If it turns out that a problem is equivalent to a known open problem, then the open-problem tag is added. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it ...

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12
votes
2answers
576 views

Are acyclic subcomplexes of finite contractible 2-complexes contractible?

Let $Y$ be a contractible finite simplicial 2-complex. Let $X$ be an acyclic subcomplex of $Y$ (i.e. $X$ connected, $H_1(X)=0$, $H_2(X)=0$). Is $X$ contractible? (Equivalently, is $\pi_1(X)$ ...
-1
votes
1answer
598 views

Question Re: Arian Berdellima's Papers On Odd Perfect Numbers [closed]

Hi everyone. I'd like to refer you to two papers by Arian Berdellima on odd perfect numbers: More Properties About Odd Perfect Numbers http://mpra.ub.uni-muenchen.de/31587/1/MPRA_paper_31587.pdf ...
18
votes
0answers
652 views

Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
4
votes
0answers
703 views

Has n^2*|sin(n)| limit? [closed]

Is there any sub-sequence $n_k$ of natural numbers such as $\lim(n_k^2|\sin(n_k)|) = 0$ ? when $k$ tends to infinity. In other words, does $\lim(n^2|\sin(n)|)$ exist (equal to infinity)? or there is ...
7
votes
1answer
257 views

Algorithmic decidability of equality in the ring of periods

Suppose two elements of the ring of periods are given by their systems of polynomial inequalities with rational coefficients. Is there a known algorithm deciding their equality? Is it known if their ...
9
votes
0answers
701 views

3-piece dissection of square to equilateral triangle?

At a workshop it was suggested that it likely remains an open problem whether or not there is a 3- or 2 -piece dissection of a square to an equilateral triangle. Can anyone confirm that this is ...
2
votes
0answers
160 views

What is the maximal density vertex subset of the 8-connected grid with induced vertex degrees $\leq 4$?

Let $G$ be the infinite graph defined by 8-connection of $\mathbb{Z}^2$. What is the maximal density vertex subset whose induced subgraph has maximum degree $\leq 4$? More precisely, what is the ...
3
votes
0answers
334 views

Compact surfaces smoothly immerse in: $\mathbb{R}^4$ or $\mathbb{R}^5$?

I wonder if someone can clarify whether it is known that every closed, orientable surface (2-manifold) has a smooth isometric immersion in $\mathbb{R}^4$? This topic has been discussed rather ...
13
votes
2answers
530 views

What is known about Ulam's problem of metric spaces with isometric squares?

Background In the book Problems in Modern Mathematics, S. Ulam asks the following question: Suppose $A$ and $B$ are metric spaces, such that $A^2$ and $B^2$ equipped with the 2-metric $d((x_1, ...
4
votes
1answer
700 views

Convergence speed of Jacobi eigenvalue algorithm for parallel ordering(Brent-Luk) ?

Is there estimate for convergence of the Jacobi eigenvalue algorithm for Hermitian matrices for "parallel ordring" (Brent-Luk ordering (see comment below)) ? For example for 4 4 matrices parallel ...
8
votes
2answers
434 views

Exact consistency-strength of “all projective sets are Ramsey”

I wonder if the exact consistency strength of "All projective sets have the Ramsey property" is still open. In Solovay's model, all sets have the Ramsey property, so the consistency strength of this ...
4
votes
1answer
482 views

What is the degree of a symmetric boolean function?

(previous title " Zero sum of binomials coefficients - a stronger version ") This is a stronger version of another question. Is there an $N\in \mathbb N$ and a sequence of non-constant functions $ ...
16
votes
0answers
743 views

When is a compact topological 4-manifold a CW complex?

Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres". Kirby showed that ...
10
votes
1answer
989 views

Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? [closed]

Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? Are there known non-smooth examples homeomorphic $CP^2$?
3
votes
0answers
1k views

Open problems in “Algebraic geometry by robin hartshorne”

Hi, Is there a list of which of the open problems in algebraic geometry by robin hartshorne are still open ? (I searched the internet and didn't find one) Thanks from advanced, Matan Fattal
10
votes
0answers
480 views

Once differentiable, piecewise degree three polynomials on triangulated planar domains

Here is an easily described, but very difficult, problem that I (and a number of other people) really would like to see solved during our life times. The basic problem is to compute the dimension of ...
14
votes
1answer
566 views

