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1500 questions
61
votes
13 answers
How do you approach your child's math education?
My son is one year old, so it is perhaps a bit too early to worry about his mathematical education, but I do. I would like to hear from mathematicians that have older children: What do you wish you'd have known early? What do you think you did…
rgrig
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61
votes
7 answers
Is the Jaccard distance a distance?
Wikipedia defines the Jaccard distance between sets A and B as $$J_\delta(A,B)=1-\frac{|A\cap B|}{|A\cup B|}.$$ There's also a book claiming that this is a metric. However, I couldn't find any explanation of why $J_\delta$ obeys the triangle…
rgrig
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- 1
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61
votes
13 answers
Example of an unnatural isomorphism
Can anyone give an example of an unnatural isomorphism? Or, maybe, somebody can explain why unnatural isomorphisms do not exist.
Consider two functors $F,G: {\mathcal C} \rightarrow {\mathcal D}$. We say that they are unnaturally isomorphic if…
Bugs Bunny
- 12,113
61
votes
8 answers
Reductio ad absurdum or the contrapositive?
From time to time, when I write proofs, I'll begin with a claim and then prove the contradiction. However, when I look over the proof afterwards, it appears that my proof was essentially a proof of the contrapositive, and the initial claim was not…
Harry Gindi
- 19,374
61
votes
5 answers
Why are abelian groups amenable?
A (discrete) group is amenable if it admits a finitely additive probability measure (on all its subsets), invariant under left translation. It is a basic fact that every abelian group is amenable. But the proof I know is surprisingly convoluted. …
Tom Leinster
- 27,167
60
votes
7 answers
Status of PL topology
I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. I'd really want to know the state of the…
Carlos Sáez
- 656
60
votes
5 answers
Can the Riemann hypothesis be undecidable?
The question is contained in the title; I mean the standard axioms ZFC. The wiki link: Riemann hypothesis. There are finite algorithms allowing one to decide if there are non-trivial zeroes of the $\zeta$-function in the domains whose union exhausts…
Shaqq
- 609
60
votes
4 answers
A question about MathSciNet etiquette
Hello,
Recently, a colleague of mine pointed me to a MathSciNet review of one of my papers that is completely off the mark - it is not negative or anything like that, but it grossly misrepresents the contents of the paper (when describing the…
60
votes
1 answer
Why "open immersion" rather than "open embedding"?
When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic geometry, except sometimes as a word for an…
Charles Staats
- 7,218
60
votes
10 answers
What do you do when you're stuck?
I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there was something missing- either the "middle" part…
It'sMe
- 767
60
votes
5 answers
Consequences of Kirti Joshi's new preprint about p-adic Teichmüller theory on the validity of IUT and on the ABC conjecture
Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635
In that preprint, Kirti Joshi claims that
he agrees with Scholze and Stix that Mochizuki's proof of ABC is…
Madeleine Birchfield
- 2,454
60
votes
11 answers
What are some open problems in algebraic geometry?
What are the open big problems in algebraic geometry and vector bundles?
More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over projective varieties/curves.
Sun
- 103
60
votes
5 answers
Is the Riemann Hypothesis equivalent to a $\Pi_1$ sentence?
1) Can the Riemann Hypothesis (RH) be expressed as a $\Pi_1$ sentence?
More formally,
2) Is there a $\Pi_1$ sentence which is provably equivalent to RH in PA?
Update (July 2010):
So we have two proofs that the RH is equivalent to a $\Pi_1$…
Kaveh
- 5,362
60
votes
15 answers
Abstract thought vs calculation
Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was the use of more abstract notions to obtain the same results with fewer…
Sergei Tropanets
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60
votes
4 answers
Is there a mathematical and information theoretic explanation for this cube packing phenomenon?
I saw this unintuitive result on dice packing:
A jumble of thousands of cubic dice, agitated by an oscillating
rotation, can rapidly become completely ordered, a result that is hard
to produce with more conventional shaking.
The paper is not very…
Turbo
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