Unanswered Questions
73
votes
2answers
2k views
Can we ascertain that there exists an epimorphism $G\rightarrow H$?
Let $G,H$ be finite groups. Suppose we have an epimorphism $G\times G\rightarrow H\times H$. Can we find an epimorphism $G\rightarrow H$?
55
votes
1answer
2k views
Connected metric spaces with disjoint open balls
Let $X$ be the $S^1$ or a connected subset thereof, endowed with the standard metric. Then every open set $U\subseteq X$ is a disjoint union of open arcs, hence a disjoint union of open balls.
Are ...
36
votes
0answers
663 views
Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G \,\,\,s.t.\,\,\,H\cong G/N$?
A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization for groups for which this ...
33
votes
0answers
597 views
How to think of the group ring as a Hopf algebra?
Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
29
votes
0answers
650 views
Is there an atlas of Algebraic Groups and corresponding Coordinate rings?
I was wondering if there was a resource that listed known algebraic groups and their corresponding coordinate rings.
Edit: The previous wording was terrible.
Given an algebraic group $G$, with Borel ...
27
votes
0answers
519 views
Can one deduce Liouville's theorem (in complex analysis) from the non-emptiness of spectra in complex Banach algebras?
As you probably know, the classical proof of the non-emptiness of the spectrum for an element $x$ in a general Banach algebra over $\mathbb{C}$ can be proven quite easily using Liouville's theorem in ...
27
votes
0answers
1k views
Simplicial homology of real projective space by Mayer-Vietoris
Consider the $n$-sphere $S^n$ and the real projective space $\mathbb{RP}^n$. There is a universal covering map $p: S^n \to \mathbb{RP}^n$, and it's clear that it's the coequaliser of $\mathrm{id}: S^n ...
25
votes
0answers
618 views
All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$
Let $P$ be a polynomial with integer coefficients such that the equation $P(x)=0$ has no positive integer solutions. Find all polynomials $P(x)$ such that for all positive integers $n$ we have ...
24
votes
0answers
1k views
Does a four-variable analog of the Hall-Witt identity exist?
Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125):
An amazing commutator formula is the Hall-Witt identity: ...
24
votes
1answer
377 views
A Bernoulli number identity and evaluating $\zeta$ at even integers
Sometime back I made a claim here that the proof for $\zeta(4)$ can be extended to all even numbers.
I tried doing this but I face a stumbling block.
Let me explain the problem in detail here. I was ...
24
votes
2answers
781 views
Eigenvalue problem: Prove that all of the eigenvalues of $A$ are 1.
Here's a cute problem that was frequently given by the late Herbert Wilf during his talks.
Problem: Let $A$ be an $n \times n$ matrix with entries from $\{0,1\}$ having all positive eigenvalues. ...
22
votes
0answers
285 views
Number of simple edge-disjoint paths needed to cover a planar graph
Let $G=(V,E)$ be a graph with $|E|=m$ of a graph class $\mathcal{G}$. A path-cover $\mathcal{P}=\{P_1,\ldots,P_k\}$ is a partition of $E$ into edge-disjoint simple paths. The size of the cover is ...
21
votes
0answers
366 views
Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$
Do you know an explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? Using the axiom of choice, every vector space admits a norm but have you an explicit formula on ...
21
votes
2answers
409 views
Operators with finite spectrum
Suppose that $T$ is a bounded operator with finite spectrum. What happens with the spectrum of $T+F$, where $F$ has finite rank? Is it possible that $\sigma(T+F)$ has non-empty interior? Is it always ...
20
votes
1answer
708 views
$4494410$ and friends
$4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number. ...
20
votes
0answers
302 views
The Ring Game on $K[x,y,z]$
I recently read about the Ring Game on Mathoverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
20
votes
3answers
411 views
Why is the topological pressure called pressure?
Let us consider a compact topological space $X$, and a continuous function $f$ acting on $X$. One of the most important quantities related to such a topological dynamical system is the entropy.
For ...
19
votes
0answers
465 views
What is the solution to Nash's problem presented in “A Beautiful Mind”?
I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve ...
19
votes
0answers
327 views
The tensor product of two Artinian modules is Artinian
user$xxxxx$ posted (and then deleted) the following question which I think deserves to be here:
Prove that the tensor product of two Artinian modules is Artinian.
19
votes
0answers
449 views
Automorphisms inducing automorphisms of quotient groups
Let $G$ be a group, with $N$ characteristic in $G$. As $N$ is characteristic, every automorphism of $G$ induces an automorphism of $G/N$. Thus, $\operatorname{Aut}(G)\rightarrow ...
18
votes
0answers
284 views
A short proof for $\dim(R[T])=\dim(R)+1$?
If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and nontrivial ...
18
votes
1answer
739 views
$\sum_{k=0}^{100} a_k x^{k}=0, a_i \in \mathbb{Z}$ How many equations?
Let $x^{100}+a_{99} x^{99}+a_{98} x^{98}+ \dots +a_{1}x +a_{0}=0, \ \ a_{100}=1, a_i \in \mathbb{Z}$, is an algebraic equation with integer coefficients.
Assume that all (100 with multiplicity) ...
18
votes
0answers
1k views
Undergraduate Research Topics in Analysis
I am thinking of doing some research (Real Analysis preferably) under a professor this year and I need ideas for interesting topics that might be accessible to an undergraduate.
Does anyone have any ...
18
votes
0answers
390 views
Computing the Chern-Simons invariant of SO(3)
I am an undergraduate learning about gauge theory and I have been tasked with working through the two examples given on pages 65 and 66 of "Characteristic forms and geometric invariants" by Chern and ...
17
votes
0answers
358 views
Generalizing Ramanujan's proof of Bertrand's Postulate: Can Ramanujan's approach be used to show a prime between $4x$ and $5x$ for $x \ge 3$
Perhaps, I've been thinking too long about Ramanujan's proof, but it appears to me that his argument can be generalized beyond $x$ and $2x$. My argument below attempts to show that for $x \ge 1331$, ...
17
votes
0answers
314 views
Ambiguous Curve: can you follow the bicycle?
Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance ...
17
votes
0answers
245 views
What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?
I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
17
votes
0answers
277 views
Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field
If $K$ is a number field, whose Galois closure over the rationals has degree 24 or so, and whose discriminant is around $163^4$, then what is a numerically efficient way of computing the first few ...
16
votes
2answers
403 views
A question about connected sets in $\mathbb{R}^2$
Let $C\subseteq\mathbb{R}^2$ be connected, open and have a bounded complement. Let $u\in C$ and $f:[0,1]\rightarrow \mathbb{R}^2$ be a continuous injective function such that $f(0)=u$. It is also ...
16
votes
0answers
222 views
Is there a categorical definition of submetry?
(Updated to include effective epimorphism.)
This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ...
