Fundamental problems whose solution seems completely out of reach In many areas of mathematics there are fundamental problems that are embarrasingly natural or simple to state, but whose solution seem so out of reach that they are barely mentioned in the literature even though most practitioners know about them. I'm specifically looking for open problems of the sort that when one first hears of them, the first reaction is to say: that's not known ??!! As examples, I'll mention three problems in geometry that I
think fall in this category and I hope that people will pitch in either more problems of this type, or direct me to the literature where these problems are studied.
The first two problems are "holy grails" of systolic geometry---the study of inequalities involving the volume of a Riemannian manifold and the length of its shortest periodic geodesic---, the third problem is one of the Busemann-Petty problems and, to my mind, one of the prettiest open problems in affine convex geometry.
Systolic geometry of simply-connected manifolds. Does there exist a constant $C > 0$ so that for every Riemannian metric $g$ on the three-sphere, the volume of $(S^3,g)$ is bounded below by the cube of the length of its shortest periodic geodesic times the constant $C$?
Comments.

*

*For the two-sphere this is a theorem of Croke.

*Another basic test for studying this problem is $S^1 \times S^2$. In this case the
fundamental group is non-trivial, but in some sense it is small (i.e., the manifold is not essential in the sense of Gromov).

*There is a very timid hint to this problem in Gromov's Filling Riemannian manifods.

Sharp systolic inequality for real projective space. If a Riemannian metric in projective three-space has the same volume as the canonical metric, but is not isometric to it, does it carry a (non-contractible) periodic geodesic of length smaller than $\pi$?
Comments.

*

*For the real projective plane this is Pu's theorem.

*In his Panoramic view of Riemannian geometry, Berger hesitates in conjecturing that this is the case (he says it is not clear that this is the right way to bet).

*In a recent preprint with Florent Balacheff, I studied a parametric version of this problem. The results suggest that the formulation above is the right way to bet.

Isoperimetry of metric balls. For what three-dimensional normed spaces are metric balls solutions of the isoperimetric inequality?
Comments.

*

*In two dimensions this problem was studied by Radon. There are plenty of norms on the plane for which metric discs are solutions of the isoperimetric problem. For example, the normed plane for which the disc is a regular hexagon.


*This is one of the Busemann-Petty problems.


*The volume and area are defined using the Hausdorff $2$ and $3$-dimensional measure.


*I have not seen any partial solution, even of the most modest kind, to this problem.


*Busemann and Petty gave a beautiful elementary interpretation of this problem:
Take a convex body symmetric about the origin and a plane supporting it at some point $x$. Translate the plane to the origin, intersect it with the body, and consider the solid cone formed by this central section and the point $x$. The conjecture is that if the
volume of all cones formed in this way is always the same, then the body is an ellipsoid.
Additional problem: I had forgotten another beautiful problem from the paper of Busemann and Petty: Problems on convex bodies, Mathematica Scandinavica 4: 88–94.
Minimality of flats in normed spaces. Given a closed $k$-dimensional polyhedron in an $n$-dimensional normed space with $n > k$, is it true that the area (taken as $k$-dimensional Hausdorff measure) of any facet does not exceed the sum of the areas of the remaining facets?
Comments.

*

*When $n = k + 1$ this is a celebrated theorem of Busemann, which convex geometers are more likely to recognize in the following form: the intersection body of a centrally symmetric convex body is convex. A nice proof and a deep extension of this theorem was given by G. Berck in Convexity of Lp-intersection bodies, Adv. Math. 222 (2009), 920-936.

*When $k = 2$ this has "just" been proved by D. Burago and S. Ivanov: https://arxiv.org/abs/1204.1543

*It is not true that totally geodesic submanifolds of a Finsler space (or a length metric space) are minimal for the Hausdorff measure. Berck and I gave a counter-example in What is wrong with the Hausdorff measure in Finsler spaces,  Advances in Mathematics, vol. 204, no. 2, pp. 647-663, 2006.

