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1500 questions
49
votes
6 answers
Open affine subscheme of affine scheme which is not principal
I'm not sure whether this is non-trivial or not, but do there exist simple examples of an affine scheme $X$ having an open affine subscheme $U$ which is not principal in $X$? By a principal open of $X = \mathrm{Spec} \ A$, I mean anything of the…
Wanderer
- 5,123
49
votes
4 answers
Which functions of one variable are derivatives ?
This is motivated by this recent MO question.
Is there a complete characterization of those functions $f:(a,b)\rightarrow\mathbb R$ that are pointwise derivative of some everywhere differentiable function $g:(a,b)\rightarrow\mathbb R$ ?
Of course,…
Denis Serre
- 51,599
49
votes
4 answers
The maximum of a polynomial on the unit circle
Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself.
Suppose we are given $n$ points $z_1,...,z_n$ on the unit circle…
Seva
- 22,827
49
votes
4 answers
Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
I'm looking for an elegant proof that any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$.
Kevin Wray
- 1,689
49
votes
1 answer
Is it best to run or walk in the rain?
According to the Norwegian meterological institute, the answer is that it is best to run. According to Mythbusters (quoted in the comments to that article), the answer is that it is best to walk.
My guess would be that this is something that can be…
Andrew Stacey
- 26,373
49
votes
6 answers
Generating finite simple groups with $2$ elements
Here is a very natural question:
Q: Is it always possible to generate a finite simple group with only $2$ elements?
In all the examples that I can think of the answer is yes.
If the answer is positive, how does one prove it? Is it possible to prove…
Hugo Chapdelaine
- 7,521
49
votes
3 answers
The Hardy Z-function and failure of the Riemann hypothesis
David Feldman asked whether it would be reasonable for the Riemann hypothesis to be false, but for the Riemann zeta function to only have finitely many zeros off the critical line. I very rashly predicted that this question would be essentially as…
David Hansen
- 13,018
49
votes
4 answers
If the Riemann Hypothesis fails, must it fail infinitely often?
That is must there either be no non-trivial zeros off the critical line or
infinitely many?
I'm sure that no one believes otherwise, but I've never seen a theorem in the
literature addressing this. Folklore perhaps?
David Feldman
- 17,466
49
votes
28 answers
Problems where we can't make a canonical choice, solved by looking at all choices at once
It's a common theme in mathematics that, if there's no canonical choice (of basis, for example), then we shouldn't make a choice at all. This helps us focus on the heart of the matter without giving ourselves arbitrary stuff to drag around.
However,…
Zev Chonoles
- 6,722
49
votes
1 answer
What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?
I am a research mathematician at a university in the United States. My training is in pure mathematics (geometry). However, for the past couple of months, I have been supervising some computer science undergrads in trying to reproduce DeepMind's…
Ryan Hendricks
- 675
49
votes
7 answers
Zorn's lemma: old friend or historical relic?
It is often said that instead of proving a great theorem a mathematician's fondest dream is to prove a great lemma. Something like Kőnig's tree lemma, or Yoneda's lemma, or really anything from this list.
When I was first learning algebra, one of…
Pace Nielsen
- 18,047
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- 133
49
votes
8 answers
Classification problem for non-compact manifolds
Background
It is well-known that the compact two-dimensional manifolds are completely classified (by their orientability and their Euler characteristic).
I'm also under the impression that there is also a classification for compact three-dimensional…
Victoria Flat
- 1,841
49
votes
30 answers
Taking a theorem as a definition and proving the original definition as a theorem
Gian-Carlo Rota's famous 1991 essay, "The pernicious influence of mathematics upon philosophy" contains the following passage:
Perform the following thought experiment. Suppose that you are given two formal presentations of the same mathematical…
Timothy Chow
- 78,129
49
votes
3 answers
Explicit metrics
Every surface admits metrics of constant curvature, but there is usually a disconnect between
these metrics, the shapes of ordinary objects, and typical mathematical drawings of surfaces.
Can anyone give an explicit and intuitively meaningful…
Bill Thurston
- 24,832
49
votes
3 answers
Is each squared finite group trivial?
A semigroup $S$ is defined to be squared if there exists a subset $A\subseteq S$ such that the function $A\times A\to S$, $(x,y)\mapsto xy$, is bijective.
Problem: Is each squared finite group trivial?
Remarks (corrected in an Edit).
I learned…
Taras Banakh
- 40,791