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1500 questions
48
votes
6 answers
Is there an "elegant" non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?
Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway...
Consider this problem. I've been trying to find a formula to expand the "regular iteration" of "exp". Regular iteration is a special kind of complex…
The_Sympathizer
- 1,677
48
votes
8 answers
When are there enough projective sheaves on a space X?
This question is being asked on behalf of a colleague of mine.
Let $X$ be a topological space. It is well known that the abelian category of sheaves on $X$ has enough injectives: that is, every sheaf can be monomorphically mapped to an injective…
Pete L. Clark
- 64,763
48
votes
26 answers
Never appeared forthcoming papers
This has been inspired by this MO question: Harmonic maps into compact Lie groups
Just for joking: which is your favourite never appeared forthcoming paper?
(do not hesitate to close this question if unappropriate)
domenico fiorenza
- 6,719
48
votes
11 answers
Cures for mathematician's block (as in writer's block)
What kind of things do you find that help you get the "creative juices flowing," to use a tired cliche, when you're stuck or burnt out on a problem? I've read about some studies that suggest listening and playing music can stimulate mathematical…
Josh
- 1,402
48
votes
2 answers
Can I wrap a suitcase with hair ties
Is there a nontrivial link in a big solid torus that is trivial in the ambient Euclidean space such that each circle is unknot and has a sufficiently small length?
It is motivated by a question that bothers me from my childhood:
Is it possible to…
Anton Petrunin
- 43,739
48
votes
8 answers
Ideas for introducing Galois theory to advanced high school students
Briefly, I was wondering if someone can suggest an angle for introducing the gist of Galois groups of polynomials to (advanced) high school students who are already familiar with polynomials (factorisation via Horner, polynomial division,…
user929304
- 649
- 6
- 12
48
votes
2 answers
Irrationality of $ \pi e, \pi^{\pi}$ and $e^{\pi^2}$
What is known about irrationality of $\pi e$, $\pi^\pi$ and $e^{\pi^2}$?
Vladimir Reshetnikov
- 6,709
48
votes
4 answers
Sheaf-theoretic approach to forcing
Inspired by the question here, I have been trying to understand the sheaf-theoretic approach to forcing, as in MacLane–Moerdijk's book "Sheaves in geometry and logic", Chapter VI.
A general comment is that sheaf-theoretic methods do not a priori…
Peter Scholze
- 19,800
- 3
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- 118
48
votes
3 answers
What happens if you strip everything but the “between” relation in metric spaces
Given a metric space $(X,d)$ and three points $x,y,z$ in $X$, say that $y$ is between $x$ and $z$ if $d(x,z) = d(x,y) + d(y,z)$, and write $[x,z]$ for the set of points between $x$ and $z$.
Obviously, we have
$x,z\in[x,z]$;
$[x,z]=[z,x]$;
$y \in…
user148575
- 877
48
votes
5 answers
Algebraically closed fields of positive characteristic
I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far as intuition goes, I might as well add "...of…
Harrison Brown
- 12,543
48
votes
2 answers
Kunneth formula for sheaf cohomology of varieties
What is a good reference for the following fact (the hypotheses may not be quite right):
Let $X$ and $Y$ be projective varieties over a field $k$. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $X$ and $Y$, respectively. Let…
Charles Staats
- 7,218
48
votes
0 answers
How many algebraic closures can a field have?
Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is algebraically closed, then $K\cong\overline F$…
Asaf Karagila
- 38,140
48
votes
4 answers
Are there more Nullstellensätze?
Over which fields $k$ is there a reasonable analogue of Hilbert's Nullstellensatz?
Here is a more precise formulation: let $k$ be an arbitrary field, $n$ a positive integer, and $R = k[t_1,..,t_n]$. There is a natural relation between $k^n$ and…
Pete L. Clark
- 64,763
48
votes
5 answers
Is Lebesgue's "universal covering" problem still open?
The following problem has been attributed to Lebesgue. Let "set" denote any subset of the Euclidean plane. What is the greatest lower bound of the diameter of any set which contains a subset congruent to every set of diameter 1? There are a number…
Garabed Gulbenkian
- 6,115
48
votes
6 answers
Are there examples of conjectures supported by heuristic arguments that have been finally disproved?
The idea for this comes from the twin prime conjecture, where the heuristic evidence seems just so overwhelming, especially in the light of Zhang's famous result from 2014 about Bounded gaps between primes and its subsequent improvements.
Are…
Wolfgang
- 13,193