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1500 questions
55
votes
2 answers
Polynomials having a common root with their derivatives
Here is a question someone asked me a couple of years ago. I remember having spent a day or two thinking about it but did not manage to solve it. This may be an open problem, in which case I'd be interested to know the status of it.
Let $f$ be a one…
algori
- 23,231
55
votes
1 answer
IMO 2017/6 via arithmetic geometry
The (very nice) final problem of IMO 2017 asked contestants to show:
If $S$ is a finite set of lattice points $(x,y)$ with $\gcd(x,y)=1$, then there is a nonconstant homogeneous polyonmial $f \in \mathbb Z[x,y]$ such that $f(x,y) = 1$ for all…
Evan Chen
- 1,167
55
votes
4 answers
An interesting integral expression for $\pi^n$?
I came on the following multiple integral while renormalizing elliptic multiple zeta values:
$$\int_0^1\cdots \int_0^1\int_1^\infty {{1}\over{t_n(t_{n-1}+t_n)\cdots (t_1+\cdots+t_n)}} dt_n\cdots dt_1.$$
Only the variable $t_n$ goes from $1$ to…
Leila Schneps
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55
votes
10 answers
How should a "working mathematician" think about sets? (ZFC, category theory, urelements)
Note that "a working mathematician" is probably not the best choice of words, it's supposed to mean "someone who needs the theory for applications rather than for its own sake". Think about it as a homage to Mac Lane's classic. I'm in no way…
Jxt921
- 1,095
55
votes
3 answers
Duality between compactness and Hausdorffness
Consider a non-empty set $X$ and its complete lattice of topologies
(see also this thread).
The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also Hausdorff. A minimal Hausdorff topology is such that no…
yada
- 1,731
55
votes
5 answers
Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?
Choose unit quaternions $q_0, q_1, q_2, q_3, q_4$ that form the vertices of a regular 4-simplex in the quaternions. Assume $q_0 = 1$. Let the other four generate a group via quaternion multiplication. Is this a free group on 4 generators?
I heard…
John Baez
- 21,373
55
votes
2 answers
How do you *state* the Classification of finite simple groups?
From the point of view of formal math, what would constitute an appropriate statement of the classification of finite simple groups? As I understand it, the classification enumerates 18 infinite families and 26 sporadic groups and asserts that a…
Mario Carneiro
- 1,142
55
votes
1 answer
Does every smooth, projective morphism to $\mathbb{C}P^1$ admit a section?
Possibly this has already been asked, but it came up again in this question of Daniel Litt. Does every smooth, projective morphism $f:Y\to \mathbb{C}P^1$ admit a section, i.e., a morphism $s:\mathbb{C}P^1\to Y$ such that $f\circ s$ equals…
Jason Starr
- 4,091
55
votes
10 answers
Why differential forms are important?
Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated without them: for example if you want to learn the…
asv
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55
votes
21 answers
Linear algebra proofs in combinatorics?
Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, the Frankl-Wilson theorem, and Fisher's…
Tony Huynh
- 31,500
55
votes
5 answers
Advice for pure-math Phd students
Pursuing a Phd in pure math can be a daunting task. A number of students who begin a Phd in pure math don't complete it, and there are high-quality dissertations and those which are not so high quality.
My question is: What advice do you, or would…
mdg
- 366
55
votes
18 answers
How can an extremely mathematically talented young person be helped to fulfill his/her potential?
Obviously, this question is not a research level mathematics question at all. But, I've just met an extremely mathematically talented $11$ years old student and I don't know how I can help him. For years I was working at a special school for young…
Amir Asghari
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55
votes
30 answers
What are examples of good toy models in mathematics?
This post is community wiki.
A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to understand it better, but I don't know of too many…
Qiaochu Yuan
- 114,941
55
votes
5 answers
Bizarre operation on polynomials
There I was, innocently doing some category theory, when up popped a totally outlandish operation on polynomials. It seems outlandish to me, anyway. I'd like to know if anyone has seen this operation before, in any context.
The categorical…
Tom Leinster
- 27,167
55
votes
1 answer
Intersecting family of triangulations
Let $\cal T_n$ be the family of all triangulations on an $n$-gon using $(n-3)$ non-intersecting diagonals. The number of triangulations in $\cal T_n$ is $C_{n-2}$ the $(n-2)$th Catalan number. Let $\cal S \subset \cal T_n$ be a subfamily of…
Gil Kalai
- 24,218