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1500 questions
46
votes
6 answers
Universal definition of tangent spaces (for schemes and manifolds)
Both schemes and manifolds are local ringed spaces which are locally isomorphic to spaces in some full subcategory of local ringed spaces (local models). Now, there is the inherent notion of the Zariski tangent space in a point (dual of maximal…
user717
- 5,153
46
votes
10 answers
What kid-friendly math riddles are too often spoiled for mathematicians?
Some math riddles tend to be spoiled for mathematicians before they get a chance to solve them. Three examples:
What is $1+2+\cdots+100$?
Is it possible to tile a mutilated chess board with dominoes?
Given a line $\ell$ in the plane and two…
Dustin G. Mixon
- 7,563
46
votes
3 answers
Class Numbers and 163
This is a bit fluffier of a question than I usually aim for, so apologies in advance if this doesn't pass the smell test for suitability.
Likely my favorite fun fact in all of number theory is the juxtaposition of two "extremal and opposite"…
Cam McLeman
- 8,417
46
votes
4 answers
How to rewrite mathematics constructively?
Many mathematical subfields often use the axiom of choice and proofs by contradiction. I heard from people supporting constructive mathematics that often one can rewrite the definitions and theorems so that both the axiom of choice and proofs by…
user877505
- 571
46
votes
11 answers
Reference request: Examples of research on a set with interesting properties which turned out to be the empty set
I've seen internet jokes (at least more than 1) between mathematicians like this one here about someone studying a set with interesting properties. And then, after a lot of research (presumably after some years of work), find out such set couldn't…
FeedbackLooper
- 344
46
votes
1 answer
Summing infinitely many infinitesimally small variables makes sense in algebra
There is an identity $e^x=\lim_{n\to \infty} (1+x/n)^n$, and I always thought it is a purely analytic statement. But then I discovered its curious interpretation in pure algebra:
Consider the ring of formal infinite sums of monomials in infinitely…
Anton Mellit
- 3,572
46
votes
5 answers
Have the tides ever turned twice on any open problem?
Oftentimes open problems will have some evidence which leads to a prevailing opinion that a certain proposition, $P$, is true. However, more evidence is discovered, which might lead to a consensus that $\neg P$ is true. In both cases the evidence…
Mark S
- 2,143
46
votes
7 answers
What is an explicit bijection in combinatorics?
A standard way of demonstrating that two collections of combinatorial objects have the same cardinality is to exhibit a bijection between them. Browsing through some examples (here, there, yonder) quickly reveals that combinatorialists call such…
Andrej Bauer
- 47,834
46
votes
2 answers
Why is Voevodsky's motivic homotopy theory 'the right' approach?
Morel and Voevoedsky developed what is now called motivic homotopy theory, which aims to apply techniques of algebraic topology to algebraic varieties and, more generally, to schemes. A simple way of stating the idea is that we wish to find a model…
Patriot
- 1,038
46
votes
3 answers
Does knight behave like a king in his infinite odyssey?
The Knight's Tour is a well-known mathematical chess problem. There is an extensive amount of research concerning this question in two/higher dimensional finite boards. Here, I would like to tackle this problem on the unbounded infinite board,…
46
votes
2 answers
What are the potential applications of perfectoid spaces to homotopy theory?
This year's Arizona Winter School was on perfectoid spaces, and there were quite a few homotopy theorists in the audience. I'd like to get a "big list" of reasons homotopy theorists might care about perfectoid spaces. Of course, a general answer is…
Yuri Sulyma
- 1,513
46
votes
2 answers
What interesting/nontrivial results in Algebraic geometry require the existence of universes?
Brian Conrad indicated a while ago that many of the results proven in AG using universes can be proven without them by being very careful (link). I'm wondering if there are any results in AG that actually depend on the existence of universes (and…
Harry Gindi
- 19,374
46
votes
2 answers
Is the following identity true?
Calculation suggests the following identity:
$$
\lim_{n\to \infty}\sum_{k=1}^{n}\frac{(-1)^k}{k}\sum_{j=1}^k\frac{1}{2j-1}=\frac{1-\sqrt{5}}{2}.
$$
I have verified this identity for $n$ up to $5000$ via Maple and find that the left-hand side…
Chitsai Liu
- 2,153
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46
votes
5 answers
Mathematicians with aphantasia (inability to visualize things in one's mind)
Are there any mathematicians with aphantasia? If so, could they please elaborate upon what their experience with mathematics is like?
I realize that this question probably falls outside of the scope of Mathoverflow, but it's so shocking that such a…
Trent
- 999
46
votes
8 answers
Non-examples of model structures, that fail for subtle/surprising reasons?
An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's limits lie, especially in cases where that's not…
Peter LeFanu Lumsdaine
- 18,946