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1500 questions
53
votes
2 answers
How to add essentially new knots to the universe?
A knot is an embedding of a circle $S^{1}$ in $3$-dimensional Euclidean space, $\mathbb{R}^3$. Knots are considered equivalent under ambient isotopy. There are two different types of knots, tame and wild. A tame knot is any knot equivalent to a…
53
votes
5 answers
Distribution of square roots mod 1
I was wondering about the distribution of $\sqrt{p}$ mod $1$ this morning, as one does while brushing one's teeth. I remembered the paper of Elkies and McMullen (Duke Math. J. 123 (2004), no. 1, 95–139.) about $\sqrt{n}$ mod $1$, but hadn't really…
Marty
- 13,104
53
votes
2 answers
What is quantum algebra?
This might be a very naive question. But what is quantum algebra, really?
Wikipedia defines quantum algebra as "one of the top-level mathematics categories used by the arXiv". Surely this cannot be a satisfying definition. The arXiv admins didn't…
Najib Idrissi
- 4,840
53
votes
13 answers
Good differential equations text for undergraduates who want to become pure mathematicians
Alright, so I have been taking a while to soak in as much advanced mathematics as an undergraduate as possible, taking courses in algebra, topology, complex analysis (a less rigorous undergraduate version of the usual graduate course at my…
lambdafunctor
- 1,489
- 4
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53
votes
4 answers
Why is Quantum Field Theory so topological?
I understand that my question suffers from my lack of knowledge about the field, but as a mathematician without much knowledge of physics I have been wondering much about the following and I always felt a bit stupid to ask this in real:
Many parts…
53
votes
4 answers
Are the rationals homeomorphic to any power of the rationals?
I asked myself, which spaces have the property that $X^2$ is homeomorphic to $X$. I started to look at some examples like $\mathbb{N}^2 \cong \mathbb{N}$, $\mathbb{R}^2\ncong \mathbb{R}, C^2\cong C$ (for the cantor set $C$). And then I got stuck,…
HenrikRüping
- 10,279
53
votes
3 answers
Why are parabolic subgroups called "parabolic subgroups"?
Over the years, I have heard two different proposed answers to this question.
It has something to do with parabolic elements of $SL(2,\mathbb{R})$. This sounds plausible, but I haven't heard a really convincing explanation along these…
Timothy Chow
- 78,129
53
votes
7 answers
How to know if somebody else is also working on your problem?
Once you have spotted a mathematical problem that (presumably) fits your degree of expertise, whether you are a phd student or an established professor, you have to deal with the following non mathematical problems:
How to know if somebody else in…
Qfwfq
- 22,715
53
votes
37 answers
What is your favorite "strange" function?
There are many "strange" functions to choose from and the deeper you get involved with math the more you encounter. I consciously don't mention any for reasons of bias. I am just curious what you consider strange and especially like.
Please also…
vonjd
- 5,875
53
votes
9 answers
How does a mathematician choose on which problem to work?
Main question:
How does a mathematician choose on which problem to work?
An example approach to framing one's answer:
What is a mathematical problem - big or small - that you solved or are working on, how did you choose this problem, and why…
jawe
- 99
53
votes
34 answers
Most intriguing mathematical epigraphs
Good epigraphs may attract more readers. Sometimes it is necessary.
Usually epigraphs are interesting but not intriguing.
To pick up an epigraph is some kind of nearly mathematical problem: it should be unexpectedly relevant to the content.
What …
Alexey Ustinov
- 11,875
53
votes
1 answer
Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and even an approach using cofiltered limits to…
Nick R
- 1,047
53
votes
5 answers
How much of the ATLAS of finite groups is independently checked and/or computer verified?
In a recent talk Finite groups, yesterday and today Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the CFSG and said…
David Roberts
- 33,851
53
votes
4 answers
When has the Borel-Cantelli heuristic been wrong?
The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many Fermat Primes, that is primes of the form…
Eric Naslund
- 11,255
53
votes
20 answers
Math paper authors' order
It seems in writing math papers collaborators put their names in the alphabetical order of their last name. Is this a universal accepted norm? I could not find a place putting this down formally.
lemega
- 478