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1500 questions
48
votes
4 answers
How many ways can you inscribe five 24-cells in a 600-cell, hitting all its vertices?
You can inscribe five tetrahedra in a dodecahedron so that each vertex of the dodecahedron is the vertex of just one tetrahedron, as drawn here by Greg Egan:
Warmup question: How many ways can you do this?
I believe there are just two: the way…
John Baez
- 21,373
48
votes
4 answers
Twin primes conjecture and extrapolation method
Let $(p_1, p_2)$ be a twin prime pair, where we include $(2, 3)$. If $p_1 \equiv 1$ mod $4$ then we let $t_{(p_1, p_2)} := p_1 ^ 2 / p_2 ^ 2$ otherwise, we let $t_{(p_1, p_2)} := p_2 ^ 2 / p_1 ^ 2$.
I conjecture that the product
$$
\prod_{(p_1,…
Dimitris Valianatos
- 1,305
48
votes
3 answers
Thurston's 24 questions: All settled?
Thurston's 1982 article on three-dimensional manifolds1 ends with $24$ "open questions":
$\cdots$
Two naive questions from an outsider:
(1) Have all $24$ now been resolved?
(2) If so, were they all resolved in his lifetime?
1Thurston,…
Joseph O'Rourke
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48
votes
0 answers
What is the current understanding regarding complex structures on the 6-sphere?
In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and realized I lacked the background to understand. The…
jdc
- 2,984
48
votes
4 answers
How to constructively/combinatorially prove Schur-Weyl duality?
How is Schur-Weyl duality (specifically, the fact that the actions of the group ring $\mathbb{K}\left[ S_{n}\right] $ and the monoid ring
$\mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\right) \right] $ on the tensor power $V^{\otimes…
darij grinberg
- 33,095
48
votes
4 answers
Do set-theorists use informal set theory as their meta-theory when talking about models of ZFC?
Here, Noah Schweber writes the following:
Most mathematics is not done in ZFC. Most mathematics, in fact, isn't done axiomatically at all: rather, we simply use propositions which seem "intuitively obvious" without comment. This is true even when…
user98009
- 489
48
votes
5 answers
How do you mentor undergraduate research?
Lets say you had an undergraduate who wanted to do some advanced work and some research, possibly for a thesis, or things like that.
There are two slightly more specific groups of questions I have about this process:
How would you go about…
48
votes
2 answers
Grothendieck says: points are not mere points, but carry Galois group actions
Apologies in advance if this question is too elementary for MO. I didn't find an explanation of these ideas in any algebraic geometry books (I don't know French).
The following is an excerpt from this archive:
Thierry Coquand recently asked me
"In…
Arrow
- 10,335
48
votes
4 answers
Are there primes of every Hamming weight?
Are there primes of every Hamming weight? That is, for every integer $n \in \mathbb{Z}_{>0}$ does there exist a prime which is the sum of $n$ distinct powers of $2$?
In this case, the Hamming weight of a number is the number of $1$s in its binary…
dakota
- 583
48
votes
6 answers
Smooth linear algebraic groups over the dual numbers
It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but assume smoothness for what follows.) The proof…
BCnrd
- 6,978
48
votes
7 answers
How to correct an error in a submitted paper?
Suppose that after submitting a paper for publication, but before hearing anything back from the referee, I discover an error in the paper that needs to be corrected, or an omission that needs to be rectified. What is the best course of action? …
Mike Shulman
- 65,064
48
votes
2 answers
Research situation in the field of Information Geometry
I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field.
I have read (4) and parts of (3).
As the comment in (1)'s answer said"I am…
Henry.L
- 7,951
48
votes
6 answers
What is the difference between homology and cohomology?
In intuitive terms, what is the main difference? We know that homology is essentially the number of $n$-cycles that are not $n$-boundaries in some simplicial complex $X$. This is, more or less, the number of holes in the complex. But what is the…
Thomas J
- 429
48
votes
5 answers
Definition and meaning of the conductor of an elliptic curve
I never really understood the definition of the conductor of an elliptic curve.
What I understand is that for an elliptic curve E over ℚ, End(E) is going to be (isomorphic to) ℤ or an order in a imaginary quadratic field ℚ(√(-d)), and that this…
Sam Derbyshire
- 5,440
48
votes
5 answers
Algebraic proof of 4-colour theorem?
4-colour Theorem. Every planar graph is 4-colourable.
This theorem of course has a well-known history. It was first proven by Appel and Haken in 1976, but their proof was met with skepticism because it heavily relied on the use of computers. The…
Tony Huynh
- 31,500