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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 3k 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 ... 1answer 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 ... 0answers 2k 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 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 ### a naive question about the second 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 ... 0answers 891 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 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 ... 0answers 728 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,$$\# ... 0answers 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 ... 0answers 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 ...

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