Totally rational polytopes

Define a convex polytope in $\mathbb{R}^d$ as totally rational (my terminology) if its vertex coordinates are rational, its edge lengths are rational, its two-dimensional face areas are rational, ...
15
votes
1answer
430 views

Convex bodies with constant maximal section function in odd dimensions

In 1970 or so, Klee asked if a convex body in $\mathbb R^n$ ($n\ge 3$) whose maximal sections by hyperplanes in all directions have the same volume must be a ball. The counterexample in $\mathbb R^4$ ...
28
votes
0answers
1k views

Shortest closed curve to inspect a sphere

Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and exterior to $S$ which has the property that every point $x$ on $S$ is visible to some point $y$ of ...
15
votes
1answer
871 views

Axiom of choice and bases of vector spaces over a fixed field

Let $k$ be a field. In 1984 Andreas Blass proved that the axiom "for every extension $K|k$, every vector space over $K$ has a basis" implies the axiom of choice. He also raised the question Does ...
7
votes
1answer
367 views

A toy model for the t-section problem

Let $S(x)$ be the area of the yellow curvilinear triangle. I'd like to find a graph for which $S(x)=H(x)$ where $H $ is some prescribed function (small, smooth, vanishing near the endpoints to any ...
46
votes
20answers
12k views

Open problems with monetary rewards

Since the old days, many mathematicians have been used to attaching monetary rewards to problems they admit are difficult. Their reasons could be to draw other mathematicians' attention, to express ...
28
votes
8answers
3k views

Series whose convergence is not known

For most of the mathematical concepts I learn, it has more or less always been possible to find (at least google and find) unsolved problems pertaining to that specific concept. Keeping a bag of ...
11
votes
4answers
8k views

The Ramanujan Problems.

I originally thought of asking this question at the Mathematics Stackexchange, but then I decided that I'd have a better chance of a good discussion here. In the Wikipedia page on Ramanujan, there is ...
19
votes
1answer
776 views

Majorization and Schur Polynomials

Let me first define the majorization order (or dominance order) on partitions as $\lambda \succeq \mu$ iff $$\sum _{i=1}^{k}\lambda_i \geq \sum_{i=1}^{k}\mu_i$$ for all $1\le k\le l-1$ and ...
34
votes
4answers
2k views

The maximum of a polynomial on the unit circle

Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself. Suppose we are given $n$ ...
8
votes
3answers
1k views

Zero divisor conjecture for finite fields

I believe that the most attractive "zero-divisor" conjecture is the existence of non-trivial zero-divisors in a group ring $\mathbb{C}[G]$ for a torsion free group $G$. For the sake of knowledge let ...
13
votes
3answers
1k views

Meromorphic 1-form and Picard's theorem

Let $D$ be the open unit disk in the complex plane and $U_1,U_2,\,\ldots\,,U_n$ be an open cover of the puntured disk $D^*= D\setminus\{0\}$. Suppose on each open set $U_j$ there is an injective ...
2
votes
0answers
532 views

Fixed point theorem for convex, closed multivalued mapping

There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem: Suppose, that $X$ is compact, convex and metrizable in locally convex Hausdorff ...
7
votes
1answer
375 views

Symplectic structures on a homotopy complex projective space

For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form. My question is as ...
4
votes
1answer
674 views

Unsolved problems concerning Artinian Rings and Artinian Modules

I am preparing a write-up on Topics on Artinian Rings and Modules for a project. I hope to mention some unsolved problems in the domain of these objects along the way. Till now, I have been able to ...
22
votes
4answers
1k views

Trees in groups of exponential growth

Question: Let $G$ be a finitely generated group with exponential growth. Is there a finite generating set $S \subset G$, such that the associated Cayley graph $Cay(G,S)$ contains a binary tree? ...
4
votes
1answer
352 views

A question concerning products of finite cyclic groups

Let $m_1,\ldots, m_n$ be pairwise coprime natural numbers $\geq 1$. We consider the product $$G(m_1,\ldots,m_n) := \prod_{i=1}^{n} \mathbb{Z} / m_i \mathbb{Z}.$$ We define $M(n)$ as the set $n$-tuple ...
23
votes
1answer
1k views

Can you flip the end of a large exotic $\mathbb{R}^4$

Can you flip the end of a large exotic $\mathbb{R}^4$ Background Definition (Exotic $\mathbb{R}^4$): An exotic $\mathbb{R}^4$ is a smooth manifold $R$ homeomorphic but not diffeomorphic to ...
3
votes
3answers
727 views

Bounds on number of conjugacy classes in terms of number of elements of a group ?

