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Results tagged with big-picture
Search options questions only
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user 2051
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.
73
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6
answers
25k
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What is a cumulant really?
A cumulant is defined via the cumulant generating function
$$ g(t)\stackrel{\tiny def}{=} \sum_{n=1}^\infty \kappa_n \frac{t^n}{n!},$$
where
$$
g(t)\stackrel{\tiny def}{=} \log E(e^{tX}).
$$
Cumulants …
- 22.1k
65
votes
16
answers
8k
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What is the high-concept explanation on why real numbers are useful in number theory?
The utopian situation in mathematics would be that the statement and the proof of every result would live "in the same world", at the same level of mathematical complexity (in a broad sense), unless t …
42
votes
26
answers
8k
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Where can square roots come from when they are not distances?
In a recent survey "Supergeometry in Mathematics and Physics", Kapranov points out cases in which observable quantities of immediate interest are represented as bilinear combinations of more fundamen …
29
votes
10
answers
3k
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Are infinite dimensional constructions needed to prove finite dimensional results?
Infinite dimensional constructions, such as spaces of diffeomorphisms, spectra, spaces of paths, and spaces of connections, appear all over topology. I rather like them, because they sometimes help me …
- 22.1k
24
votes
3
answers
3k
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Elevator pitch for the Virtual Fibering Theorem
There has been a great deal of excitement among topologists about the proof of the Virtual Haken Theorem, and in fact of the Virtual Fibering Theorem (for closed hyperbolic 3-manifolds, but I'm guess …
- 22.1k
23
votes
6
answers
6k
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Why chain homotopy when there is no topology in the background?
Given two morphisms between chain complexes $f_\bullet,g_\bullet\colon\,C_\bullet\longrightarrow D_\bullet$, a chain homotopy between them is a sequence of maps $\psi_n\colon\,C_n\longrightarrow D_{n …
- 22.1k
22
votes
4
answers
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Natural setting for characteristic classes?
In my mind, algebraic topology is comprised of two components:
Chain complex information, which is intrinsic information concerning how your object may be built up out of simple "lego blocks".
Charac …
- 22.1k
18
votes
0
answers
492
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What do tangles teach us about braids?
A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$ …
- 22.1k
15
votes
1
answer
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Why are Witten-Reshetikhin-Turaev invariants expected to be integral?
A Witten-Reshetikhin-Turaev (WRT) Invariant $\tau_{M,L}^G(\xi)\in\mathbb{C}$ is an invariant of closed oriented 3-manifold $M$ containing a framed link $L$, where $G$ is a simple Lie group, and $\xi$ …
- 22.1k
13
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Proof by `universal receiver'
Anyone following the news knows about the major breakthoughs that have taken place recently in $3$-manifold topology. These have come via a route whose big-picture I find to be conceptually interestin …