Unanswered Questions

18
votes
1answer
651 views

transcendental Galois theory

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 …
17
votes
0answers
146 views

Cayley graphs of finitely generated groups

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 …
16
votes
0answers
626 views

Errata to Principles of Algebraic Geometry

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 …
15
votes
0answers
359 views

Disjoint stable sets in tournaments

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 …
14
votes
0answers
165 views

To what extent does Spec R determine Spec of the Witt ring on R?

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 …
14
votes
0answers
291 views

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 …
13
votes
1answer
412 views

What does the incidence algebra of the lattices in C tell us about modular forms?

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 …
13
votes
0answers
279 views

Fundamental groups of the spaces of rational functions

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 …
13
votes
0answers
276 views

Elementary Aspects of Galois Deformation

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 …
13
votes
0answers
370 views

Area of the star of the difference set

(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 …
13
votes
0answers
155 views

Semiadditivity and dualizability of 2

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 …
13
votes
0answers
232 views

Almost complex 4-manifolds with a “holomorphic” vector field

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 …
12
votes
0answers
314 views

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 …
12
votes
0answers
226 views

constants in Gamma factors in functional equation for zeta functions.

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 …
12
votes
0answers
174 views

Which pairs of complex reflection groups have isomorphic braid groups?

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 …

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