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Results tagged with big-picture
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user 2530
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.
19
votes
Describe a topic in one sentence.
Representation theory of Lie groups: there is a whole world between $\mathrm{Sym}^n V$ and $\wedge^n V$. (Okay, this is an oversimplication - I am talking about the representations of $\mathrm{GL}\lef …
1
vote
What are examples of good toy models in mathematics?
Trees (resp. labelled trees) are a toy example for Young diagrams (resp. Young tableaux). Feel free to elaborate (I don't have time to).
10
votes
Structures that turn out to exhibit a symmetry even though their definition doesn't
I always found $\mathrm{Tor}_R\left(M,N\right) \cong \mathrm{Tor}_R\left(N,M\right)$ for a commutative ring $R$ and two $R$-modules $M$ and $N$ to be mysterious. Then again I have no idea about homolo …
15
votes
"Algebraic" topologies like the Zariski topology?
The $I$-adic topology on a commutative ring $A$ (with unity), where $I$ is an ideal of $A$. The closed sets are intersections of finite unions of sets of the form $a+I^n$ with $a\in A$ and $n\in\mathb …
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5
votes
What notions are used but not clearly defined in modern mathematics?
Left/right derived functors. If $F$ is an additive functor from a category $A$ to another category $B$, then the left/right derived functors of $F$ go from $A$ to... where? Not to $B$ certainly, becau …