# Unanswered Questions

2answers
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### Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}\$ such that $f:\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?
0answers
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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 ...
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### 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 ...
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### 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 ...
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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 ...
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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 ... 0answers 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 ... 0answers 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 ... 1answer 543 views ### Continuous maps which send intervals of$\mathbb{R}$to convex subsets of$\mathbb{R}^2$Let$f : \mathbb{R} \longrightarrow \mathbb{R}^2$be a continuous map which sends any interval$I \subseteq \mathbb{R}$to a convex subset$f(I)$of$\mathbb{R}^2$. Is it true that there must be a ... 0answers 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, ... 0answers 3k 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 ... 1answer 1k views ### Transitivity on$\mathbb{N}_0$— a 42 problem Let$r(m)$denote the residue class$r+m\mathbb{Z}$, where$0 \leq r < m$. Given disjoint residue classes$r_1(m_1)$and$r_2(m_2)$, let the class transposition$\tau_{r_1(m_1),r_2(m_2)}$be the ... 0answers 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 ... 0answers 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 ... 0answers 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 ...

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