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Results tagged with gm.general-mathematics
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user 25510
Questions about mathematics which don't fall into the other arXiv categories. If you have a general question about mathematics but it is not research level, it's off-topic but it might be welcomed on Mathematics Stack Exchange.
12
votes
When is 2 qualitatively different from 3?
Complex numbers exist only in dimension 2. That is the only multiplication laws on $R^n$ which satisfy all field axioms exist for $n=1$ (real numbers) and $n=2$ (complex numbers).
- 91.8k
67
votes
Useless math that became useful
The most famous example is conic sections. Conic sections were of great interest to
Greek mathematicians, and their theory was highly developed in the 2-nd century BC.
However I don't know of any appl …
- 91.8k
62
votes
Accepted
Is spherical trigonometry a dead research area?
It is not. As a proof, I will mention three relatively recent papers where I am a co-author:
M. Bonk and A. Eremenko, Covering properties of meromorphic functions, negative curvature and spherical geo …
34
votes
Theorems with many distinct proofs
"what is worth proving
is worth proving again" (Attributed to N. Katz in D. Ruelle's paper, The nature of properly human mathematics.)
You are asking for a very long list: most deep and important th …
- 91.8k
61
votes
Is amateur research in mathematics viable?
This is possible. I have at least two friends who studied mathematics (in the graduate school), did not defend their PhD, and found some jobs not related to mathematics. Still they do research, and pu …
- 91.8k
14
votes
What are some examples of understanding a space by studying the functions on this space?
The idea goes back to the 1930s when algebraic geometers understood that points
of an algebraic variety "are" maximal (or prime) ideals of the ring of regular functions
on it. The counterpart of this …
- 91.8k
4
votes
Golden ratio in contemporary mathematics
Yes, it does:
Lyubich, Mikhail; Milnor, John The Fibonacci unimodal map. J. Amer. Math. Soc. 6 (1993), no. 2, 425–457.
and two more papers of the same authors studying what they call Fibonacci map.
- 91.8k
9
votes
When did you "meet Polya"?
As a high school student, I read his "Mathematical Discovery". Then, as a freshman, I met my future adviser
(A. A. Goldberg) and he recommended "Problems and Theorems in Analysis", saying that
this bo …
- 91.8k
5
votes
Accepted
Do mathematicians use notebooks to keep their results these days?
Yes, mathematicians keep notebooks. Sometimes, after their death, notebooks are published.
Here is an example: http://www.claymath.org/publications/quillen-notebooks
Here is another example, though …
- 91.8k
31
votes
Counterexamples against all odds
The most famous example is the so-called Riemann-Hilbert problem, which has a long and complicated history which I don't explain in detail. As it happens Hilbert's own formulation was not very exact, …
- 91.8k
30
votes
The origin(s) of the word "elliptic"
The origin of all these uses is very different. Joe Silverman explained the genesis of the sequence ellipse $\rightarrow$ elliptic integral $\rightarrow$ elliptic function $\rightarrow$ elliptic curve …
- 4,713
7
votes
Recreational mathematical papers
Two of my favorite recreational papers are:
J-F Mestre, R Schoof, L Washington and D Zagier,
Quotients homophones des groupes libres. Homophonic quotients of free groups
Experiment. Math. 2 (1993), …
- 91.8k
20
votes
Is the field of q-series 'dead'?
The opinions that certain areas of mathematics are dead are frequently stated but in many cases incorrect. Some areas experience declines in activity and then revivals. Many examples can be given.
On …
- 91.8k
23
votes
Fascinating moments: equivalent mathematical discoveries
An example which always puzzled me is J. Milnor's paper entitled
Eigenvalues of the Laplace operator on certain manifolds,
Proc. Nat. Acad. Sci. U.S.A. 51 1964, 542.
The whole paper occupies about ha …
- 43.2k
29
votes
Where can square roots come from when they are not distances?
$i=\sqrt{-1}$ has no apparent relation with any distance.
Also $\int_{-\infty}^\infty e^{-x^2}dx=\sqrt{\pi}.$
- 91.8k