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Results tagged with big-picture
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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.
3
votes
Visualizing how Cech cohomology detects holes
Here's an idea; I'm not sure if all of the steps are possible, but I think it should work.
Take a simplicial complex of dimension $n$ (all simplices are of dimension $\leq n$). "Thicken" the simplici …
- 21k
53
votes
Accepted
Heuristic behind the Fourier-Mukai transform
First, recall the classical Fourier transform. It's something like this: Take a function $f(x)$, and then the Fourier transform is the function $g(y) := \int f(x)e^{2\pi i xy} dx$. I really know almos …
- 21k
7
votes
What's a groupoid? What's a good example of a groupoid?
Let me expand a bit on what Dave said.
The Yoneda lemma tells us that given an object $X$ of a category $\mathcal C$, the (covariant, contravariant, whatever) functor $h_X : \mathcal C \to \mathsf{Se …
9
votes
What is a cohomology theory (seriously)?
Urs Schreiber has already addressed parts of this question here. There's lots of good stuff to mine through in the nLab entry on cohomology.
- 21k
11
votes
Mathematics of path integral: state of the art
Recently I have been reading Kevin Costello's book (draft) Renormalization of Quantum Field Theories, which claims to work out some foundations of perturbative quantum field theory following the "Wils …
18
votes
Accepted
Doing geometry using Feynman Path Integral?
Try:
Witten, Quantum field theory and the Jones polynomial
Witten, The index of the Dirac operator in loop space
I have found both of these papers quite difficult to understand. I don't know any ea …
- 21k
36
votes
Fundamental Examples
Someone has already mentioned tori, but I think elliptic curves in algebraic geometry merit their own separate mention.
Answered by Kevin Lin
28
votes
Why is it useful to study vector bundles?
I think many of the other answers boil down to the same underlying idea: Sections of vector bundles are "generalized functions" or "twisted functions" on your manifold/variety/whatever.
For example, …
- 21k