MO-Hard Questions

Question

The term "MO-Hard" has entered the mathematical vocabulary, describing interesting questions that were asked on MathOverflow but haven't been answered in spite of efforts by many community members. Each one of these is a MathOverflow success story! Open problems are precious commodities that have driven mathematical research for centuries. While these are not necessarily at the same level as the Riemann Hypothesis and the Jacobian Conjecture, we should celebrate them and promote them!

While the unanswered tab is a good way to find such gems, questions that have interesting partial answers do not appear there. In the same spirit as Best of MathOverflow, let's collect MO-Hard questions here. Questions that still appear in the unanswered list are welcome since they might disappear any day. Comments, anecdotes and other remarks are also encouraged.

Answer 1

Here is a prime example of a MO-Hard question that doesn't appear on the unanswered list.

Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

The title says it all. In fact, the body of the question is just one line. It has had 16 answer attempts though most have been deleted after an error was found...

Answer 2

This question is a very nice topological one, a cousin of Brouwer's fixed point theorem and related to several questions in the literature:

Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk to itself. Does there always exist a point $x$ such that $f(x)=g(x)$?

Answer 3

The question Polynomial representing all nonnegative integers asked by Poonen has been on MO for a very long time. (It should possibly be mentioned that it was asked elsewhere earlier, in particular on Poonen's website.) It received various interesting answers, but none of them solved the problem.

The question is short so I reproduce it:

Is there a 2-variable polynomial $f(x,y) \in \mathbf{Q}[x,y]$ such that $f(\mathbf{Z} \times \mathbf{Z})=\mathbf{N}$?

For $3$ (and more variables) it is well known such a polynomial exists (basically going back to Gauss).

Answer 4

I make no claim to the significance of any of these problems, nor that they are even "hard"—some are too peripheral to attract interest. With those caveats, here are a few of mine (in chronological order):

• "Which convex bodies roll along closed geodesics?" (Apr 2011): An Easter posting :-) Some examples provided by Robert Bryant.

• "Homometric ⇒ isometric?" (May 2011): Are homometric manifolds necessarily isometric? Anton Petrunin followed-up with a question whose positive resolution would answer mine.

• "Ratio of circumscribed/inscribed $(n{−}1)$-gons" (Jan 2012). Asks for extreme value of area$(P^+)/$area$(P^-)$. Lower bound on area$(P^-)$ established by domotorp (Pálvölgyi Dömötör).

• "Gaussian prime spirals" (Mar 2012). An instance where I unwittingly hit a known open problem with my question of whether every such spiral forms a cycle, as François Brunault observed.

• "Building a genus-n torus from cubes" (Jun 2012). Some bounds offered. Douglas Zare: the "2-dimensional version of this question ... is not trivial." Update: Now largely solved by Vivek Shende: the min # of cubes approaches $2 g$ for a genus-$g$ handlebody.

• "A curious sequence of rationals: finite or infinite?" (May 2013). $f(a/b) = (a b) / (a + b - 1)$ in lowest terms. Beautiful analyses by Gottfried Helms, Barry Cipra, and Karl Fabian. I would say the evidence points toward some infinite sequences...

• "Reversing shortest paths among unit disks" (Dec 2013). Some promising ideas, but still unclear whether a shortest path ever reverses in a hexagonal penny packing.

• "Blocking light with mirrored convex objects" (Feb 2014). A possibly easier version of an unsolved problem of Janos Pach.

• "Surfaces with many (but not solely) closed geodesics?" (Feb 2014). Jointly posed with alvarezpaiva (J.-C. Alvarez Paiva).

• "Generating random finite groups" (May 2014). Here I wandered into a problem known to the experts to be inaccessible.

• "Packing space by cones: Translates best?" (Jun 2104). Wlodek Kuperberg believes the question is open for generic cones.

• "Simple, closed geodesics in $\mathbb{S}^3$ manifold" (Sep 2014). Clarify this ambiguous situation: One respected differential geometer (Klinenberg) proves $4$ closed geodesics in 1981, but only in 2009 is it acknowledged that there are at least $2$ closed geodesics. And simple geodesics seems not specified in this literature, as alvarezpaiva (J.-C. Alvarez Paiva) points out.

• "Packing disks on a cone, or: Garlands on a tree" (Dec 2014). A Christmas-eve posting . :-)

• "Mod sequences that seem to become constant; and the number 316" (Dec 2014). Do all mod sequences $A(s)$ settle to a constant? Gjergji Zaimi found related OEIS sequences also highlighting "the mysterious 316."

