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Which small area of mathematics had fully developed already and thus no more research in this area? For example, no more PHD research in Euclidean Geometry anymore.

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    $\begingroup$ Google open problems in euclidean geometry. $\endgroup$
    – Lucian
    Feb 12, 2014 at 2:55

3 Answers 3

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Some additional examples that may be of interest:

  1. Spherical trigonometry.
  2. Quaternion analysis (in the 19th century a rather hot and active area of research).
  3. Synthethic projective geometry (in the tradition of Steiner and Poncelet).
  4. Analytical theory of generalized continued fractions.
  5. Geometry of straight-line linkages (popular from the 1870s to the 1890s); mathematical kinematics in general.

Personally, I'd say quaternion analysis is the most spectacular example of an area that was once very fashionable and now all but forgotten.

All these things are relative, of course. It would be very hard, I think, to find mathematical disciplines in which the amount of active research is exactly zero.

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    $\begingroup$ Some of this list is just plain wrong. For instance, enumerative geometry is absolutely flourishing in the last 20 years thanks to connections to the string theory (just google "Gromov-Witten invariants" or "quantum cohomology"). Item 6 is also alive and well, just check out en.wikipedia.org/wiki/Carpenter%27s_rule_problem. $\endgroup$ Jan 10, 2015 at 23:02
  • $\begingroup$ I stand corrected on the enumerative geometry. Removed it. $\endgroup$ Jan 10, 2015 at 23:08
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    $\begingroup$ Permit me to differ re linkages. See, e.g., the 2011 book, Geometric Design of Linkages (Springer link.) $\endgroup$ Jan 11, 2015 at 0:41
  • $\begingroup$ And linkages should be also removed from your list; just take a look at Joseph's link or Igor Pak's book math.ucla.edu/~pak/geompol8.pdf or simply at the list of publications of Robert Connelly at Cornell University! $\endgroup$ Jan 11, 2015 at 0:52
  • $\begingroup$ That's why I added the caveat about such things being relative (and more or less subjective too). I do not for a moment doubt that there are still books being written about linkages or that some people still research them. The same can be said about classical Euclidean geometry, though (Floor van Lamoen comes to mind). $\endgroup$ Jan 11, 2015 at 1:05
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Here are some "fully developed areas" (and not having PhDs awarded in), which once upon a time were active research areas:

  1. Squaring the circle and trisecting the angle using compass and ruler.

  2. Proving the 5th postulate from the rest of Euclidean axioms.

  3. Proving Fermat's last theorem using elementary algebra/number theory.

  4. More recent: constructing homotopy 3-spheres which are not homeomorphic to the standard 3-sphere.

(Of course, areas described in items 1, 2 and 3 are very still active among math cranks.)

I can go on with examples, but I think, you got my point by now.

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No one area in mathematics can be considered as fully developed. This is because some day, some one will come up with some brilliant ideas that will

extend our research to a new dimension; or even disprove what we have accepted as must-be-true.

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    $\begingroup$ This is a comment, not an answer. $\endgroup$
    – Emily
    Feb 12, 2014 at 4:40
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    $\begingroup$ @Arkamis: This is also a comment, not an answer. :-) $\endgroup$
    – Lucian
    Feb 12, 2014 at 6:33
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    $\begingroup$ This 'answer' says "no one area.....". It gives a direct answer to the question "which area...." $\endgroup$
    – Mick
    Feb 12, 2014 at 9:46
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    $\begingroup$ It's not an answer because it bears no substance, no supporting evidence, and bears nothing but an opinion predicated on no apparent foundation. $\endgroup$
    – Emily
    Feb 12, 2014 at 15:34
  • $\begingroup$ From simple complex numbers, Euler came up with a formula that relates complex exponential function with trigonometric functions. Fifty years later, Caspar Wessel described the geometrical interpretation of the complex numbers as points in the complex plane. But before this, none of the mathematician realized it. $\endgroup$
    – Mick
    Feb 13, 2014 at 3:43

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