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1500 questions
54
votes
10 answers
The "sensitivity" of 2-colorings of the d-dimensional integer lattice
Consider the $d$-dimensional integer lattice, $Z^d$. Call two points in $Z^d$ "neighbors" if their Euclidean distance is 1 (i.e., if they differ by 1 on exactly one coordinate).
Let $C$ be a two-coloring of $Z^d$, which makes each point either red…
Scott Aaronson
- 9,743
54
votes
2 answers
How were modular forms discovered?
When modular forms are usually introduced, it is by: "We have the standard action of $SL(2,\mathbb Z)$ on the upper half-plane, so let us study functions which are (almost) invariant under such transformations".
But modular forms were discovered in…
FusRoDah
- 3,680
54
votes
6 answers
What is the etymology of the term "perverse sheaf"?
Grothendieck famously objected to the term "perverse sheaf" in Récoltes et Semailles, writing "What an idea to give such a name to a mathematical thing! Or to any other thing or living being, except in sternness towards a person—for it is evident…
Daniel Litt
- 22,187
54
votes
0 answers
What did Gelfand mean by suggesting to study "Heredity Principle" structures instead of categories?
Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the following intriguing statement (Page 7, section 1.3, end of the…
Dmitri Zaitsev
- 1,228
54
votes
0 answers
Uniformization over finite fields?
The following is a question I've been asking people on and off for a few years, mostly out of idle curiosity, though I think it's pretty interesting. Since I've made more or less no progress, I figured I'd ask it here.
Let $C_1, C_2$ be smooth…
Daniel Litt
- 22,187
54
votes
12 answers
Examples of advance via good definitions
In my research I came across a case where I could derive a known theorem with rather straightforward way by choosing "non-standard" definitions using my knowledge from a related field. This particular case does not seem to be interesting to wider…
user39297
54
votes
5 answers
Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
Suppose you have a tetrahedron $T$ in Euclidean space with edge lengths $\ell_{01}$, $\ell_{02}$, $\ell_{03}$, $\ell_{12}$, $\ell_{13}$, and $\ell_{23}$. Now consider the tetrahedron $T'$ with edge lengths
$$\begin{aligned}
\ell'_{02} &= \ell_{02}…
Dylan Thurston
- 10,033
54
votes
3 answers
What is precisely still missing in Connes' approach to RH?
I have read Connes' survey article http://www.alainconnes.org/docs/rhfinal.pdf
and I am somewhat familiar with his classic paper on the trace formula: http://www.alainconnes.org/docs/selecta.ps
Very roughly speaking the idea is to describe a…
santker heboln
- 1,318
54
votes
16 answers
Why do we need random variables?
In this MathStackExchange post the question in the title was asked without much outcome, I feel.
Edit: As Douglas Zare kindly observes, there is one more answer in MathStackExchange now.
I am not used to basic Probability, and I am trying to prepare…
Filippo Alberto Edoardo
- 5,630
54
votes
5 answers
Can an arbitrary collection of circles of total area 1/2 fit into a circle of area 1?
Assume the circles are actually open disks, otherwise two circles each of area $\frac{1}{4}$ wouldn't fit into the circle of area 1.
This seems like it should be true, thinking about packing density, but I've not been able to find an algorithm that…
Henry Segerman
- 1,886
- 20
- 25
54
votes
2 answers
Does every ring of integers sit inside a monogenic ring of integers?
Given a number field $K/\mathbf{Q}$ whose ring of integers $\mathcal{O}_K$ is, in general, not of the form $\mathbf{Z}[\alpha]$ (not monogenic), does there exist an extension $L/K$ which has $\mathcal{O}_L= \mathbf{Z}[\alpha]$ (is monogenic)?
This…
Eins Null
- 1,579
54
votes
6 answers
"Why the heck are the homotopy groups of the sphere so damn complicated?"
This is a quote from a dear friend asking the rest of us on Facebook. I gave him some half-baked response, but the truth is I don't really know enough about this to give him a good response.
So why ARE they so complicated? The topologists here want…
54
votes
6 answers
Publication rates in Mathematics
Have there been any studies of publication rates in Mathematics?
We are trying to construct a workload model for the Faculty of Science and Engineering at my institution. Part of this involves assigning a fixed number of "points" for each published…
Gerry Myerson
- 39,024
54
votes
3 answers
The view from inside of a mirrored tetrahedron
Suppose you were standing inside a regular tetrahedron $T$ whose
internal face surfaces were perfect mirrors.
Let's assume $T$'s height is $3{\times}$ yours, so that your
eye is roughly at the centroid,
and that you look perpendicular to a…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
54
votes
3 answers
If any open set is a countable union of balls, does it imply separability?
If a metric space is separable, then any open set is a countable union of balls. Is the converse statement true?
UPDATE1. It is a duplicate of the question…
Fedor Petrov
- 102,548