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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17
votes
1answer
890 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
477 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
361 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 ...
14
votes
2answers
763 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 ...
59
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 ...
14
votes
1answer
622 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
455 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 ...
21
votes
2answers
1k views

unboundedness of number of integral points on elliptic curves?

If $E/\mathbf{Q}$ is an elliptic curve and we put it into minimal Weierstrass form, we can count how many integral points it has. A theorem of Siegel tells us that this number $n(E)$ is finite, and ...
6
votes
0answers
314 views

Real approximation for homogeneous spaces of linear algebraic groups

Let $X$ be a smooth geometrically integral variety over $\mathbf{Q}$, having a $\mathbf{Q}$-point. We say that $X$ has the real approximation property if $X(\mathbf{Q})$ is dense in $X(\mathbf{R})$. ...
8
votes
3answers
1k views

Prepending strings to primes.

Hello, we all know that 31,331,3331,33331,333331,3333331,33333331 all are primes, and that 333333331 is not. Here we prepend the digit 3 to 31, to get a list of 7 primes.This gives me the following ...
8
votes
1answer
2k views

The Invariant Subspace Problem: examples

Question. Is there a concrete example of a bounded linear operator on a Hilbert space for which it is not known if it has a non-trivial closed invariant subspace? [Added 24.01.2011: According to ...
22
votes
1answer
2k views

Iwasawa main conjectures vs Bloch-Kato conjectures

Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the ...
0
votes
1answer
814 views

Perfect Numbers - On Mersenne and Euler Primes

Hi, I apologize if there is already an (obvious) answer to my question, but please bear with me for the moment as I find it hard to see a good way to answer this question: In the same way that the ...
19
votes
2answers
5k views

Is the Invariant Subspace Problem interesting?

There's an amusing comment in Peter Lax's Functional Analysis book. After a brief description of the Invariant Subspace Problem, he says (paraphrasing) "...this question is still open. It is also an ...
1
vote
1answer
487 views

Infinitely many pairs of primes?

Hi. I want to know how many (infinitely many) pairs of primes are known. For convinience, let me give two definitions. For any nonconstant polynomial $f(x)\in \mathbb{Z}[x]$, define $A_{f}=\lbrace ...
3
votes
3answers
2k views

Re: Mordell's Equation $y^2 = x^3 + k$ and Perfect Numbers

I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on. My question is: When does Mordell's Equation $$y^2 = x^3 ...
2
votes
1answer
543 views

Reference Request - Sharp Estimates for a Logarithmic Sum

Can anybody suggest a good (e.g. "non-technical") introduction to estimating bounds for logarithmic sums of the form $$\sum_{i=1}^{r}{{\alpha_i}{\log(q_i)}}$$ where the $$\alpha_i$$ are positive ...
4
votes
2answers
956 views

On Sorli's Conjecture Re: OPNs (Circa 2003)

In the PhD dissertation titled "Algorithms in the Study of Multiperfect and Odd Perfect Numbers" (hyperlinked here) and completed in 2003, Ronald Sorli conjectured that the exponent $k$ on the Euler ...
6
votes
1answer
979 views

A Game of Knights and Queens

Let $m,n,u,v \in \mathbb{N}$ be parameters with $m,n \geq 3$. Suppose two players play a game on a $m \times n$ chess board and we denote the squares of the board by the set of points $ (i,j) $ such ...
4
votes
2answers
691 views

Artin's conjecture for n=2

I am interested in the following question: Is it known that $2$ is a primitive root modulo $p$ for infinitely many primes $p$? there is some information about Artin's conjecture in ...
16
votes
1answer
1k views

Nonnegative to Positive Curvature.

This questions asks for your intuition and insight as I'm surprised by how little is known about the difference between nonnegative and positive curvature. I don't want to be completely vague, so I ...
37
votes
3answers
3k views

Difficult examples for Frankl's union-closed conjecture

Frankl's well-known union-closed conjecture states that if F is a finite family of sets that is closed under taking unions (that is, if A and B belong to the family then so does $A\cup B$), then there ...
16
votes
1answer
851 views

Covering the primes by arithmetic progressions

Define the length of a set of arithmetic progressions of natural numbers $A=\lbrace A_1, A_2, \ldots \rbrace$ to be $\min_i | A_i |$: the length of the shortest sequence among all the progressions. ...
1
vote
0answers
433 views

Zeroes of a tricky function.

I am attempting to show that there does not exist an N past which every open unit interval (k, k+1) -where k is an integer- contains a zero of the following function: $h(x)=\sum_{n=2}^{[\sqrt(x)]} ...
2
votes
0answers
361 views

Natural numbers n which satisfy gnu(n)=n?

Are there any natural numbers $n$ (other than 1) for which $gnu(n)=n$? We define $gnu(n)$ to be the number of isomorphism classes of groups of order $n$. This question popped into my head today, and ...
5
votes
0answers
550 views

Are there an infinite number of prime Euclid numbers?

A number defined as the product of first $n$ prime numbers $+1$ is called $n$th Euclid number. Are there any survey on the progress for answering the following question: are there an infinite number ...
32
votes
1answer
1k views

Density of values of polynomials in two variables

This question is a reposting of a comment I made on Polynomial representing all nonnegative integers. Let $f(x,y)\in \mathbb Q[x,y]$ such that $f(\mathbb Z\times \mathbb Z)$ is a subset of $\mathbb ...
10
votes
1answer
830 views

What polynomials biject from $\mathbb{N}^{2}$ to $\mathbb{N}$?

