**13**

votes

**2**answers

272 views

### All compact surfaces $S\subseteq \mathbb{R}^3$ are rigid?

Recently I've come across a lecture in differential geometry by Fernando Codá (in portuguese!) in which he stated that the following problem is (at least, up to 2014) open:
Given ...

**1**

vote

**2**answers

206 views

### Looking for (information about) long diamonds

I was given an open problem as a birthday present recently. While I can probably handle spoilers at this point, what I really want are literature and other references. Also acceptable would be ...

**4**

votes

**0**answers

170 views

### What is behind the Hodge conjecture? [duplicate]

My question is quite naive, and my knowledge limited on the subject. I heard lot of talks about Hodge conjecture. I wanted to ask about an intuitive way to figure out why we should care about Hodge ...

**10**

votes

**0**answers

280 views

### Progress in Guy's “Unsolved problems in Number Theory”? [closed]

I often peruse through Guy's book whenever I'm not being boggled down by my research. It crossed my mind today if any of these "unsolved problems" have become indeed solved. I thought about doing a ...

**5**

votes

**1**answer

154 views

### Does $ZF$+$LM$+$A_{\lt2^{\aleph_0}}$ imply $\lnot$$WCH$?

In what follows, I will use the following acronyms:
$AS$:= Freiling's Axiom of Symmetry
$LM$:="Every set of reals is Lebesgue measurable."
$WCH$:="every uncountable subset of $\mathbf R$ can be put ...

**11**

votes

**2**answers

289 views

### Gerstenhaber conjecture for free loop space

I- Is the following statement still a conjecture see this article ?
Conjecture (?)
Let $M$ be a simply connected compact oriented $d$-manifold (smooth), then $HH^{\ast}(C^{\ast}(M))$ the Hochschild ...

**7**

votes

**0**answers

107 views

### Bang's open question strengthening Tarski's planks problem

Tarski's Planks problem,
solved by Thøger Bang in 1951, says (in a simplified $\mathbb{R}^2$ version) that it requires
"planks" (parallel strips) of total width $\ge d$ in order to completely cover
a ...

**4**

votes

**1**answer

159 views

### Sets not containing the vertices of unit triangles (Question posed by Erdős)

Following this post, I have been thinking about the problem posed by Erdős,
Does there exist a constant $c > 0$ such that every subset $A$ of the plane of area more than $c$ contains the ...

**5**

votes

**0**answers

88 views

### Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x

Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form
$$
g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i ...

**6**

votes

**1**answer

262 views

### Conway's subprime Fibonacci sequences

I want to be certain I have the latest information on
Conway's subprime Fibonacci sequences,
arXiv-posted a year ago; I am referencing the status in
a review.
To wit, starting with $(0,1)$:1
$$
0, 1, ...

**20**

votes

**1**answer

910 views

### Why is it so hard to prove Toeplitz' conjecture?

I'm a layman in mathematics, so please excuse me in advance for anything in this question that may be inappropriate :D. Well: Four years ago, I was reading (and working to solve the puzzles on) ...

**-5**

votes

**3**answers

133 views

### Collecting sufficient conditions for Sorli's conjecture on odd perfect numbers [closed]

Sorli's conjecture predicts that, for an odd perfect number $N$ given in the Eulerian form $N = {q^k}{n^2}$ (where $q$ is prime with $\gcd(q, n) = 1$ and $q \equiv k \equiv 1 \pmod 4$), the condition ...

**3**

votes

**1**answer

251 views

### Open problems books [closed]

As the title might indicate , I would like to look for recommendations for mathematical book that present open problems in depth with commentary.
The only book of this type that I've come across is ...

**1**

vote

**0**answers

192 views

### On Descartes / spoof odd perfect numbers

Descartes, Frenicle, and subsequently Sorli, conjectured that $k = 1$, if $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and ...

**2**

votes

**0**answers

188 views

### Are there any interesting open questions having to do with submodularity, specially in the intersection of theoretical machine learning? [closed]

I was interested in knowing about open research topics related with sub modularity, specially within its intersection with theoretical machine learning (and related topics).
It seems to me that much ...

