# All Questions

**17**

votes

**1**answer

4k views

### About Goldbach's conjecture

let's consider a composite natural number $n$ greater or equal to $4$. Goldbach's conjecture is equivalent to the following statement: "there is at least one natural number $r$ such as $(n-r)$ and ...

**8**

votes

**1**answer

1k views

### The cyclic subfactors theory: a quantum arithmetic?

Context: First recall some results:
- Actions of finite groups on the hyperfinite type $II_{1}$ factor $R$ (Jones 1980).
- A Galois correspondence for depth 2 irreducible subfactors ...

**144**

votes

**79**answers

25k views

### Not especially famous, long-open problems which anyone can understand

Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list ...

**425**

votes

**185**answers

108k views

### Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...

**123**

votes

**33**answers

31k views

### Widely accepted mathematical results that were later shown wrong?

I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time ...

**66**

votes

**36**answers

24k views

### Most interesting mathematics mistake?

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincare's 3d sphere charaterization or the search to prove ...

**21**

votes

**2**answers

3k views

### Does the curvature determine the metric?

Hello,
I ask myself, whether the curvature determines the metric.
Concretely: Given a compact Riemannian manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that ...

**12**

votes

**1**answer

739 views

### Non-“weakly group theoretical” integral fusion categories?

Is there an integral fusion category of global dimension $210$, such that the simple objects have dimensions $\{1,5,5,5,6,7,7\}$ and the following fusion matrices?
$\small{\begin{smallmatrix}
1 & ...

**4**

votes

**3**answers

423 views

### Jordan-Hölder theorem for subfactors?

All the subfactors $(N\subset M)$ are irreducible and finite index inclusions of II$_1$ factors.
First recall that in this paper, D. Bisch characterizes the Jones projections $e_K$ of the ...

**130**

votes

**35**answers

28k views

### Why is a topology made up of 'open' sets? [closed]

I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...

**91**

votes

**15**answers

17k views

### Why worry about the axiom of choice?

As I understand it, it has been proven that the axiom of choice is independent of the other axioms of set theory. Yet I still see people fuss about whether or not theorem X depends on it, and I don't ...

**50**

votes

**1**answer

5k views

### The amplituhedron minus the physics

Is it possible to appreciate the geometric/polytopal properties of the amplituhedron without delving into the physics that gave rise to it?
All the descriptions I've so far encountered assume ...

**21**

votes

**2**answers

2k views

### Similarities between Post's Problem and Cohen's Forcing

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated ...

**11**

votes

**2**answers

1k views

### Euler characteristic of a manifold and self-intersection

This is probably quite easy, but how do you show that the Euler characteristic of a manifold M (defined for example as the alternating sum of the dimensions of integral cohomology groups) is equal to ...

**5**

votes

**1**answer

301 views

### coloring in lattice

This is a mathematical question raised from engineering and physics:
Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For ...

**4**

votes

**0**answers

291 views

### Reference for Wang Tile

I am working on projects in solving ground state of generalized ising models. One recent work involves tiling with basic tiles that filled the whole lattice. For example, we could obtain results:
...

**0**

votes

**0**answers

175 views

### Lifting a quadratic system to a non vanishing vector field on $S^{3}$ or $T^{1} S^{2}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...

**108**

votes

**75**answers

34k views

### Best online mathematics videos?

I know of two good mathematics videos available online, namely:
Sphere inside out (part I and part II)
Moebius transformation revealed
Do you know of any other good math videos? Share.

**60**

votes

**9**answers

16k views

### solving f(f(x))=g(x)

This question is of course inspired by the question How to solve f(f(x))=cosx
and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...

**38**

votes

**2**answers

12k views

### Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those ...

**33**

votes

**5**answers

3k views

### Is the Riemann Hypothesis equivalent to a $\Pi_1$ sentence?

1) Can the Riemann Hypothesis (RH) be expressed as a $\Pi_1$ sentence?
More formally,
2) Is there a $\Pi_1$ sentence which is provably equivalent to RH in PA?
(This is mentioned in P. ...

**11**

votes

**7**answers

2k views

### Bijection between irreducible representations and conjugacy classes of finite groups

Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of $S_n$)?

**19**

votes

**4**answers

21k views

### Eigenvalues of Matrix Sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum?
What about the special case when they are Hermitian and positive-definite?
I am investigating ...

**11**

votes

**3**answers

1k views

### Is there a purely group-theoretic reformulation of an equivalence of subgroups?

There is an equivalence relation between inclusion of finite groups coming from the world of subfactors:
Definition: $(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$ if $(R^{G_{1}} \subset ...

**11**

votes

**1**answer

427 views

### Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?

Let the proof theoretic ordinal $\alpha$ of a theory $T$ be the least recursive ordinal such that $T$ does not prove that $\alpha$ is well-founded. This ordinal is intended to quantify in some sense ...

**3**

votes

**0**answers

183 views

### Maximization of a total variation distance subject to another total variation distance in Markov chain

Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...

