Consolidation: Aftermathematics of fads

Question

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 ?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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!

Answer 5

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?

Answer 6

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.

If I'm wrong about this, someone please point me to a reference!!!