I studied applied math, so each course (except abstract algebra) was dedicated to solution of a similar problems. After those courses it seems that every branch of mathematics has a developed theory with a few unsolved problems. I believe that is wrong. I found out that there is no general approach to Diophantine equations (the 10th Hilbert problem), to problems like "Collatz conjecture" and to nonlinear differential equations.

And so the question is: what is the other branches of mathematics (collections of similar problems) without a general method to solve them?

  • $\begingroup$ There's still no unified way to represent solutions to transcendental equations in closed form, IIRC. $\endgroup$ – J. M. is a poor mathematician Sep 20 '10 at 21:53
  • $\begingroup$ You are mixing two things: being «underdeveloped» and «not having a general method to solve» are completely different things! $\endgroup$ – Mariano Suárez-Álvarez Sep 21 '10 at 11:07
  • $\begingroup$ @Mariano Suárez-Alvarez, thank you I've change the title. $\endgroup$ – rystsov Sep 21 '10 at 12:16

Diophantine equations are not special; any branch of mathematics in which it is possible to ask sufficiently strong questions will suffer a similar fate due to the existence of problems like the halting problem and the possibility of asking a question equivalent to the halting problem (or something similar). For example, group theory has such problems: it is generally impossible to decide from a presentation of a group whether it is trivial or not. See this list of problems on MathOverflow.

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    $\begingroup$ This is not "undeveloped" so much as "undevelopable." Mathematics is much less complete than the average undergraduate education would have you believe! $\endgroup$ – Qiaochu Yuan Sep 20 '10 at 22:01

There is no general method for solving differential equations. There are general methods for proving existence and uniqueness of solutions, but not for computing solutions (if that's even possible) or for studying detailed properties of solutions.

You can make a tiny change to a differential equation and it can go from being well understood to being an area of research. Or it can go from something one group of mathematicians study to something another group studies.

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    $\begingroup$ Even from the numerics viewpoint, it only takes a tiny change in the initial conditions for some DEs to see a change in the behavior of the solutions. Lorenz's set comes to mind. $\endgroup$ – J. M. is a poor mathematician Sep 21 '10 at 14:28
  • $\begingroup$ If there are general methods for proving existence and uniqueness of solutions, why is Navier–Stokes existence and smoothness problem still unsolved? $\endgroup$ – Ruslan Dec 24 '15 at 17:41

Irrationality of certain numbers is still though problem for mathematics. Even though irrationality of some numbers like square roots of integers that are not perfect squares and irrationality of $\pi$ and $e$ has long been known, it is still unknown whether $\pi + e$, $2^e$, $\pi^{e}$, $\pi^{\sqrt{2}}$, $\pi \cdot e$ , $\pi / e$, Catalan's constant, Euler-Mascheroni constant, etc. are irrational or not, even though all are highly suspected to be.


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