TITLE: On progress in mathematics: some long-open problems and long-standing conjectures
QUESTION [10 upvotes]: I would like to ask a question here on Math Stack Exchange taking inspiration (and therefore combining) from two well-known threads on MathOverflow: (1)
Not especially famous, long-open problems which anyone can understand; (2)
The resolution of which conjecture/problem would advance Mathematics the most?
My question is:

Could you list (giving a sound motivation and reference papers) some
  long-standing conjectures or long-open problems in mathematics (or even in mathematical physics) that are intrinsically interesting and 
(1) are not
  so "mainstream" (by which I mean: there is no need to mention the
  Millenium Problems or Riemann Hypothesis or similarly famous questions); 
(2) are extremely important
  because their solution would imply a major progress in an area of
  mathematics and also in mathematics as a whole; 
(3) can be stated in some appropriate, but reasonably terse, form
  without involving extremely abstruse concepts and terms;
(4) have been object of some (even slight) progress towards a solution
  in recent years.

REPLY [3 votes]: Given a closed topological manifold $M$, we can ask  whether $M$ admits a smooth structure and whether that structure is unique. It's easy to show that it's the case in dimensions $1$ and $2$, and it's nontrivial but also true in dimension $3$. It fails in dimension $4$, but there has been a lot of progress on trying to determine which $4$-manifolds are smoothable. Take $M$ to be a simply-connected, closed $4$-manifold. For purely algebraic-topological reasons (although there's also a straightforward geometric proof; see, for example, Scorpan's very readable Wild World of $4$-Manfiolds), $M$ is determined up to homotopy equivalence by its intersection form $i_M:H_2(M) \otimes H_2(M) \to H_4(M) = \mathbb{Z}$. (Since $\pi_1 M = 0$, the manifold $M$ is clearly orientable, and we take it to be oriented.) There's another invariant, called the Kirby-Siebenmann invariant $\operatorname{ks}(M)\in H^4(M)$, that's an obstruction to smoothability. If we assume that $\operatorname{ks}(M)$ vanishes, we can now ask whether which forms $i_M$ correspond to smooth manifolds.
The definite case is taken care of by results of Freedman and Donaldson from the '$80$s: If $i_M$ is definite, then $M$ is smoothable iff $i_M$ is diagonalizable. (Recall that we're working over $\mathbb{Z}$ throughout.) If $i_M$ is indefinite and odd, then it's automatically smooth. The case of $i_M$ indefinite and even, though, is currently unknown. Such an $i_M$ must be of the form $i_M = H^{\oplus n} \oplus E^{\oplus 2m}$, where $H$ and $E$ are certain forms with
\begin{align*}
\dim H &= 2 & \sigma(H) &= 0;\\
\dim E &= 8 & \sigma(E) &= \pm 8
\end{align*}
(The sign of the signature $\sigma(E)$ depends on some orientation conventions that I've been too lazy about above to define rigorously now.) If $m \leq 2n$, then it's known that $M$ is not smoothable. If $m\geq 3n$, then it's known that $M$ is smoothable. The general case, though, is unknown. (There are a few other pairs $(n, m)$ for which the answer is known.) Since $\dim i_M = 2n + 16m$ and $\sigma(i_M) = \pm 16m$, we can restate the results above as saying that $M$ is smoothable if $\dim i_M \geq \frac{11}{8} |\sigma(M)|$ and not smoothable if $\dim i_M \leq \frac{10}{8} |\sigma(M)|$. The $11/8$-conjecture states that the $\frac{11}{8}$ is the best constant possible in the former half of the previous sentence: any $M$ with $\dim M < \frac{11}{8} |\sigma(M)|$ has no smooth structure. 
What makes this particular situation so interesting is that a lot of the tools available in higher dimensions--- the $h$-cobordism theorem, handlebody decompositions, surgery theory, etc.--- don't apply or aren't as pervasively useful in dimension $4$. On the other hand, the low dimension simplifies matters, as in the determination of $M$ up to homotopy equivalence by just the form $i_M$. There are quite a bit of odd phenomena that are exclusive to dimension $4$, such as exotic $\mathbb{R}^4$s. The $\frac{11}{8}$-conjecture is an attempt to figure out what exactly is going on in these odd cases.