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Results tagged with big-picture
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user 8628
Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and focused, it can be helpful to keep in mind that the design of MathOverflow does not make it a good fit for questions that are too broad.
4
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1
answer
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Maximality without Zorn
When confronted with finding an object that is maximal with regard to some ordering relation, most of us have the reflex to use Zorn's Lemma.
I am interested in instances of proving the existence of …
- 48.9k
11
votes
Proposals for polymath projects
A conjecture that can be stated in so simple terms that it is hard to classify, is Frankl's Union-Closed Sets Conjecture. It would be fantastic to see this solved.
1
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What are examples of good toy models in mathematics?
Boolean algebras are toy models for distributive lattices, which in turn are toy models for lattices in general (and partially ordered sets).
11
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Examples of eventual counterexamples
One could reasonably conjecture that there are no positive integers $a,b,c$ satisfying
$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4.$$
I say "reasonably", because the smallest integers satisfyi …
6
votes
Concepts in topology successfully transferred to graph theory and combinatorics with non-tri...
Infinite graphs have been used as a "discrete version" of topological spaces, for instance infinite Cayley graphs as a discretisation of homogeneous spaces). Gromov constructed homogeneous spaces out …
9
votes
Accepted
Continuous relations?
Here's a different and quite generic approach: Let $X,Y$ be topological spaces. Then we topologize $\mathcal{P}(Y)$ and say that $R\subseteq X\times Y$ is continuous if and only if the function $f_R: …
- 48.9k
-1
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Each mathematician has only a few tricks
A trick that is used daily is Zorn's Lemma. Sure, every mathematician knows it, but it certainly helped prove non-trivial propositions and it is in daily use. I would consider it in a list of the Top …