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In history (for last 150 years), which math research fields were hot (popular) at their time , but whose results became insignificant (almost useless) later on?

The reason I ask this question is that my research group has a discussion on how to pick a research field (or topic) which will be important in long term, will have big impact in the future, and to avoid those topics which are popular now, but its results will become useless or abandoned in future.

We understand that there are research fields that people move on because all problems are solved. That is OK, as long as its research results are useful in future. We just try to avoid fields that are popular now, but their results or methods become useless in future.

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  • $\begingroup$ There was a surprising fad for catastrophe theory in the 1970's. I don't know if it is "insignificant", though. $\endgroup$ – MJD Sep 2 '14 at 23:09
  • $\begingroup$ There's likely to be a selection bias here in that relatively few people today will have heard of such fields. I'm interested to hear of good examples, though it's dubious that there will be general conclusions to be drawn applicable to making research choices today. $\endgroup$ – Kevin Carlson Sep 2 '14 at 23:22
  • $\begingroup$ Well...in the 1700s-1800s there was a big flurry of activity in trying to find numerically and computationally tractable and stable ways to determine longitude, given some measurements of star altitudes and the Greenwich Mean Time. It was pretty significant -- esp. the work of Marc St. Hilaire (did I spell that right?) --- for quite some time, but I think it's safe to say that the fad is over and done with at this point. $\endgroup$ – John Hughes Sep 2 '14 at 23:32
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    $\begingroup$ @JackM Speaking as someone who uses them on a daily basis, I'd disagree that quaternions became insignificant. They were 'abandoned' for a while, but they never stopped being useful. (And I feel like this question is essentially a duplicate of something I've read here before, but I'm hard-pressed to find any specific analogs) $\endgroup$ – Steven Stadnicki Sep 2 '14 at 23:47
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    $\begingroup$ I really like the spirit of this question, but I suppose most specialized fields you can pick that were once a fad later have turned out to be something ignored. Maybe a better question is what mathematical topics were fads in the last 50 years? $\endgroup$ – abnry Sep 3 '14 at 0:03
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Great question! I think that what drew me to mathematics in the first place was that mathematical results rarely become useless, just less used :-)

So, like the comments, I will try to focus on topics that, while still useful, are not active areas that would sustain a PhD candidate into their career (i.e., what horses were bad to hitch your proverbial wagon to?)

First, I think it's useful to classify dormant math fields according the underlying reason they are no longer active:

  1. Obsolete - Here I think of those poor folks who labored to produce ever more accurate numerical tables (e.g., logs, statistical distributions, nomograms) that we can now just punch into a computer and get more accurate answers.

  2. Overhyped - Here I'd put "chaos theory" (which was so "hot" that it got mention in Jurassic Park, of all places), and "Possibility theory"...which is supposed to be an alternative to probability theory that "fixes" its flaws. When it became apparent that Chaos Theory would not "tame chaos", nor would possiblity theory reduce risk (e.g., in building a bridge), a lot of support vaporized...although there is some useful work being done in both areas to some degree, but the research groups are pretty small compared to areas like "machine learning".

  3. Application-dependent: Unlike general mathematical theories, which are always relevant, highly applications-oriented math can actually become useless, although the lines blur between engineering and math for this area. For an example from my area: manufactured gas plants (turn coal into gas) had a lot of complex mathematical equations for predicting gas yield and such, but now that no one uses MGP's, that math is actually useless....for now.

  4. Sociological: Here I'd place the myriad of "smart manufacturing" fads that arose after the US got surprised by the Japanese manufactring superiority in the 1980's - 1990's. Example: six-sigma, total quality management, just-in-time....all useful, but we have a glut of these, and the shelf-life is dwindling at each iteration.

That brings me to my final comment: what is "useless" today may find reapplication later in a different context - that is the beauty of mathematics research. Think of the Euler-Beta Function in String Theory, or statistical mechanics and information theory...any "dead" of mathematics area can be resurrected by a fortuitous isomorphism. This is fundamentally different from the sciences, where, for example, it is unlikely that the theory of phlogiston or the aether (as metaphysical objects...not the associated math!) will make a comeback.

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  • $\begingroup$ Although the actual making of tables is obsolete, many of the mathematical techniques that were useful for this have continued to be useful, or have led to further developments that are now useful in numerical analysis. $\endgroup$ – Robert Israel Sep 3 '14 at 0:39
  • $\begingroup$ @RobertIsrael agreed...I was careful to choose my words here...Taylor Series' are a major tool of mine and play a vital role to this day. Its just hard to find truly useless mathematics... $\endgroup$ – user76844 Sep 3 '14 at 0:40

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