**1**

vote

**1**answer

191 views

### Reference request: Research done on whether the Euler prime can be the largest factor of an odd perfect number

(Note: This was cross-posted from MSE.) I posted the following reference request in MSE three (3) days ago, but was unable to elicit any responses. I am cross-posting it to MO, hoping that it is ...

**9**

votes

**2**answers

471 views

### Can any finite lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$.
Question: Can any finite lattice be realized as ...

**19**

votes

**5**answers

1k views

### Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?

**5**

votes

**1**answer

384 views

### Ruth-Aaron triples, etc

A Ruth-Aaron pair is two numbers $(n,n+1)$ such that
their sum of prime factors is equal, counting repeated prime factors.
(The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!)
So
...

**12**

votes

**0**answers

347 views

### Is every projective space curve a set-theoretic intersection of two surfaces? What is known about this question?

I am sorry if this question has been already asked, couldn't find anything similar myself. I have recently recalled this long standing open problem of whether every irreducible curve in ...

**2**

votes

**0**answers

87 views

### Long paths in the supercritical percolation.

I have a question on the length of the longest path, denoted by $\ell_n$, in the supercritical percolation on $[0,n]^d$, denoted by $C_n$.
We know that $C_n$ has a giant component whose size is of ...

**1**

vote

**2**answers

459 views

### Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form [closed]

(Note: This was cross-posted from MSE.)
Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$).
Therefore, ...

**4**

votes

**0**answers

142 views

### Reduction argument from a general vertex set V(G) to a prime power in Prof. Keevash's proof on the Existence of Designs

The proof flow of the paper "On the Existence of Designs" by Prof. Keevash as I understand it is the following:
-- Reduction from the general case to $V = \mathbb{F}_{p^a}$ (Lemma 6.3)
-- Covering ...

**6**

votes

**1**answer

224 views

### Inverted pair of complex analytic families

I read the following "problem" in an old set of notes of Morrow and Kodaira which focused on deformations of complex manifolds:
Find a pair of complex analytic families $\lbrace M_t\rbrace$ and ...

**8**

votes

**0**answers

170 views

### Randomly placing nonoverlapping unit cuboids

Suppose one places unit cuboids of dimension $d$ with min-corners
uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden.
At some point, the region is "saturated," ...

**2**

votes

**2**answers

310 views

### Gauss Codes that produce classical knots as opposed to virtual knots

I have been doing some research in Gauss codes and have been reading Kauffman's paper Virtual Knot Theory. In section 3.3, Theorem 2, he states that
If $K$ is a virtual knot whose underlying Gauss ...

**13**

votes

**0**answers

337 views

### Does $S^4$ have a “symplecto-homeomorphic” structure?

The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms ...

**33**

votes

**3**answers

948 views

### Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point.
Theorem. Suppose $P$ and $Q$ are posets ...

**3**

votes

**1**answer

111 views

### reduction to np hard ordering problem

I am trying to show a reduction from a problem of ordering problem to an np-hard problem that has approximation poly-time algorithm.
My problem is: I have M auctions and in each auction I have N ...

**1**

vote

**1**answer

101 views

### A possible minimal aperiodic set of corner Wang Tile

From one of my previous question Aperiodic set of corner Wang Tile (although it is put on hold), I realize there is a systematic way to construct aperiodic corner type of Wang tile from edge type ...

**1**

vote

**0**answers

162 views

### Both NP-hard but different [closed]

What's the fundamental difference between the Knapsack problem and the travelling salesman (TSP) problem both of which are NP-hard, while the reality is that TSP could be solved much much faster?

**7**

votes

**2**answers

456 views

### Update on list of open problems for Cherednik/Symplectic Reflection Algebras

Background:
There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...

**7**

votes

**4**answers

1k views

### Is the Manickam-Miklós-Singhi Conjecture solved? [closed]

This arXiv paper is claimed to contain a proof for the MMS conjecture. But it seems that this manuscript is not yet peer reviewed by other mathematicians. I personally tried to follow the paper, but ...

