Fundamental problems whose solution seems completely out of reach - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T09:05:11Z http://mathoverflow.net/feeds/question/101014 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach Fundamental problems whose solution seems completely out of reach alvarezpaiva 2012-06-30T18:48:02Z 2012-07-03T15:54:58Z <p>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 <em>open problems</em> 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 <em>I hope that people will pitch in either more problems of this type, or direct me to the literature where these problems are studied</em>. </p> <p>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. </p> <p><strong>Systolic geometry of simply-connected manifolds.</strong> <em>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 $C$?</em></p> <p><em>Comments.</em> </p> <ul> <li>For the two-sphere this is a theorem of Croke. </li> <li>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). </li> <li>There is a very timid hint to this problem in Gromov's <em>Filling Riemannian manifods</em>.</li> </ul> <p><strong>Sharp systolic inequality for real projective space.</strong> <em>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 $\pi$?</em> </p> <p><em>Comments.</em></p> <ul> <li>For the real projective plane this is Pu's theorem.</li> <li>In his <em>Panoramic view of Riemannian geometry</em>, Berger hesitates in conjecturing that this is the case (he says it is not clear that this is the right way to bet). </li> <li>In a recent <a href="http://front.math.ucdavis.edu/1109.4253" rel="nofollow">preprint</a> with Florent Balacheff, I studied a parametric version of this problem. The results suggest that the formulation above is the right way to bet.</li> </ul> <p><strong>Isoperimetry of metric balls.</strong> <em>For what three-dimensional normed spaces are metric balls solutions of the isoperimetric inequality?</em></p> <p><em>Comments.</em></p> <ul> <li><p>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.</p></li> <li><p>This is one of the Busemann-Petty problems.</p></li> <li>The volume and area are defined using the Hausdorff $2$ and $3$-dimensional measure.</li> <li>I have not seen any partial solution, even of the most modest kind, to this problem.</li> <li>Busemann and Petty gave a beautiful elementary interpretation of this problem: </li> </ul> <p>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 $x$. <em>The conjecture is that if the volume of all cones formed in this way is always the same, then the body is an ellipsoid.</em> </p> <p><strong>Additional problem:</strong> I had forgotten another beautiful problem from the paper of Busemann and Petty: <em>Problems on convex bodies</em>, Mathematica Scandinavica 4: 88–94. </p> <p><strong>Minimality of flats in normed spaces.</strong> 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?</p> <p><em>Comments.</em></p> <ul> <li>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 <em>Convexity of Lp-intersection bodies</em>, Adv. Math. 222 (2009), 920-936.</li> <li>When $k = 2$ this has "just" been proved by D. Burago and S. Ivanov: <a href="http://front.math.ucdavis.edu/1204.1543" rel="nofollow">http://front.math.ucdavis.edu/1204.1543</a></li> <li>It is <em>not</em> 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 <em>What is wrong with the Hausdorff measure in Finsler spaces</em>, Advances in Mathematics, vol. 204, no. 2, pp. 647-663, 2006. </li> </ul> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101027#101027 Answer by quid for Fundamental problems whose solution seems completely out of reach quid 2012-07-01T00:00:24Z 2012-07-02T09:30:35Z <p>Every finite abelian group is (isomorphic to) the class group of the ring of algebraic integers of some number field.</p> <p>Some comments:</p> <p>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.</p> <p>Also a 'geometric analog' is known (Perret, 1999). </p> <p>And every finite ablian group is at least a subgroup of a classgroup (even for a cyclotomic field).</p> <p>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.</p> <p>ps. Not sure this passes all (or any) of the criteria; I'll let you decide :)</p> <p>ps2. Searching for a reference, I found this math.SE question on exactly this <a href="http://math.stackexchange.com/questions/10949/finite-abelian-groups-as-class-groups" rel="nofollow">http://math.stackexchange.com/questions/10949/finite-abelian-groups-as-class-groups</a></p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101047#101047 Answer by Georges Elencwajg for Fundamental problems whose solution seems completely out of reach Georges Elencwajg 2012-07-01T08:00:24Z 2012-07-01T08:00:24Z <p>Is every algebraic curve in $\mathbb P^3$ the set-theoretic intersection of two algebraic surfaces ? Not known!