• "A Collatz-like sequence that bifurcates on primes" (May 2015). Does there exist any $n$ such that $f^k(n) \to \infty$? Maybe $n=397$?

• "Terrible tilers for covering the plane" (Dec 2016). Is the regular pentagon the worst (most wasteful) shape to cover the plane? What is the wastefulness of the regular pentagon? Wlodek Kuperberg showed a cover with $\theta(P) \approx 1.118$ that is minimal among double-lattice coverings, and he conjectures "that this density is minimum among all coverings with $P$, not just double-lattice ones."

• "Transport tubes in a sphere (Jul 2017). How to arrange fast transport segments to minimize the max distance btwn any two points.

• "Cut locus on a hypercube" (Mar 2017). What does it look like with respect to one corner?

• "Iteration cycles of $\mathbb{Z}_n$ weights in path graphs: Why cycles of length $182$ for a $6$-node path?" (Aug 2017). Start with any permutation of $0,\ldots,5$. In 1 step, replace each element by the sum of its neighbors $\bmod 6$. Repeat. Always cycles in $182$ steps. Why? Moritz Firsching extended my data considerably.

• "Trapping lightrays with segment mirrors" (Nov 2017). Can all the light from one candle be blocked by disjoint mirrors? Surely Not, but ... proof?

• "Tiling with incommensurate triangles" (July 2018). Is there a lattice tiling composed of pairwise incommensurate triangles?

• "Does this infinite primes snake-product converge?" (Feb 2019): $\frac{2}{3}\cdot\frac{7}{5}\cdot\frac{11}{13}\cdot\frac{19}{17}\cdot\frac{23}{29}\cdot\frac{37}{31} \cdot \cdots \;.$ It seems to converge to $\approx 0.9048$.

• "Can 4-space be partitioned into Klein bottles?." Inspired by a negative result for Möbius strips partitioning $\mathbb R^3$.

Answer 5

I posted a question about a year ago on MO (and even further back on MSE) that remains elusive:

Randomly break a stick in five places.

Question: What is the probability that the resulting six pieces can form a tetrahedron?

The two breaks to form a triangle problem has a collection of proofs on MO here; the five breaks for a tetrahedron problem is much harder, and relates to the Cayley-Menger determinant.

---

Another one is my question about whether a particular manifold admits a Riemannian metric.

Answer 6

One I like very much is

Can we color $\mathbb Z^+$ with $n$ colors such that $a, 2a, \dots, na$ all have different colors for all $a$?

Among the failed proposed attempts, the ones that appeared most promising were in essence group- or number-theoretic, for instance the question of whether the partial graph of multiplication $$\{(a,b,c)\in\{1,\dots,n\}^3\mid ab=c\}$$ could be extended to a group operation on $\{1,\dots,n\}$, with the first counterexample being $n=195$.

Answer 7

Simon Thomas asked in

Ultrafilters and automorphisms of the complex field

whether the existence of non-principal ultrafilters (over the natural numbers) suffices to imply the existence of a nontrivial automorphism of the complex field $\mathbb C$. In set theoretic terms, the question is whether (under appropriate large cardinal assumptions) there is such an automorphism in $L(\mathbb R)[\mathcal U]$ where $\mathcal U$ is a nonprincipal ultrafilter on $\mathbb N$.

Answer 8

A question that fascinates me is the problem of characterizing the subsets $S$ of the unit circle $\mathbb T$ for which there is a power series with radius of convergence $1$ that, on $\mathbb T$, converges on $S$ and diverges otherwise.

Behaviour of power series on their circle of convergence.

(The question has also been asked on MSE a couple of times.)

Answer 9

I came across this question the other day...

Order-increasing bijection from arbitrary groups to cyclic groups

... It's possible that the question has been answered in the literature by now, as it is a little old. I don't know much about it but the discussion is very interesting, and suggests that a solution would be of some significance.

Answer 10

Even though it has not been here for particularly long, I think it is fair to add Is the set $ AA+A $ always at least as large as $ A+A $? to this list.

It is a problem that at first glance seems very simple, but the partial answers so far are revealing that there might be something deep going on.

Answer 11

This question is a relative to the well-known question of the chromatic number of unit-distance graphs (so, thematically speaking, it lies between geometry and combinatorics with a Ramsey flavor). There are in fact several questions in there, please see the original question.

Answer 12

Here are my MO unsolved questions :

Intersecting Family of Triangulations

This is an Erdos-Ko-Rado type conjecture for triangulations. It was largely extended by Gjergji Zaimi to polytopes without triangular faces.

A Weak Form of Borsuk's Conjecture

Can we cover a convex polytope $P$ with $m$ facets by $m$ (or $poly(m)$) sets of smaller diameter?

