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It seems that the focus of mathematical research moves on every so often, and some areas are not proven wrong, but have just become uninteresting in the current mathematical culture. I was under the impression that solutions to quartic (and high power) equations might be one of these abandoned area, but are there others?

This question has the current areas of mathematics mapped out, but what I'm looking for would be more the changes in the map over time, and which node are excluded from the current mapping.

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  • $\begingroup$ Welcome to SE.HMS! Can you provide more details about why you consider the resolution of quartic equations (and higher powers) as a major area of mathematics? The possibly to solve them with radicals? $\endgroup$ – Laurent Duval Aug 12 '16 at 11:06
  • $\begingroup$ I remember hearing that the volume of work in this area was massive (as judged by number of papers) but I don't have a citation for this. Perhaps we can define a major abandoned area of Mathematics as an area, that at one point in time consumed a large percent of total mathematical publication output, but no longer does, and hasn't been even mentioned in a while. $\endgroup$ – mboss Aug 12 '16 at 16:48
  • $\begingroup$ Queternions might be a decent example. People do still apply them to things, but they have a tiny fraction of the mind-share they had around 1900. $\endgroup$ – Ben Crowell Jan 5 '17 at 15:28
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Geometry

I'm not sure you can really call geometry abandoned, but it certainly was much more popular a few hundred years ago (discovery of spherical geometry and hyperbolical geometry, parallel axiom debate) and a few thousand years ago (the old Greeks developed a lot of geometry). Nowadays, there are very few papers about just plain Euclidean geometry (or spherical geometry, or hyperbolical geometry, for that matter). arXiv doesn't even have a specific category for it. They do have Algebraic Geometry, Metric Geometry and Differential Geometry, but these fields are no where near just studying the properties of triangles, circles, etc, while I think there are still things to discover. It might be partially a result of having only very few open problems.

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    $\begingroup$ This doesn't mean that geometry has been abandoned, it just means that progress has been made in geometry, so the frontier is now in a different place. $\endgroup$ – Ben Crowell Aug 18 '16 at 16:10
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    $\begingroup$ @BenCrowell But Algebraic Geometry, Metric Geometry and Differential Geometry are really quite different form Geometry. $\endgroup$ – wythagoras Aug 18 '16 at 16:21
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    $\begingroup$ Possibly a clarification on the usage of "Geometry" in a classical sense? I seem to see a heck of a lot of contemporary references to topology, which is geometric at heart. $\endgroup$ – DukeZhou Jan 24 '17 at 1:02
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I would say that no area of mathematics has ever been completely abandoned. The areas go in and out of fashion, but nothing seems to be completely abandoned. For example, approximately in 1940's most mainstream mathematical journals stopped to consider papers on elementary geometry. But the area is not abandoned in any way. First of all, there are "non-mainstream" journals, second, new results in elementary geometry can be parts of the research dedicated to other subjects.

Let me give my own paper as an example: Covering properties of meromorphic functions, negative curvature and spherical geometry, Ann. of Math. (2) 152 (2000), no. 2, 551-592, arXiv:math/0009251.

It is published in a mainstream journal, and its main subject belongs to analysis. However the core argument in this paper is elementary-geometric, and it contains new results in elementary geometry. And in the reference list of this paper you can see some mid 19 century elementary geometry books.

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  • $\begingroup$ This seems like a great example to me. Are there any other areas like elementary geometry that are currently out of fashion? $\endgroup$ – mboss Aug 19 '16 at 21:17
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    $\begingroup$ Not to such extent that papers are not accepted on the basis of the area, but some areas which used to be fashionable are clearly out of fashion (abstract potential theory, classification of open Riemann surfaces). On the other hand some long forgotten areas, like 1D dynamics experienced a strong revival after a 50 year period of complete oblivion. $\endgroup$ – Alexandre Eremenko Aug 19 '16 at 22:32
  • $\begingroup$ All the stuff in Whittaker and Watson's "A course in Modern Analysis" hasn't moved much in the last century, i.e. much of the special function associated with 2nd order ODE and Sturm-Liouville theory is not exactly hot. $\endgroup$ – user5245 Feb 4 '17 at 19:10
  • $\begingroup$ @ZeroTheHero: This is simply untrue. Search on the Mathscinet. Second order ODE and Sturm Liouville theory are quite hot still. Look at the citation statistics on Whittaker-Watson, at least! $\endgroup$ – Alexandre Eremenko Feb 4 '17 at 19:20
  • $\begingroup$ @AlexandreEremenko actually I took your advice and went to Web of Science. A topic search for "Sturm-Liouville" gives 221 journal papers In 2016. I would not call this a hot topic. Also, not all of these are published in math journals: CompPhysComm or Phys. Rev. D aren't really math journal. (Admittedly this is still more than I thought) We can haggle about journal category but 221 for the year doesn't seem that much, For comparison a similar topic search for "Lie algebra" yields 1018 journal papers in 2016, "Higgs" yields 1395 , while "dark matter" yields 2755. $\endgroup$ – user5245 Feb 4 '17 at 22:13
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I suppose the prime example would be classical invariant theory (CIT), although agreeing with Alexander, and despite exaggerated news of its death it had some bursts of revival recently. It is interesting to note that in the 19th century this subject was almost inseparable from classical elimination theory (CET). There has been a revival of CIT without CET (e.g., works of Dixmier, the recent book by Olver, etc.) as well as a revival CET without CIT (e.g., the book by Gelfand, Kapranov and Zelevinsky) but not really a revival of the combination CIT & CET.

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Another example is the study of configurations; its history is given in §1.2 of Branko Grünbaum's Configurations of Points and Lines, Graduate Studies in Mathematics volume 103, American Mathematical Society, 2009.

Broadly, this area of combinatorics, though not defined in full generality until 1876 by Theodor Reye, encompasses work of Pappus and Desargues. Other names associated with work up to 1910 include Möbius, Cayley, Burnside, and Steinitz. Then there was a "dark ages" until 1990 when Grünbaum and others reinvigorated the field. (I was fortunate to attend a topics class he taught on this material at the University of Washington in the early 1990s.) It is now an active area of research.

By the way, there is a statement about configurations from Hilbert & Cohn-Vossen's Geometry and the Imagination that Grünbaum considers an overstatement:

H & C-V: "...there was a time when the study of configurations was considered the most important branch of geometry."

G: "The author would like to conjecture that this is the greatest exaggeration of the truth that can be found in any of Hilbert's writings. While it is a fact that---as mentioned above---in the "classical period" of the history of configurations there were quite a few people interested in the topic, configurations were never a central topic of mathematical (or geometric) research."

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