</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101048#101048 Answer by JeffE for Fundamental problems whose solution seems completely out of reach JeffE 2012-07-01T08:09:34Z 2012-07-01T08:09:34Z <p><strong>P vs NP</strong></p> <p>According to Leonid Levin (via Scott Aaronson), Richard Feynman could not be convinced that this was actually an open problem.</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101053#101053 Answer by Holowitz for Fundamental problems whose solution seems completely out of reach Holowitz 2012-07-01T09:29:39Z 2012-07-01T09:29:39Z <p>Normality of numbers. Is Pi normal?</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101054#101054 Answer by Yemon Choi for Fundamental problems whose solution seems completely out of reach Yemon Choi 2012-07-01T10:12:39Z 2012-07-01T10:12:39Z <p>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?</p> <p>(See for instance these <a href="http://www.maths.nottingham.ac.uk/personal/jff/Beamer/pdf/RegularityConditions-handout-web.pdf" rel="nofollow">notes of Feinstein</a> )</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101056#101056 Answer by Yemon Choi for Fundamental problems whose solution seems completely out of reach Yemon Choi 2012-07-01T10:19:10Z 2012-07-01T10:19:10Z <p>Does $H^\infty(D)$, the Banach space of all bounded holomorphic functions on the unit disc, have Grothendieck's approximation property?</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101058#101058 Answer by Pablo Shmerkin for Fundamental problems whose solution seems completely out of reach Pablo Shmerkin 2012-07-01T11:12:27Z 2012-07-01T11:12:27Z <p>Here are two from ergodic theory:</p> <ul> <li><p><strong>(The problem of smooth realizations)</strong> 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?</p></li> <li><p><strong>(Furstenberg's $\times 2 \times 3$ problem)</strong> 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 $x\to 3x\bmod 1$?</p></li> </ul> <p>For the first problem, as far as I know there has been no significant progress. </p> <p>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. </p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101073#101073 Answer by Kevin O'Bryant for Fundamental problems whose solution seems completely out of reach Kevin O'Bryant 2012-07-01T14:49:36Z 2012-07-01T15:17:23Z <p>Artin's Conjecture: There are infinitely many primes $p$ for which 2 is a primitive root, i.e., 2 generates the multiplicative group of ${\mathbb Z}/p{\mathbb Z}$.</p> <p><a href="http://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots" rel="nofollow">The conjecture</a> is actually a bit more general, but we should at least be able to say what happens with 2! The <a href="http://oeis.org/A001122" rel="nofollow">OEIS lists</a> the first several such primes.</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101074#101074 Answer by Kevin O'Bryant for Fundamental problems whose solution seems completely out of reach Kevin O'Bryant 2012-07-01T15:05:04Z 2012-07-01T15:05:04Z <p>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 $x$. </p> <ul> <li>Say anything about the sequence $\{ (3/2)^n \}.$</li> </ul> <p>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.</p> <p>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 $$\epsilon&lt; \{\phi^n \} &lt; 1-\epsilon.$$</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101075#101075 Answer by Kevin O'Bryant for Fundamental problems whose solution seems completely out of reach Kevin O'Bryant 2012-07-01T15:12:11Z 2012-07-01T15:12:11Z <p>A proof of this conjecture of Erdos would certainly turn heads, raise eyebrows, and garner the attention of the Fields Medal committee.</p> <ul> <li>If $\sum_{a \in A} \frac 1a$ diverges and $A\subseteq {\mathbb N}_{>0}$, then $A$ contains a 3-term arithmetic progression.</li> </ul> <p>Probably "diverges" can be replaced with "is bigger than 4".</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101083#101083 Answer by Holdsworth88 for Fundamental problems whose solution seems completely out of reach Holdsworth88 2012-07-01T17:04:41Z 2012-07-01T17:04:41Z <p>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 </p> <p>$$4\leq \chi \leq 7$$.