**200**

votes

**22**answers

23k views

### Thinking and Explaining

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words ...

**114**

votes

**29**answers

36k views

### Real-world applications of mathematics, by arxiv subject area?

What are the most important applications outside of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to arxiv mathematics categories, ...

**88**

votes

**8**answers

9k views

### If $f$ is infinitely differentiable then $f$ coincides with a polynomial

Let $f$ be an infinitely differentiable function on $[0,1]$ and suppose that for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{(n)}(x)=0$. Then does $f$ coincide on $[0,1]$ ...

**95**

votes

**64**answers

19k views

### Which math paper maximizes the ratio (importance)/(length)?

My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are ...

**82**

votes

**38**answers

15k views

### Examples of eventual counterexamples

Define an "eventual counterexample" to be
$P(a) = T $ for $a < n$
$P(n) = F$
$n$ is sufficiently large for $P(n) = T\ \ \forall n \in \mathbb{N}$ to be a 'reasonable' conjecture to make.
where ...

**63**

votes

**3**answers

6k views

### Does every non-empty set admit a group structure (in ZF)?

It is easy to see that in ZFC, any non-empty set $S$ admits a group structure: for finite $S$ identify $S$ with a cyclic group, and for infinite $S$, the set of finite subsets of $S$ with the binary ...

**58**

votes

**8**answers

13k views

### How to solve f(f(x)) = cos(x) ?

I found the following interesting equation on some web page I cannot remember:
$f(f(x))=\cos(x)$
Out of curiosity I tried to solve it, but realized that I do not have a clue how to approach such an ...

**84**

votes

**3**answers

8k views

### Convergence of $\sum(n^3\sin^2n)^{-1}$

I saw a while ago in a book by Clifford Pickover, that whether $\displaystyle \sum_{n=1}^\infty\frac1{n^3\sin^2 n}$ converges is open.
I would think that the question of its convergence is really ...

**58**

votes

**5**answers

5k views

### What do epimorphisms of (commutative) rings look like?

(Background: In any category, an epimorphism is a morphism $f:X\to Y$ which is "surjective" in the following sense: for any two morphisms $g,h:Y\to Z$, if $g\circ f=h\circ f$, then $g=h$. Roughly, ...

**104**

votes

**7**answers

7k views

### Does $\mathrm{Aut}(\mathrm{Aut}(…\mathrm{Aut}(G)…))$ stabilize?

Purely for fun, I was playing around with iteratively applying $\DeclareMathOperator{\Aut}{Aut}\Aut$ to a group $G$; that is, studying groups of the form
$$ {\Aut}^n(G):= \Aut(\Aut(...\Aut(G)...)) $$
...

**52**

votes

**1**answer

3k views

### What is the amplituhedron?

The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces ...

**58**

votes

**4**answers

7k views

### How small can a sum of a few roots of unity be?

Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...

**36**

votes

**37**answers

12k views

### Major mathematical advances past age fifty [closed]

From A Mathematician’s Apology, G. H. Hardy, 1940:
"I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever ...

**46**

votes

**5**answers

5k views

### Decidability of chess on an infinite board

The recent question Do there exist chess positions that require exponentially many moves to reach? of Tim Chow reminds me of a problem I have been interested in. Is chess with finitely many men on an ...

**36**

votes

**14**answers

30k views

### If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...

**32**

votes

**6**answers

3k views

### Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...

**31**

votes

**7**answers

5k views

### Arguments against large cardinals

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...

**23**

votes

**9**answers

2k views

### Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature

A curve in the plane is determined, up to orientation-preserving
Euclidean
motions, by its curvature function, $\kappa(s)$.
Here is one of my favorite examples, from
Alfred Gray's book,
Modern ...

**28**

votes

**4**answers

2k views

### Cliques, Paley graphs and quadratic residues

A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs.
If p=1 mod 4 is a prime, we can define the ...

**22**

votes

**4**answers

2k views

### Which number fields are monogenic? and related questions

A number field $K$ is said to be monogenic when $\mathcal{O}_K=\mathbb{Z}[\alpha]$ for some $\alpha\in\mathcal{O}_K$. What is currently known about which $K$ are monogenic? Which are not? From ...

**16**

votes

**1**answer

690 views

### Examples of finite groups with “good” bijection(s) between conjugacy classes and irreducible representations?

For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups ...

**15**

votes

**2**answers

2k views

### Are non-PL manifolds CW-complexes?

Can every topological (not necessarily smooth or PL) manifold be given the structure of a CW complex?
I'm pretty sure that the answer is yes. However, I have not managed to find a reference for ...

**25**

votes

**2**answers

2k views

### Number of elements in the set $\{1,\cdots,n\}\times\{1,\cdots,n\}$

Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?

**19**

votes

**2**answers

1k views

### Are most curves over Q pointless?

Fresh out of the arXiv press is the remarkable result of Manjul Bhargava saying that most hyperelliptic curves over $\mathbf{Q}$ have no rational points. Don Zagier suggests the paraphrase : Most ...