Perhaps there are none with integral coefficients; so let us admit rational coefficients. The map $(x, y) \mapsto x + \frac{1}{2}(x + y)(x + y + 1)$ is well known, and swapping $x$ and $y$ in the ...
24
votes
1answer
2k views

Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle

Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely: Question: (Furstenberg) Let $\mu$ be a ...
22
votes
2answers
2k views

Is there a 7-regular graph on 50 vertices with girth 5? What about 57-regular on 3250 vertices?

The following problem is homework of a sort -- but homework I can't do! The following problem is in Problem 1.F in Van Lint and Wilson: Let $G$ be a graph where every vertex has degree $d$. ...
13
votes
1answer
1k views

The determinant of the sum of normal matrices

Given two normal matrices $A,B\in M_n({\mathbb C})$ whose respective spectra are $(\alpha_{1},\ldots,\alpha_{n})$ and $(\beta_{1},\ldots,\beta_{n})$, is it true that $\det(A+B)$ belongs to the ...
10
votes
2answers
1k views

Midpoint geodesic polygon / Birkhoff curve shortening

I would like to know under what conditions the process of creating a midpoint piecewise geodesic polygon converges on a surface $S \subset \mathbb{R}^3$. $S$ may be assumed smooth, closed, and ...
5
votes
1answer
464 views

Milnor's isotopy invariant using spectral sequence?

I'm reading stalling's article "the augmented ideal in group ring" in Ann. Math. Studies 84(R. H. Fox memorial volume) In his final remark, he says that Milnor's link invariant could be interpreted ...
16
votes
0answers
893 views

Special values of Artin L-functions

This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so. Background: Stark's conjecture interprets ...
8
votes
2answers
636 views

Base locus of divisors on blowings up of the projective space

Let $X$ be the blowing-up of $\mathbb{P}^n$ in $r$ points in general position. Let $\{H,E_1,...,E_r\}$ be the standard basis of $Pic(X)$. Further let $D=dH-\sum m_j E_j$, be an effective divisor, with ...
30
votes
0answers
2k views

A Combinatorial Abstraction for The “Polynomial Hirsch Conjecture”

Consider $t$ disjoint families of subsets of {1,2,…,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ . Suppose that (*) For every $i \lt j \lt k$ and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$, ...
27
votes
2answers
2k views

Projective Plane of Order 12

I asked this question on the new Theoretical Computer Science "overflow" site, and commenters suggested I ask it here. That question is here, and it contains additional links, which I doubt I can ...
21
votes
6answers
2k views

What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$?

Some experts have a hunch that for any nonnegative integer $r$ there are infinitely many elliptic curves over $\Bbb Q$ with Mordell-Weil rank at least $r$. The best empirical evidence for this hunch ...
17
votes
2answers
2k views

Walking to infinity on the primes: The prime-spiral moat problem

It is an unsolved problem to decide if it is possible to "walk to infinity" from the origin with bounded-length steps, each touching a Gaussian prime as a stepping stone. The paper by Ellen Gethner, ...
34
votes
2answers
2k views

Euler and the Four-Squares Theorem

There are several questions in the Euler-Goldbach correspondence that I am unable to answer. Sometimes it does not take very much: in his letter to Goldbach dated June 9th, 1750, Euler conjectured ...
28
votes
8answers
9k views

What are some open problems in algebraic geometry?

What are the open big problems in algebraic geometry and vector bundles? More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over ...
2
votes
1answer
1k views

The importance of Poincare Conjecture or SPC4?

As well known, Perelman proved Poincare conjecture by proving Thurston's Geometrization conjecture. Somebody says that we can understand part of the universe from Poincare conjecture. As a purely ...
13
votes
2answers
1k views

Detecting almost-primes quickly

There are many fast algorithms (deterministic and probabilistic) for detecting primality. Are there any fast algorithms (probabilistic ones allowed) known for detecting whether a number is the product ...
24
votes
0answers
1k views

Finite-dimensional subalgebras of $C^\star$-algebras

Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...
11
votes
2answers
2k views

What impact would P!=NP have on the characterization of BQP?

Many complexity theorists assume that $P\ne NP.$ If this is proved, how would it impact quantum computing and quantum algorithms? Would the proof immediately disallow quantum algorithms from ever ...
12
votes
0answers
2k views

How can an approach to $P$ vs $NP$ based on descriptive complexity avoid being a natural proof in the sense of Raborov-Rudich?

EDIT: This question has been modified to make it a stand-alone question. Feel free to retract your votes for the previous version. Here are Vinay Deolalikar's paper, and Richard Lipton's first post ...
6
votes
1answer
726 views

Reconstruction Conjecture: Group theoretic formulation

As we read from wiki, informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. Is there a group-theoretic formulation of this ...
8
votes
3answers
1k views

Conjugacy classes and reduced group $C^*$-algebra of an amenable group

The reduced $C^*$-algebra of a non-abelian free group $G$ has a unique trace. Hence, there is no chance to separate conjugacy classes of group elements using traces on $C^\star_{red} G$. On the other ...
11
votes
3answers
2k views

Permute Wada Lakes keeping the coastline intact? (still open in dim >2)

Wada Lakes are three disjoint open subsets of $\mathbb R^2$ with common boundary. Originally they were constructed by hand, but they also arise naturally in the real life, that is, theory of dynamical ...
16
votes
1answer
1k views

Are all primes in a PAP-3?

Van der Corput [1] proved that there are infinitely many arithmetic progressions of primes of length 3 (PAP-3). (Green & Tao [2] famously extended this theorem to length $k$.) But taking this in ...