 A: Gelfand's problem: can you find a closed, proper unital subalgebra A of C[0,1] such that the natural map from [0,1] to the character space of A is bijective?
(See for instance these notes of Feinstein )
A: Is every complemented subspace of $C[0.1]$ isomorphic to $C(K)$ for some compact metric space $K$?
Is every infinite dimensional complemented subspace of $L_1[0.1]$ isomorphic either to $L_1[0.1]$ or to $\ell_1$?
A: A proof of this conjecture of Erdos would certainly turn heads, raise eyebrows, and garner the attention of the Fields Medal committee.


*

*If $\sum_{a \in A} \frac 1a$ diverges and $A\subseteq {\mathbb N}_{>0}$, then $A$ contains a 3-term arithmetic progression.


Probably "diverges" can be replaced with "is bigger than 4".
A: Is every  algebraic curve in $\mathbb P^3$  the set-theoretic intersection of two algebraic surfaces ? Not known!
A: Is every finite lattice a congruence lattice of a finite (universal) algebra? 
Astonishingly, by Pálfy and Pudlák, this question is equivalent to a question in group theory: is every finite lattice isomorphic to an interval of the subgroup lattice of a finite group? 
A: This comes up in Waring's Problem, but it is so freakishly simple that it has taken on a life of its own. Let $\{ x \} = x \mod 1 = x-\lfloor x \rfloor$ be the fractional part of $x$.

*

*Say anything about the sequence $\{ (3/2)^n \}.$
Computations support the thought that the sequence should uniformly distributed in $[0,1]$, as for almost all $x$ the sequence $\{x^n\}$ is u.d. But with $x=3/2$, there is no value known to be a limit point, nor any value known to not be a limit point, it's unknown if there are two limit points, unknown if the sequence is infinitely often in $[0,1/2)$, or that it is infinitely often not in $[0,1/2)$. Really, nothing is known.
Edit: it is known that there are infinitely many limit points! See the comments below for citations.
As a final comment on this problem, the golden ratio is special. With $x=\phi=(1+\sqrt 5)/2$, for every $\epsilon>0$ there are only finitely many $n$ with $$\epsilon< \{\phi^n \} < 1-\epsilon.$$
A: Here is a variation of Georges Elencwajg's question, due to Gennady Lyubeznik. Is every closed point (of arbitrary degree over $\mathbb{Q}$) in $\mathbb{P}^2_{\mathbb{Q}}$ set-theoretically the intersection of two curves?
A: Let $G$ be a finite group. We define $r(G)$ to be the smallest number of relations possible in a presentation of $G$ with the minimal number of generators. If $G$ is a $p$-group, we can also consider "pro-$p$ presentations" of $G$ (using the free objects in the category of pro-$p$ groups); we write $r_p(G)$ for the smallest number of relations possible in a pro-$p$ presentation with the minimal number of generators.
Does $r(G) = r_p(G)$?
A: It is still not known whether the problem of determining whether a linear integer recurrence (of which the Fibonacci recurrence $F_n = F_{n-1}+F_{n-2}$, $F_1=F_0=1$ is the most well known) contains a zero is decidable or not.  Even the case of recurrences of depth 6 is currently open.  (I discussed this problem at http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/ .)  We do have the famous Skolem-Mahler-Lech theorem that gives a simple criterion as to when the number of zeroes is finite, but nobody knows how to get from that to deciding when there is a zero at all.  (This is perhaps the simplest example of a large family of results in number theory in which one has an ineffective finiteness theorem for the number of solutions to a certain number-theoretic problem (in this case, an exponential Diophantine problem), but no way to determine if a solution exists at all.  Other famous examples include Faltings' theorem and Siegel's theorem.)
EDIT:  See also this survey of Halava-Harju-Hirvensalo-Karhumäki from 2005 on this problem: http://tucs.fi/publications/view/?id=tHaHaHiKa05a
A: Artin's Conjecture: There are infinitely many primes $p$ for which 2 is a primitive root, i.e., 2 generates the multiplicative group of ${\mathbb Z}/p{\mathbb Z}$.
The conjecture is actually a bit more general, but we should at least be able to say what happens with 2! The OEIS lists the first several such primes.
A: Can we exactly calculate Ramsey numbers? Erdős once famously remarked:

"Suppose aliens invade the earth and threaten to obliterate it in a year's time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world's best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack."

