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Inspired by the comparison of programming languages by GitHub and Stack Overflow activity (e.g. this one for 2015) I decided to look at the popularity of mathematical disciplines by using data from both arXiv and MathOverflow (see also my motivation for getting a dump of arXiv metadata). Here it is:

arXiv vs MathOverflow - popularity of disciplines

It's based on all data until January 2015. For MO there is little dependence of popularity of topics over time; for arXiv there is some, but it does not change the plot in any drastic way (due to arXiv growth the lifetime average is close the the average from the last 5 years).

There is some (positive) correlation between the number of questions here and the number of preprints on arXiv, per discipline. Of course, there are many factors that come into play:

  • the number of people interested in a given field,
  • coverage of people (not all mathematicians are here, not all - are posting to arXiv),
  • difficulty to ask a question,
  • difficulty to write a paper,
  • etc.

I am not a (real) mathematician, and not even a frequent MO user; so I may be missing some explanations, which are obvious for everyone in a given field.

Do you know plausible explanations why certain fields lie above, or below, the regression line?

(I have some guesses, but don't want to mix this answer with a very partial response.)

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    $\begingroup$ For various topics below the line in your figure, there are concurring sites -- e.g. math.SE, stats.SE, physics.SE, Physics Overflow, cstheory.SE, while -- besides mailing lists -- there are likely not so many good alternative places to ask advanced questions on, say, group theory, number theory or algebraic geometry. $\endgroup$
    – Stefan Kohl Mod
    Commented Jul 22, 2015 at 12:46
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    $\begingroup$ What's the significance of the dark gray area? $\endgroup$
    – Gerry Myerson
    Commented Jul 22, 2015 at 12:56
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    $\begingroup$ @GerryMyerson Confidence interval (at 95%) for the linear regression parameters (vide docs.ggplot2.org/current/stat_smooth.html). $\endgroup$
    – Piotr Migdal
    Commented Jul 22, 2015 at 14:04
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    $\begingroup$ When you count "logic" on MathOverflow, do you also count [set-theory] and other related tags which often appear without the [lo.logic] tag? $\endgroup$
    – Asaf Karagila Mod
    Commented Jul 22, 2015 at 14:16
  • $\begingroup$ @AsafKaragila lo.logic -> logic. I use only arXiv tags. $\endgroup$
    – Piotr Migdal
    Commented Jul 22, 2015 at 14:37
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    $\begingroup$ mathoverflow.net/questions/6292 $\endgroup$
    – Steve Huntsman
    Commented Jul 22, 2015 at 14:41
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    $\begingroup$ I think that you're missing out on a substantial part of MathOverflow, then. I think that including more tags (which I'm sure are relevant to other fields, not just logic) might change the outcome, perhaps for more than one topic. $\endgroup$
    – Asaf Karagila Mod
    Commented Jul 22, 2015 at 15:05
  • $\begingroup$ @AsafKaragila I stick to arXiv tags as otherwise mapping would be hard, inconsequential or not 1-1. Of course, it is possible that different fields have different levels of diligence (use of at least 1 arXiv tag is highly advised). $\endgroup$
    – Piotr Migdal
    Commented Jul 22, 2015 at 15:29
  • $\begingroup$ Out the the ~2100 set theory questions, only ~750 have the logic tag. This means that however many questions that you counted under logic, there are significantly more. And I am more than certain this situation is not uncommon in other fields. The arXiv tags are not as used as they were five years ago. For better and for worse. $\endgroup$
    – Asaf Karagila Mod
    Commented Jul 22, 2015 at 15:33
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    $\begingroup$ @AsafKaragila So, do you have a better idea to make a systematic mapping of questions to arXiv fields? (Some may be to more than one, some may be to none - especially not research-level questions (which are discouraged, but happen nonetheless).) $\endgroup$
    – Piotr Migdal
    Commented Jul 22, 2015 at 15:55
  • $\begingroup$ It should be possible to poll which tag have only set theory, and not lo.logic, I guess. And you should be able to have these sort of tag grouping. Like one or two dominant tags for each topic. $\endgroup$
    – Asaf Karagila Mod
    Commented Jul 22, 2015 at 16:01
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    $\begingroup$ @AsafKaragila Sure, I can single out set-theory with no effort. But what about all other tags? (And I prefer to have a uniform undercounting rather than manual and subjective case-by-case approach, generating artifacts.) $\endgroup$
    – Piotr Migdal
    Commented Jul 23, 2015 at 7:31
  • $\begingroup$ Piotr, I'd write a code that accepts a list of tags. Then poll the community for help, or at least look what are the top 1-2 tags used in conjunction with each arXiv tag. But you should also take into account what quid wrote in the answer below, especially the last point. $\endgroup$
    – Asaf Karagila Mod
    Commented Jul 23, 2015 at 8:05
  • $\begingroup$ A different option along the lines of what @Asaf proposes: go through the list of tags by popularity. Those that seem to fit clearly an arXiv category take into account there (avoidind double counts); forget the rest; stop when you are tired of it. Like, real-analysis should go to ca.analysis-and-ode graph-theory to co.combinatorics; yet matrices is likely too vague to go anywhere. $\endgroup$
    – user9072
    Commented Jul 23, 2015 at 10:33
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    $\begingroup$ its natural for there to be a difference & to expect one, because mathoverflow and stackexchange in general is not really a site for presenting new research aka terra incognita, but instead to work with existing research aka terra cognita. short Q/A by experts/ general internet audience is much different than long-form papers by experts/ academics. one would expect the users/ audiences to be somewhat different/ overlapping also. $\endgroup$
    – vzn
    Commented Jul 23, 2015 at 20:57

2 Answers 2

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Some observations:

  • Some subjects seem to draw more amateur-interest and also "idle curiosity" from professionals than others. (Number theory, foundational questions (logic/set-theory), questions on history come to mind, which are all over-represented and as Asaf remarked "logic" could have more questions on MO if just the tag was used).

  • Some subjects are "fundamental" in the sense that researchers in another field might have use for some technical result and ask about it while they hardly would consider writing paper in that field. (General topology, and maybe category theory comes to mind.) Relatedly there are some subjects that are more common in graduate curricula than others.

  • MO always was a bit biased towards more pure math (see the link by Steve Huntsman), and the relative lack of applied analysis is well-known.

  • The existence of other sites as mentioned by Stefan Kohl (information theory, statistics, and mathematical physics might be affected by this, and in addition to the preceding point).

  • Technical artifacts. Like, arXiv enforces one of those categories, MO does not, and (thus?) hardly anybody tags general mathematics while on arXiv it is sometimes used. Or, consistency and clarity of the use of the tag (e.g., algebraic geometry might benefit from this).

It is hard to tell what is rationalization of the data, as I saw it; but I would have said some of the things even without the data.

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I guess that MO is more forgiving about discussing curious problems, or recreational mathematics, such as Stanleys question about certain chess positions for example. Problems which are easy to state without a lot of overhead are easier to post on MO, and more people can chip in on these. This might explain why discrete math (combinatorics) and algebraic geometry are prominent. Every mathematician know (or should know) what a graph, a permutation and what a polynomial is.

Then number theory of course, draws attention for a similar reason; primes are understood by everyone and is a hot amateur mathematical topic. Same with logic, which attracts a lot of amateur mathematicians.

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