Questions asking for the intuition behind some definition, conjecture, proof etc. In other words, questions designed to improve or to acquire understanding on a conceptual or intuitive level, as opposed to on a technical or formal level. When asking such a question it can be helpful to include a ...

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14
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6answers
1k views

Can the “physical argument” for the existence of a solution to Dirichlet's problem be made into an actual proof?

Caveat: I don't really know anything about PDEs, so this question might not make sense. In complex analysis class we've been learning about the solution to Dirichlet's problem for the Laplace ...
13
votes
15answers
1k views

Most helpful heuristic?

What's the most useful piece of mathematical "folk wisdom" you've encountered? I'm talking here about things that aren't theorems, or even conjectures, or even shadows of conjectures -- just broad ...
24
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6answers
6k views

Intuition for Integral Transforms

It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace ...
11
votes
3answers
752 views

An intuitive reason why the “Rule 30” CA is random/pseudorandom?

I'm a little bit hesitant to ask this here, so please notice the tag. My hope is that someone will have a more satisfying answer than what I've heard before... A long time ago I read (perhaps ...
17
votes
3answers
2k views

Some intuition behind the five lemma?

Slightly simplified, the five lemma states that if we have a commutative diagram (in, say, an abelian category) $$\require{AMScd} \begin{CD} A_1 @>>> A_2 @>>> A_3 @>>> A_4 ...
37
votes
5answers
4k views

Intuition about the cotangent complex?

Does anyone have an answer to the question "What does the cotangent complex measure?" Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as ...
15
votes
3answers
897 views

elementary Ext^1 intuition

I am wondering what sort of basic basic intuitive meaning Ext1(M,N) has. As a base case: if ...
26
votes
6answers
2k views

How to think about model categories?

I've read about model categories from an Appendix to one of Lurie's papers. What are the examples of model categories? What should be my intuition about them? E.g. I understand the typical examples ...
25
votes
11answers
21k views

Why is the gradient normal?

This is a somewhat long discussion so bear with me. There is a theorem that I have always been curious about from an intuitive standpoint. This is an issue that has been glossed over in most ...
33
votes
4answers
4k views

Is there a good way to think of vanishing cycles and nearby cycles?

Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...
10
votes
5answers
3k views

Examples and intuition for arithmetic schemes

How should a beginner learn about arithmetic schemes (interpret this as you wish, or as a regular scheme, proper and flat over Spec(Z))? What are the most important examples of such schemes? Good ...
27
votes
10answers
7k views

What is (co)homology, and how does a beginner gain intuition about it?

This question comes along with a lot of associated sub-questions, most of which would probably be answered by a sufficiently good introductory text. So a perfectly acceptable answer to this question ...