A: Is there an algebraic irrational number in the Cantor set?  
More generally: Are algebraic irrational numbers normal in all bases?  
A: Normality of numbers. Is Pi normal?
A: Here are two from ergodic theory:


*

*(The problem of smooth realizations) Let $X$ be a Lebesgue space with measure $\mu$, and let $T:X\to X$ be a transformation preserving the measure $\mu$. If the entropy $h_\mu(T)$ is finite, is $(X,T,\mu)$ always measurably isomorphic to a smooth system $(M,f,v)$, where $M$ is a compact manifold, $f$ is a diffeomorphism of $M$ and $v$ is a smooth volume?

*(Furstenberg's $\times 2 \times 3$ problem) Does there exist a Borel probability measure $\mu$ on the unit circle $\mathbb{R}/\mathbb{Z}$, which is neither discrete nor Haar measure, and which is invariant under both $x\to 2x \bmod 1$ and $x\to 3x\bmod 1$?
For the first problem, as far as I know there has been no significant progress. 
For Furstenberg's conjecture, Furstenberg himself solved the analog question for sets (answer is negative), and Rudolph proved that the answer is negative under an extra positive entropy assumption. While there has been a huge amount of progress in the positive entropy case since, the zero entropy case remains untractable despite the simplicity of the statement. 
A: Does $H^\infty(D)$, the Banach space of all bounded holomorphic functions on the unit disc, have Grothendieck's approximation property?
A: Here is an old question by Borel: is there any a priori growth restriction on entire functions $f(z)$ satisfying polynomial differential equations $P(z,f(z),\dots,f^n(z))=0$ where $P$ is a polynomial with complex coefficients in $n+2$ variables? 
A: Chromatic Number of the Plane (Hadwiger-Nelson Problem): What is the minimum number of colors required to color the plane so that no two points which are unit distance apart are the same color? Let $\chi$ denote this number. The current bounds on $\chi$ are 
$$4\leq \chi \leq 7$$.
A: Every finite abelian group is  (isomorphic to) the class group of the ring of algebraic integers of some number field.
Some comments:
For Dedekind domains this is well-known (even for any abelian group); due to Claborn and Pete L. Clark has an alternate proof/a refinement.
Also a 'geometric analog' is known (Perret, 1999). 
And every finite ablian group is at least a subgroup of a classgroup (even for a cyclotomic field).
It can also be shown that, for a fixed prime $p$, every finite abelian $p$-group is the $p$-Sylow of the class group of the ring of algebraic integers of some number field (by Yahagi, Tokyo J. of Math 1978) and that every finite $p$-group is the Galois group of the maximal unramified $p$-extension of a number field (Ozaki, Inventiones 2011); note that this Galois group coincides with the class group if one adds the condition that it be abelian, by Class Field Theory.
ps. Not sure this passes all (or any) of the criteria; I'll let you decide :)
ps2. Searching for a reference, I found this math.SE question on exactly this https://math.stackexchange.com/questions/10949/finite-abelian-groups-as-class-groups
A: The Bunyakovsky conjecture (or Bouniakowsky conjecture) stated in 1857 by the Russian mathematician Viktor Bunyakovsky, claims that 

  
*
  
*an irreducible polynomial of degree two or higher with integer coefficients generates for natural arguments either an infinite set of numbers with greatest common divisor (gcd) exceeding unity, or infinitely many prime numbers.
  

A: Are there infinitely many regular primes? We know there are infinitely many irregular ones, and that their percentage should be much smaller than the regular ones, still it is unproven that the latter are infinite.
Let me recall that a prime $p$ is irregular if it divides the class number of $\mathbb{Q}(\zeta_p)$, the cyclotomic field.
Similarly, we cannot prove that there are infinitely many real quadratic fields of class number $1$.
A: An obvious problem in algebraic topology would be the computation of the homotopy groups of spheres.
A: Sendov's Conjecture

For a polynomial $$f(z) = (z-r_{1}) \cdot (z-r_{2}) \cdots (z-r_{n}) \quad \text{for} \ \ \ \ n \geq 2$$ with all roots $r_{1}, ..., r_{n}$ inside the closed unit disk $|z| \leq 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point of $f$.

