Suppose we define an arbitrary field extension $K/F$ to be Galois if, for all subextensions $L$ of $K/F$, we have $K^{\operatorname{Aut}(K/L)} = L$. In words: for any element $x$ o …

Let $\approx$ be the binary relation on the class of finitely generated groups such that $G \approx H$ iff $G$ and $H$ have isomorphic (unlabeled nondirected) Cayley graphs with r …

The Principles of Algebraic Geometry is a great book with, IMHO, many typos and mistakes. Why don't we collaborate to write a full list of all of its typos, mistakes etc? My sugg …

Let $(V,A)$ be a tournament. A subset of vertices $V'\subseteq V$ is stable if there exists no $v\in V\setminus V'$ such that $V'\cup${$v$} contains an inclusion-maximal transitiv …

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

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 …

I have two different and probably unrelated questions that can both be superficially described by the title, so I hope you'll forgive me if I ask them together. They both fall und …

Here is a question which I asked myself (and couldn't answer) while reading "The topology of spaces of rational functions" by G. Segal. Let $X$ be a smooth complete complex curve …

Galois deformations are an important tool in Wiles' arsenal for proving FLT. Are there any more elementary aspects (I'm thinking of 1-dimensional Galois representations attached t …

(Question was edited; the previous version concerned the diameter instead of area and appears at the end.) Background. The Brunn-Minkowski(-Lusternik) inequality expresses the co …

Short version: Let (C, ⊗, 1) be a locally presentable closed symmetric monoidal category with a zero object, and write 2 = 1 ∐ 1. Suppose the object 2 has a dual. Does it …

What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$? Apperently all orientable 4-ma …

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 …

Usually the Riemann zeta function $\zeta(s)$ gets multiplied by a "gamma factor" to give a function $\xi(s)$ satisfying a functional equation $\xi(s)=\xi(1-s)$. If I changed this g …

Background I'm thinking about this because it would be neat to understand "Catalan phenomena" for complex reflection groups in a uniform way. One remaining piece of the algebraic …