**10**

votes

**1**answer

304 views

### Subspaces of $l_{1}$ are not Lipschitz complemented in $l_{1}$

I have thought about the following question for several years. This question may be stupid or not interesting. My question is: Is there a subspace $U$ of $l{_1}$ such that the quotient $l_{1}/U$ is ...

**10**

votes

**1**answer

283 views

### Is there a non-zero ghost map between finite suspension spectra?

A morphism $f\colon X\to Y$ of spectra such that for every integer $n$ the induced map $\pi_n(f)\colon\pi_n(X)\to\pi_n(Y)$ on stable homotopy groups is zero is called a ghost map.
Not every ghost map ...

**10**

votes

**1**answer

271 views

### A special case of the integer Hodge conjecture

Let $X$ be a projective complex manifold of dimension $n$.
Are torsion cohomology classes in $H^{2n-2}(X,\mathbb{Z})$ algebraic?
(We may assume, without loss of generality, that $n=3$, because of the ...

**4**

votes

**0**answers

263 views

### Packing space by cones: Translates best?

Let $C$ be a right circular cone, the convex hull of a unit-radius disk
and a point directly above the disk center at height $h$.
Is the ...

**0**

votes

**1**answer

115 views

### Rost Correspondence and minimal Pfister-Neighbors

In http://www.math.uiuc.edu/K-theory/0357/ Karpenko utters the following:
Conjecture 1.6. If an anisotropic quadric $X = Q$ possesses a Rost correspondence,
then the quadratic form (defining ...

**19**

votes

**2**answers

662 views

### Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$.
If there are $g(n)$ non-isomorphic groups of order $n$,
ideally each group would occur with probability ...

**1**

vote

**1**answer

100 views

### simple cycle analog in 2D (with application in tiling)

We know that any closed cycle of a graph could be decomposed into sum of simple cycles. To translate this theorem into tiling of 1D (Wang tile). We know that any 1D periodic tiling could be ...

**6**

votes

**1**answer

437 views

### Is there a precise notion of “almost all” such that almost all finite groups are Galois groups of extensions of the rationals?

I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to ...

**1**

vote

**1**answer

227 views

### relationship between corner tile and edge tile of wang tile

It is clear that any corner type of Wang Tile could be converted to edge type of Wang Tile by defining the edge color according to the corner color.
However, could we convert edge type of Wang Tile ...

**37**

votes

**1**answer

1k views

### improving known bounds for Pierce expansions; cash prize

Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have ...

**6**

votes

**1**answer

188 views

### Integral straight-line embeddings of planar graphs

Wikipedia says (in the article on Fáry's theorem),
"Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...

**17**

votes

**3**answers

1k views

### What is Chern-Simons theory expected to assign to a point?

Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a ...

**3**

votes

**2**answers

201 views

### Conjecture of a subset of Wang tile which might be decidable

From the two papers proving the undecidability of Wang tile in 1966 by Berger and in 1971 by RM Robinson, the tiles used in proving undecidability has a general common feature:
The left color and ...

**36**

votes

**0**answers

606 views

### Which region in the plane with a given area has the most domino tilings?

I just finished teaching a class in combinatorics in which I included a fairly easy upper bound on the number of domino tilings of a region in the plane as a function of its area. So this led to ...

**7**

votes

**1**answer

662 views

### What keeps asymptotic Goldbach's conjecture out of reach of current technology?

Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...

**5**

votes

**6**answers

845 views

### practical algorithms for np complete problems

Inspired by:
Conjecture on NP-completeness of tesselation of Wang Tile up to finite size
And the practicality of this topic (solving tessellation on a lattice):
coloring in lattice
Computational ...

**16**

votes

**1**answer

801 views

### Paul Erdős: Determine or estimate the number of maximal triangle-free graphs on n vertices

Among the collections of the open problems of Paul Erdős on the website of
Professor Fan Chung, there is one called "number of triangle-free graphs".
...