A curious generalization of Helly's theorem

Helly-type theorems for pairs of convex sets.

Infinitely many primes, and Mobius randomness in sparse sets

Is it possible to prove Mobious randomness or PNT for a subset $A$ of integers with $|A \cap \{1,2,\dots,n\}| \le n^{1/2-\epsilon}$?

The next items appeared as open questions earlier so I knew they are not easy but I thought they had a chance.

Volumes of Sets of Constant Width in High Dimensions

Can we have sets of constant width with exponentially (in the dimension) smaller volumes than balls? (Oded Schramm)

A Combinatorial Abstraction for The "Polynomial Hirsch Conjecture"

The shortest path in first passage percolation Update: solved by Daniel Ahlberg and Christopher Hoffman.

Optimal Monotone Families for the Discrete Isoperimetric Inequality

Answer 13

This question from Riemannian geometry:

Consider a two-dimensional sphere with a Riemannian metric of total area $4\pi$. Does there exist a subset whose area equals $2\pi$ and whose boundary has length no greater than $2\pi$?

Answer 14

The question that asks if a square of area $4n^2$ is the geometric figure with the most number of domino tilings, of all figures of the same area, here, is still unsolved. I find this question quite intriguing.

Answer 15

There are several MO-hard problems about smooth and proper schemes over $\mathbb Z$, including:

Smooth proper schemes over $\mathbb Z$ with points everywhere locally

Non-simply-connected smooth proper scheme over $\mathbb Z$

What can be the dimension of a pointless smooth proper $\mathbb Z$-scheme?

All these were inspired by Poonen's (solved) question, asking whether such a scheme necessarily has a section:

Smooth proper scheme over $\mathbb Z$

Answer 16

Self-promotion, but I would like to mention my question Meager subspaces of a Banach space and weak-* convergence. It contains two questions. I have resolved Q1 and posted it as an answer (hence the question is no longer "unanswered") but Q2 has not been resolved. I quote it here for visibility.

Q2. Let $X$ be a Banach space. Let us say a linear subspace $E \subset X$ determines weak-* convergence (of sequences) if for every sequence $\{f_n\} \subset X^*$ such that $f_n(x) \to 0$ for every $x \in E$, we have $f_n(x) \to 0$ for every $x \in X$. Is every such $E$ nonmeager?

The converse is an easy exercise with the uniform boundedness principle.

Update Q2 has now been resolved. The answer is No (though I would still be interested in a separable counterexample). Should I delete this?

Answer 17

This question describes a combinatorial game based on commutative algebra where the game state is a ring and a move is taking the quotient by a nonzero element:

A Game on Noetherian Rings

The hard problem is to either determine which positions are winning and which are losing, or to show that the previous problem has no solution.

Answer 18

Three of my MO questions appear to be quite hard:

1. Is there a simple discrete subgroup of $SL(n, {\mathbb R})$?.

2. The existence of good covers of topological manifolds. In addition to MO, I asked Ciprian Manolescu (specifically, the 4-dimensional case), he did not know.

3. Open immersions of open manifolds. In addition to MO I asked Yasha Elishberg: he does not know but expects a negative answer.

Answer 19

About discontinuties of a function and its Fourier transform : Is this statement which relates the Fourier transform of a function to its singularities correct?

Answer 20

I think that this question is may be non easy. it is about a certain generalization of the Poincare Bendixon theorem

A certain generalization of the Poincare Bendixon theorem

Answer 21

Here are some number theory questions that (so far as I know) are original with me and appeared on MathOverflow first . They may be hard; at this writing none of them are satisfactorily answered.

Jumping prime sieve: How do these primes jump?

Playing prime leapfrog: Playing leapfrog with primes

Semiprime cover: How much do these interval collections cover?

Combinatorial ring toss: Has anyone seen this version of ring toss (combinatorial object) before?

Square difference sequences: How many sequences of rational squares are there, all of whose differences are also rational squares?

Gerhard "Will Accept Fame From Questions" Paseman, 2017.09.02.

Answer 22

This question entitled "Limit cycles as closed geodesics in negatively or positively curved space" is not answered yet. So may be the above question and its linked question entitled "Limit cycles as closed geodesics(2)" can be counted as a non trivial question.

Answer 23

All of my 25 questions (except for one) from July 19, 2016 to now which is October 24, 2017 are still unanswered (and more are on the way). These questions have covered diverse topics such as forcing, braid groups, large cardinals, cellular automata, linear algebra, and cryptography but the main these of these questions is the elusive notion of self-distributivity.