</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101086#101086 Answer by Andrew for Fundamental problems whose solution seems completely out of reach Andrew 2012-07-01T18:56:29Z 2012-07-01T18:56:29Z <p>Can one transform a particular closed knotted piece of rope into another one?</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101090#101090 Answer by fedja for Fundamental problems whose solution seems completely out of reach fedja 2012-07-01T19:37:06Z 2012-07-01T19:37:06Z <p>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? </p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101113#101113 Answer by Chandrasekhar for Fundamental problems whose solution seems completely out of reach Chandrasekhar 2012-07-02T03:15:28Z 2012-07-02T03:15:28Z <p><a href="http://oeis.org/wiki/User%3APeter_Luschny/SchinzelSierpinskiConjectureAndCalkinWilfTree" rel="nofollow"><strong>Schinzel-Sierpinski Conjecture</strong></a></p> <p>Taken from this <a href="http://mathoverflow.net/questions/53736/on-a-conjecture-of-schinzel-and-sierpinski" rel="nofollow">MathOverflow link.</a></p> <p>Melvyn Nathanson, in his book <em>Elementary Methods in Number Theory</em> (Chapter 8: Prime Numbers) states the following:</p> <ul> <li>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 $3$.</li> </ul> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101115#101115 Answer by Chandrasekhar for Fundamental problems whose solution seems completely out of reach Chandrasekhar 2012-07-02T03:20:38Z 2012-07-02T03:20:38Z <p>The <a href="http://en.wikipedia.org/wiki/Bunyakovsky_conjecture" rel="nofollow"><em>Bunyakovsky conjecture</em></a> (or Bouniakowsky conjecture) stated in 1857 by the Russian mathematician <a href="http://en.wikipedia.org/wiki/Viktor_Bunyakovsky" rel="nofollow">Viktor Bunyakovsky</a>, claims that </p> <blockquote> <ul> <li>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.</li> </ul> </blockquote> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101117#101117 Answer by Filippo Alberto Edoardo for Fundamental problems whose solution seems completely out of reach Filippo Alberto Edoardo 2012-07-02T03:33:38Z 2012-07-02T03:33:38Z <p>Are there infinitely many <em>regular primes</em>? We know there are infinitely many <em>irregular</em> ones, and that their percentage should be much smaller than the regular ones, still it is unproven that the latter are infinite.</p> <p>Let me recall that a prime $p$ is irregular if it divides the class number of $\mathbb{Q}(\zeta_p)$, the cyclotomic field.</p> <p>Similarly, we cannot prove that there are infinitely many real quadratic fields of class number $1$.</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101118#101118 Answer by Filippo Alberto Edoardo for Fundamental problems whose solution seems completely out of reach Filippo Alberto Edoardo 2012-07-02T03:35:18Z 2012-07-02T09:23:29Z <p>What about Goldbach conjecure asking if every even natural number is the sum of two primes?</p> <p>Another quite famous problem is Collatz' conjecture (also known as $3n+1$ problem), see <a href="http://en.wikipedia.org/wiki/Collatz_conjecture" rel="nofollow">http://en.wikipedia.org/wiki/Collatz_conjecture</a>: 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 $n$.</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101119#101119 Answer by Spice the Bird for Fundamental problems whose solution seems completely out of reach Spice the Bird 2012-07-02T04:55:18Z 2012-07-02T04:55:18Z <p>An obvious problem in algebraic topology would be the computation of the homotopy groups of spheres.</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101132#101132 Answer by Alex R. for Fundamental problems whose solution seems completely out of reach Alex R. 2012-07-02T10:09:27Z 2012-07-02T12:45:39Z <p>Can we exactly calculate <a href="http://en.wikipedia.org/wiki/Ramsey_theory" rel="nofollow">Ramsey numbers</a>? Erdős once famously remarked:</p> <blockquote> <p>"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."</p> </blockquote> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101133#101133 Answer by Fred Kline for Fundamental problems whose solution seems completely out of reach Fred Kline 2012-07-02T10:15:16Z 2012-07-02T10:15:16Z <p>From the Overview of the Royal Danish Sciences Institution's work and its members' work in the year 1882.</p> <p>In the notes from a meeting on March 9th 1877, after discussing papers by Legendre, J. W. L. Glaisher, and Meissel, Oppermann stated:</p> <blockquote> <p>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 $n(n+1)$.