A: Schinzel-Sierpinski Conjecture
Taken from this MathOverflow link.
Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) states the following:


*

*A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can be represented as a quotient of shifted primes, that $x=\frac{p+1}{q+1}$ for primes $p$ and $q$. It is known that the set of shifted primes, generates a subgroup of the multiplicative group of rational numbers of index at most $3$.

A: From the Overview of the Royal Danish Sciences Institution's work and its members' work in the year 1882.
In the notes from a meeting on March 9th 1877, after discussing papers by Legendre, J. W. L. Glaisher, and Meissel,  Oppermann stated:

At the same occasion, I made people aware of the not yet proven conjecture, that when $n$ is a whole number $>1$, at least one prime number lies between $n(n-1)$ and $n^2$ and also between $n^2$ and $n(n+1)$.

A solution to Oppermann's Conjecture leads to simple solutions to Legendre's, Brocard's, and Andrica's Conjectures.  
Proof of Andrica assuming Oppermann
A: The Whitehead asphericity conjecture.  Let $X$ be a $2$-dimensional aspherical simplicial complex and let $Y \subset X$ be a connected subcomplex.  The conjecture then is that $Y$ is aspherical.
Very little is known about this, but a deep theorem of Bestvina and Brady says that the Eilenberg-Ganea conjecture and the Whitehead asphericity conjecture cannot both be true.
A: I'm not sure what your threshold for "barely mentioned in the literature" is, since some of the highly-voted answers seem rather well known to me, but here's one that is certainly fundamental, seemingly out of reach, and perhaps not so well known except to complexity theorists.

Describe explicitly a Boolean function whose minimum circuit size is superlinear.

A simple counting argument shows that almost all Boolean functions require exponentially large circuits to express.  However, giving explicit examples is another matter.  Here, "explicit" is a bit vague, but let's say for example that it means that the truth table can be computed in time polynomial in the size of the truth table.  Thus NP-complete Boolean functions count as "explicit," and proving superpolynomial circuit lower bounds for them would separate P from NP, but even if we weaken the requirement to a superlinear lower bound on any explicit function, nobody seems to have any clue.
A: The Eilenberg-Ganea conjecture.  Recall that the cohomological dimension $\text{cd}(G)$ of a discrete group $G$ is the maximal $n$ such that there exists a $G$-module $M$ with $H^n(G;M) \neq 0$.  The geometric dimension $\text{gd}(G)$ of $G$ is the smallest $n$ such that $G$ has a $K(G,1)$ which is an $n$-dimensional CW complex.  It is elementary that $\text{cd}(G) \leq \text{gd}(G)$.  Moreover, if $\text{cd}(G) \neq 2$, then it is classical that $\text{cd}(G) = \text{gd}(G)$.  The Eilenberg-Ganea conjecture says that this also holds if $\text{cd}(G)=2$.  It is known, by the way, that if $\text{cd}(G)=2$ then $2 \leq \text{gd}(G) \leq 3$.
The only progress that I know of concerning this is a deep theorem of Bestvina and Brady that says that at the Eilenberg-Ganea conjecture and the Whitehead asphericity conjecture cannot both be true.
A: Can one transform a particular closed knotted piece of rope into another one?
A: P vs NP
According to Leonid Levin (via Scott Aaronson), Richard Feynman could not be convinced that this was actually an open problem.
A: What about Goldbach conjecure asking if every even natural number is the sum of two primes?
Another quite famous problem is Collatz' conjecture (also known as $3n+1$ problem), see
http://en.wikipedia.org/wiki/Collatz_conjecture: consider the algorithm taking $n\in\mathbb{N}$ and sending it to $n/2$ if $n$ is even, and to $3n+1$ if $n$ is odd, iteratively. The question is whether the algorithm always ends up producing the loop $1\mapsto 3\cdot 1+1=4\mapsto 2\mapsto 1\mapsto 4\dots$ regardless of the initial input $n$.
