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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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### 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 ...
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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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### 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 ...
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### Vector bundle $L$ admits connection if and only if degree of every direct summand of $L$ divisible by $\text{char}\,k$, intuition

Consider the following theorem of Atiyah. Let $X$ be a connected smooth projective curve over an algebraically closed field $k$. Then a vector bundle $L$ on $X$ admits a connection if and only if ...
### Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?
This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here. $\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an ...
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 ...