Which small area of mathematics had fully developed already and thus no more research in this area?

Question

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.

Answer 1

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.

Answer 2

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.