# Unanswered Questions

**103**

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### 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 $G_{\...

**73**

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4k 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 ...

**68**

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### Hironaka's proof of resolution of singularities in positive characteristics

Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier...
Unlike Atiyah's paper, Hironaka's paper does not have a ...

**65**

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1k views

### Topological cobordisms between smooth manifolds

Wall has calculated enough about the cobordism ring of oriented smooth manifolds that we know that two oriented smooth manifolds are oriented cobordant if and only if they have the same Stiefel--...

**62**

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### 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 Tx-Ty\...

**60**

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1k views

### Converse to Euclid's fifth postulate

There is a fascinating open problem in Riemannian Geometry which I would like to advertise here because I do not think that it is as well-known as it deserves to be. Euclid's famous fifth postulate, ...

**60**

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2k views

### Why do combinatorial abstractions of geometric objects behave so well?

This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference: http://www.math.harvard.edu/cdm/.
Here are two examples of the kind of combinatorial ...

**58**

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2k views

### The exponent of Ш of $y^2 = x^3 + px$, where $p$ is a Fermat prime

For $d$ a non-zero integer, let $E_d$ be the elliptic curve
$$
E_d : y^2 = x^3+dx.
$$
When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD,
$$
\# Ш(E_p)...

**55**

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### Normalizers in symmetric groups

Question: Let $G$ be a finite group. Is it true that there is a subgroup $U$ inside some symmetric group $S_n$, such that $N(U)/U$ is isomorphic to $G$? Here $N(U)$ is the normalizer of $U$ in $S_n$.
...

**55**

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### 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 ...

**51**

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### Which region in the plane with a given area has the most domino tilings?

I just finished teaching a class in combinatorics in which I included a fairly easy upper bound on the number of domino tilings of a region in the plane as a function of its area. So this led to ...

**47**

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11k views

### Atiyah's May 2018 paper on the 6-sphere

A couple years ago Atiyah published a claimed proof that $S^6$ has no complex structure. I've heard murmurs and rumors that there are problems with the argument, but just a couple months ago he ...

**47**

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### Dualizing the Notion of Topological Space

$\require{AMScd}$
Defining a topological space on a set $X$ is equivalent to designating certain subobjects of $X$ in ${\bf Set}$ (monomorphisms into $X$ up to equivalence) as open. The requirements ...

**47**

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### Are there periodicity phenomena in manifold topology with odd period?

The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$:
$n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...

**47**

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### 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 ...