</p> </blockquote> <p>A solution to Oppermann's Conjecture leads to simple solutions to Legendre's, Brocard's, and Andrica's Conjectures.</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101141#101141 Answer by Chandrasekhar for Fundamental problems whose solution seems completely out of reach Chandrasekhar 2012-07-02T12:31:56Z 2012-07-02T12:31:56Z <p><a href="http://en.wikipedia.org/wiki/Sendov_conjecture" rel="nofollow"><strong>Sendov's Conjecture</strong></a></p> <blockquote> <p>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 $f$.</p> </blockquote> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101153#101153 Answer by Gerald Edgar for Fundamental problems whose solution seems completely out of reach Gerald Edgar 2012-07-02T15:16:26Z 2012-07-02T15:16:26Z <p>Is there an algebraic irrational number in the Cantor set? </p> <p>More generally: Are algebraic irrational numbers normal in all bases? </p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101155#101155 Answer by Gejza Jenča for Fundamental problems whose solution seems completely out of reach Gejza Jenča 2012-07-02T16:10:28Z 2012-07-02T16:10:28Z <p>Is every finite lattice a congruence lattice of a finite (universal) algebra? </p> <p>Astonishingly, by <a href="http://rd.springer.com/article/10.1007/BF02483080" rel="nofollow">Pálfy and Pudlák</a>, 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? </p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101156#101156 Answer by Hunter Brooks for Fundamental problems whose solution seems completely out of reach Hunter Brooks 2012-07-02T16:21:28Z 2012-07-02T16:21:28Z <p>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.</p> <p>Does $r(G) = r_p(G)$?</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101172#101172 Answer by Timothy Chow for Fundamental problems whose solution seems completely out of reach Timothy Chow 2012-07-02T19:01:49Z 2012-07-02T19:01:49Z <p>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.</p> <blockquote> <p>Describe explicitly a Boolean function whose minimum circuit size is superlinear.</p> </blockquote> <p>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 <i>superlinear</i> lower bound on <i>any</i> explicit function, nobody seems to have any clue.</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101179#101179 Answer by Terry Tao for Fundamental problems whose solution seems completely out of reach Terry Tao 2012-07-02T21:00:40Z 2012-07-02T21:17:51Z <p>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 <a href="http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/" rel="nofollow">http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/</a> .) 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.)</p> <p>EDIT: See also this survey of Halava-Harju-Hirvensalo-Karhumäki from 2005 on this problem: <a href="http://tucs.fi/publications/view/?id=tHaHaHiKa05a" rel="nofollow">http://tucs.fi/publications/view/?id=tHaHaHiKa05a</a></p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101189#101189 Answer by Bill Johnson for Fundamental problems whose solution seems completely out of reach Bill Johnson 2012-07-03T00:05:25Z 2012-07-03T00:05:25Z <p>Is every complemented subspace of $C[0.1]$ isomorphic to $C(K)$ for some compact metric space $K$?</p> <p>Is every infinite dimensional complemented subspace of $L_1[0.1]$ isomorphic either to $L_1[0.1]$ or to $\ell_1$?</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101206#101206 Answer by Andy Putman for Fundamental problems whose solution seems completely out of reach Andy Putman 2012-07-03T04:13:16Z 2012-07-03T04:13:16Z <p>The Eilenberg-Ganea conjecture. Recall that the <em>cohomological dimension</em> $\text{cd}(G)$ 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 <em>geometric dimension</em> $\text{gd}(G)$ 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 $2 \leq \text{gd}(G) \leq 3$.</p> <p>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.</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101207#101207 Answer by Andy Putman for Fundamental problems whose solution seems completely out of reach Andy Putman 2012-07-03T04:17:45Z 2012-07-03T04:17:45Z <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/101014/fundamental-problems-whose-solution-seems-completely-out-of-reach/101238#101238 Answer by Mohan for Fundamental problems whose solution seems completely out of reach Mohan 2012-07-03T15:54:58Z 2012-07-03T15:54:58Z <p>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?</p>