What are bounds on number of conjugacy classes in terms of number of elements of a group ? (I allowed myself to edit the question in spirit of remarkable answers given to it by Gerry Myerson and ...
19
votes
3answers
4k views

Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

The Diophantine equation $$x^3+y^3+z^3=3$$ has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
7
votes
3answers
1k views

Kronecker Approximation theorem and Fibonacci numbers

There is a famous old theorem by Kronecker that for every positive real $\alpha$ and $\epsilon>0$ there exists a positive integer n such that $\alpha n$ is within $\epsilon$ of an integer. ...
0
votes
2answers
505 views

A simple question regarding the sum-of-divisors function

A good day to everyone. Consider the following "Conjecture": If $a, b \in M \subset \mathbb{N}$, then $1 < a < b$ and ... [plus some more conditions on $a, b$ and $M$...] if and only if ...
0
votes
1answer
359 views

Existence of Solutions to an Equation Involving the Sum-of-Divisors Function [Reference Request]

Let $\sigma(x) = \sigma_1(x)$ denote the sum of all the positive divisors of $x$. If $n \in \mathbb{N}$ is odd and $\gcd(n, \sigma(n)) = 1$, then do there exist any solutions to the following ...
2
votes
1answer
299 views

A question on a special type of function

Suppose I have a function $f$, and positive integers $x$ and $y$ such that $x$ is a square, $x \ne y$ and $y \ne \sqrt{x}$. Now, assume that: $|\frac{f(x)}{y} - \frac{f(y)}{x}| > 2$ ...
43
votes
0answers
2k views

Alternating colors on a line: infinitely often or converge?

Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / ...). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all ...
12
votes
3answers
894 views

Card game / options pricing / Brownian bridge question

We play a game. I shuffle a deck of cards and start dealing them face up. After any card you can say "stop", at which point I pay you 1 dollar for every red card dealt and you pay me 1 for every black ...
16
votes
1answer
790 views

What's the Kirby Diagram of a universal $\mathbb{R}^4$?

What's the Kirby diagram of a universal $\mathbb{R}^4$? Background Define $\mathcal{R}$ as the set of smoothings of $\mathbb{R}^4$. For two oriented elements $R_1$, $R_2$ in $\mathcal{R}$ we can ...
13
votes
0answers
463 views

Comparing the Kazhdan-Lusztig and Steinberg pre-orders

Both Kazhdan-Lusztig and Steinberg have defined pairs of preorders on $S_n$. Kazhdan and Lusztig's preorders come from their basis: We write $x\leq_L y$ if any left ideal spanned by K-L basis ...
0
votes
1answer
341 views

Something new in old question about sums of three polynomial cubes ?

An old problem asks whether or not the polynomial $$ t \in \mathbb{Q}[t] $$ is a sum of three cubes, (of polynomials in $\mathbb{Q}[t]$). Question: Something new known now ? Somebody has an idea of ...
13
votes
2answers
720 views

Free subgroups vs law

Consider the following two conditions for a group $G$: (1) $G$ does not satisfy a nontrivial law. (2) $G$ contains a non-abelian free subgroup. Obviously (2) implies (1) and it is easy to ...
55
votes
1answer
3k views

Nontrivial finite group with trivial group homologies?

I stumbled across this question in a seminar-paper a long time ago: Does there exist a positive integer $N$ such that if $G$ is a finite group with $\bigoplus_{i=1}^NH_i(G)=0$ then $G=\lbrace ...
13
votes
1answer
585 views

Amenability of groups in terms of a perturbation condition

Let $G$ be a countable group and $\lambda \colon G \to U(\ell^2 G)$ its left-regular representation. Suppose that there exists a constant $C>0$ such that for all $T \in B(\ell^2 G)$ $$\inf ...
4
votes
1answer
413 views

Implication of Polignac's conjecture on prime distribution in models of PA

Polignac's conjecture (PC) is that there exists infinitely many pairs of consecutive prime numbers that are a distance $d$ apart for some natural number $d$. The twin prime conjecture is the ...
15
votes
9answers
4k views

Open Questions in Riemannian Geometry

What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.