# Unanswered Questions

**126**

votes

**0**answers

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### Ultrafilters and automorphisms of the complex field

It is well-known that it is consistent with $ZF$ that the only automorphisms of the complex field $\mathbb{C}$ are the identity map and complex conjugation. For example, we have that ...

**74**

votes

**0**answers

6k views

### Dropping three bodies

Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...

**57**

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**0**answers

3k views

### Volumes of Sets of Constant Width in High Dimensions

Background
The n dimensional Euclidean ball of radius 1/2 has width 1 in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between them ...

**53**

votes

**0**answers

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### The topology of Arithmetic Progressions of primes

The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner ...

**51**

votes

**0**answers

2k views

### 2, 3, and 4 (a possible fixed point result ?)

The question below is related to the classical Browder-Goehde-Kirk fixed point theorem.
Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$
be a mapping such that $\Vert ...

**50**

votes

**0**answers

4k views

### Grothendieck-Teichmuller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmuller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here ...

**46**

votes

**0**answers

2k views

### Complex structure on $S^6$ gets published in Journ. Math. Phys

A paper by Gabor Etesi was published that purports to solve a major outstanding problem:
Complex structure on the six dimensional sphere from a spontaneous symmetry breaking
Journ. Math. Phys. 56, ...

**44**

votes

**0**answers

2k views

### A naive question about the double dual of a vector space

Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor
$$
V\mapsto V^{**}/V
$$
of the category of $K$-vector spaces?
I asked a related question on Mathematics Stackexchange, but ...

**41**

votes

**0**answers

3k views

### What is an étale theta function?

Let me start out by urging you to take seriously that whatever I write about the papers surrounding IUTT really are questions. If you would like to use it as a guide to the mathematics in any way, ...

**40**

votes

**1**answer

4k views

### Open map D⁴ → S²

Is it possible to construct an embedding $D^4\hookrightarrow S^2\times
\mathbb R^2$
such that the projection $D^4\to S^2$ is an open map?
Here $D^n$ denotes closed $n$-ball.
An open map D⁴ → S².
It ...

**39**

votes

**0**answers

2k views

### A short proof for $\dim(R[T])=\dim(R)+1$

For a commutative ring $R$ we clearly have $\dim(R[T]) \geq \dim(R)+1$. If $R$ is noetherian, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...

**39**

votes

**0**answers

2k views

### Constructing non-torsion rational points (over Q) on elliptic curves of rank > 1

Consider an elliptic curve $E$ defined over $\mathbb Q$. Assume that the rank of $E(\mathbb Q)$ is $\geq2$. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank $=$ algebraic ...

**38**

votes

**0**answers

1k views

### Intersecting Family of Triangulations

Let $\cal T_n$ be the family of all triangulations on an $n$-gon using $(n-3)$ non-intersecting diagonals. The number of triangulations in $\cal T_n$ is $C_{n-2}$ the $(n-2)$th Catalan number. Let ...

**36**

votes

**0**answers

1k views

### What does the theta divisor of a number field know about its arithmetic?

This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link).
Let me first ...

**36**

votes

**0**answers

1k views

### To what extent does Spec R determine Spec of the Witt vector ring over R?

Let $R$ be a perfect $\mathbb{F}_p$-algebra and write $W(R)$ for the Witt ring [i.e., ring of Witt vectors -- PLC] on $R$. I want to know how much we can deduce about $\text{Spec } W(R)$ from ...