best way is an ongoing area of research. – Konrad Voelkel Jun 20 '10 at 9:49

It's funny you mention that physics can deal with this, because, as a physicist, I see the opposite of this all the time. I was actually just having a discussion with a friend the other day about how physics is desperately in need of cleaning up, organization, and consolidation! I think that a lot of mathematicians have this (wrong) impression of physics, though, because they tend to get their physics from books with titles like "Quantum Mechanics for Mathematicians." (Let me assure you most physicists would find such books largely incomprehensible!)

A big problem with the way physicists are currently educated is that there's a lot of needless redundancy, with topics presented in completely different ways, and with different methods to solve identical problems in different contexts for purely historical reasons.

For example, most physicists never realize that many of the tools they use in field theory are identical to tools used in general relativity. If they learn both, most have to learn the same tools twice and never realize they're identical because they look so different!

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It is a way different culture. Quinn is a Hilbert fan, duplication in presentation is lèse-Bourbaki, and the argument being put forward makes sense in those terms too (no Bourbaki Seminar on an area, need it exist at all?) Well, I may be too harsh here, but as Tom Körner once explained to me, the academic politics of euthanasia for topic areas is quite different from that of natural death. Physics does have better surveys, and it also has a turnover period of c. 3 years rather than a decade. Classical analysis and "natural death" conservatism go together, in a debate active since 1945. – Charles Matthews May 26 '10 at 13:45
This is an interesting answer. I had the following (somewhat tangential) reaction to it: learning "different methods to solve identical problems" does not necessarily sound like a waste of time to me at all! If anything it may be less of a waste of time than applying the same method to superficially different problems. But from the next paragraph it seems that you mean that physicists learn superficially different methods several times as if from scratch without realizing their essential similarity. Yes, that sounds like a waste of people's time. – Pete L. Clark Aug 23 '11 at 15:48

One thing that comes directly to mind is the calculus of variations, in the classical sense, where the point is to get rigorous results by mathematical analysis.

Now, there are probably several typical kinds of objection here. Firstly the area really is not dormant: physicists use it in the same fashion as ever; there are various kinds of "variational formalism" discussed, for example in soliton theory; and mathematicians have "gone round" this area by the use of Morse theory and moment maps, to break out of the traditional formulation into areas of geometry.

But in terms of identifying a "break in tradition" (I don't entirely agree with Quinn's framing of the issue, but it is real enough when those who wrote the papers are no longer around) I would guess there is no line of textbooks that continues from the early twentieth century treatments. Few people may know what was considered important in that line of development. I'm aware of work on variational problems (e.g. the Plateau problem) that is pretty much current, but that illustrates one tendency, to make a given problem into a theory of its own. Anyway, do mathematicians in general know why Jesse Douglas got a Fields Medal in 1936? How many could read his papers?

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It may be unpopular to say this, but the theory of subfactors needs a consolidating account. (I don't think of this theory as a fad, but I do think it may be in danger of being difficult to unravel without some consolidating effort.)

This is a major reason why I asked the question here.