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1500 questions
101
votes
21 answers
Proofs of the uncountability of the reals
Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem (Wayback Machine). At first, I was excited to see a variant proof (as it did not use the diagonal argument explicitly). However, as…
Unknown
- 2,815
101
votes
30 answers
Errata for Atiyah–Macdonald
Is there a good list of errata for Atiyah–Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists inaccuracies and gaps?
Tim Campion
- 60,951
101
votes
3 answers
What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity categories that are weak infinity groupoid.
In 1998…
Simon Henry
- 39,971
101
votes
34 answers
What is the most useful non-existing object of your field?
When many proofs by contradiction end with "we have built an object with such, such and such properties, which does not exist", it seems relevant to give this object a name, even though (in fact because) it does not exist. The most striking example…
user56097
- 402
101
votes
15 answers
Have you solved problems in your sleep?
I have hit upon major (for me—relative to my trivial accomplishments)
insights in my research
in various sleep-deprived altered states of consciousness,
e.g., long solo car-drives extending through the night into the morning.
But I have never…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
101
votes
4 answers
How feasible is it to prove Kazhdan's property (T) by a computer?
Recently, I have proved that Kazhdan's property (T) is theoretically provable
by computers (arXiv:1312.5431,
explained below), but I'm quite lame with computers and have
no idea what they actually can do. So, my question is how feasible is it…
Narutaka OZAWA
- 9,621
101
votes
1 answer
Dropping three bodies
Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time later by a theorem
I proved some time ago. (See the…
Richard Montgomery
- 8,118
100
votes
8 answers
Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?
Is $$ \sum_{n=0}^\infty {x^n \over \sqrt{n!}} > 0 $$ for all real $x$?
(I think it is.) If so, how would one prove this? (To confirm: This is the power
series for $e^x$, except with the denominator replaced by $\sqrt{n!}$.)
J Russell
- 1,103
100
votes
6 answers
Light rays bouncing in twisted tubes
Imagine a smooth curve $c$ sweeping out a unit-radius disk that is
orthogonal to the curve at every point.
Call the result a tube.
I want to restrict the radius of curvature of $c$ to be at most 1.
I am interested in the behavior of a light ray…
Joseph O'Rourke
- 149,182
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- 342
- 933
100
votes
2 answers
Riemann hypothesis via absolute geometry
Several leading mathematicians (e.g. Yuri Manin) have written or said publicly that there is a known outline of a likely natural proof of the Riemann hypothesis using absolute algebraic geometry over the field of one element; some like Mochizuki and…
Zoran Skoda
- 5,162
- 2
- 42
- 35
100
votes
10 answers
Why do Bernoulli numbers arise everywhere?
I have seen Bernoulli numbers many times, and sometimes very surprisingly. They appear in my textbook on complex analysis, in algebraic topology, and of course, number theory. Things like the criteria for regular primes, or their appearance in the…
36min
- 3,758
100
votes
5 answers
Is there a high level reason why the inverse square law of gravitation yields periodic orbits without precession?
Given a spherically symmetric potential $V: {\bf R}^d \to {\bf R}$, smooth away from the origin, one can consider the Newtonian equations of motion
$$ \frac{d^2}{dt^2} x = - (\nabla V)(x)$$
for a particle $x: {\bf R} \to {\bf R}^d$ in this potential…
Terry Tao
- 108,865
- 31
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- 517
100
votes
6 answers
Is there an analogue of curvature in algebraic geometry?
I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as curvature in the former context while curvature…
Paul Siegel
- 28,772
99
votes
5 answers
New arXiv procedures?
Recently I encountered a new phenomenon when I tried to submit a paper to arXiv. The paper was an erratum to another, already published, paper and will be published separately. I got a message from arXiv saying that I need to join the erratum with…
user6976
99
votes
2 answers
Extent of “unscientific”, and of wrong, papers in research mathematics
This question is cross-posted from academia.stackexchange.com where it got closed with the advice of posting it on MO.
Kevin Buzzard's slides (PDF version) at a recent conference have really unsettled me.
In it, he mentions several examples in…
Archie
- 883