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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24
votes
6answers
2k views

Why is addition of observables in quantum mechanics commutative?

I am no expert in the field. I hope the question is suitable for MO. Background/Motivation I once followed a quantum mechanics course aimed at mathematicians. Instead of the usual motivations coming ...
29
votes
6answers
2k views

Why do Littlewood-Richardson coefficients describe the cohomology of the Grassmannian?

I'm looking for a "conceptual" explanation to the question in the title. The standard proofs that I've seen go as follows: use the Schubert cell decomposition to get a basis for cohomology and show ...
16
votes
3answers
2k views

Stacks and sheaves

I'm a bit confused by the double role which sheaves play in the theory of stacks. On the one hand, sheaves on a site are the obvious generalization of a sheaf on a topological space. On the other ...
18
votes
5answers
1k views

Why does the group law commute with morphisms of elliptic curves?

I know this should be pretty simple, but right now the only way I can see how to prove it is to sit down and write out explicit formulae for the group law, and see that everything works out. What's ...
34
votes
9answers
7k views

Intuition for Group Cohomology

I'm beginning to learn cohomology for cyclic groups in preparation for use in the proofs of global class field theory (using ideal-theoretic arguments). I've seen the proof of the long exact sequence ...
93
votes
16answers
12k views

How do I make the conceptual transition from multivariable calculus to differential forms?

One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on ...
50
votes
9answers
6k views

How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2?

My apologies if this is too elementary, but it's been years since I heard of this paradox and I've never heard a satisfactory explanation. I've already tried it on my fair share of math Ph.D.'s, and ...
163
votes
61answers
85k views

Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...
5
votes
2answers
382 views

What are natural transformations in 1-categories?

It's well-known that, for lots of concrete categories (but by no means all), we can think of the objects as themselves being small categories, and morphisms are the functors between these categories. ...
20
votes
5answers
2k views

tips on cohomology for number theory

I am curious about what is a good approach to the machinery of cohomology, especially in number-theoretic settings, but also in algebraic-geometric settings. Do people just remember all the rules and ...
50
votes
15answers
6k views

What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?

Before you close for "homework problem", please note the tags. Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about ...
0
votes
5answers
2k views

How to teach addition of negative numbers? [closed]

I have a friend with dyscalculia and was teaching her some some mathematics (namely, solving a linear equation, simplifying certain expressions, and what (affine linear) functions are). She ...
8
votes
3answers
1k views

Intuition about schemes over a fixed scheme

I am taking a first course on Algebraic Geometry, and I am a little confused at the intuition behind looking at schemes over a fixed scheme. Categorically, I have all the motivation in the world for ...
20
votes
4answers
1k views

Heuristic explanation of why we lose projectives in sheaves.

We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking? This question was asked(and I found it very helpful) but I ...
14
votes
4answers
2k views

Etale cohomology and l-adic Tate modules

$\newcommand{\bb}{\mathbb}\DeclareMathOperator{\gal}{Gal}$ Before stating my question I should remark that I know almost nothing about etale cohomology - all that I know, I've gleaned from hearing off ...
15
votes
2answers
1k views

What does primary decomposition of (sub) modules mean geometrically?

I want to know how I should visualize modules in algebraic geometry. The way we visualize rings, via their spectra, automatically (or by the beauty of its design) depicts primary decomposition of ...
27
votes
3answers
4k views

Why are planar graphs so exceptional?

As compared to classes of graphs embeddable in other surfaces. Some ways in which they're exceptional: Mac Lane's and Whitney's criteria are algebraic characterizations of planar graphs. (Well, ...
68
votes
8answers
5k views

Why are flat morphisms “flat?”

Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason! Is ...
26
votes
6answers
3k views

Why does one think to Steenrod squares and powers?

I'm studying Steenrod operations from Hatcher's book. Like homology, one can use them only knowing the axioms, without caring for the actual construction. But while there are plenty of intuitive ...
78
votes
25answers
21k views

What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
11
votes
5answers
1k views

Abstract nonsense versions of “combinatorial” group theory questions

In particular, I'm just curious whether there's a version of the Sylow theorems (which are very combinatorially-flavored) which allows horizontal and/or vertical categorification? Or at least can be ...
18
votes
5answers
2k views

Flips in the Minimal Model Program

In order get a minimal model for a given a variety $X$, we can carry out a sequence of contractions $X\rightarrow X_1\ldots \rightarrow X_n$ in such a way that that every map contracts some curves on ...
28
votes
3answers
2k views

What do higher Chow groups mean?

Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each ...
23
votes
7answers
2k views

What examples of distributions should I keep in mind?

I'm learning a bit about the theory of distributions. What examples of distributions will help me develop good intuition? Definitions: Let $U$ be an open subset of $\mathbb{R}^n$. Write ...
4
votes
1answer
644 views

Explanation for Satake correspondence

Some time ago I was told there's an interesting classical Satake correspondence which I will write as $$[\mathop{\mathrm{disk}} \Rightarrow G] \\,\backslash\\, [\mathop{\mathrm{disk}^\times} ...
24
votes
3answers
3k views

What is the “intuition” behind “brave new algebra”?

Y.I. Manin mentions in a recent interview the need for a “codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”. This makes me puzzle, because I ...
35
votes
12answers
5k views

Cures for mathematician's block (as in writer's block) [closed]

What kind of things do you find that help you get the "creative juices flowing," to use a tired cliche, when you're stuck or burnt out on a problem? I've read about some studies that suggest listening ...
46
votes
52answers
15k views

Colloquial catchy statements encoding serious mathematics

As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as ...
27
votes
6answers
4k views

Algebraically closed fields of positive characteristic

I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
4
votes
3answers
2k views

Intuitive Example of a Jacobson Radical

Can anyone explain what a Jacobson radical is using an intuitive example? I can't quite understand Wikipedia's explanation.
25
votes
6answers
3k views

Intuition for the last step in Serre's proof of the three-squares theorem

Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer a is the sum of three squares if and only if it is not of the form 4^m (8n ...
54
votes
11answers
9k views

“Philosophical” meaning of the Yoneda Lemma

The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward. Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning ...
13
votes
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 ...
12
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 ...
22
votes
6answers
5k 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
667 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
1k 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 ...
33
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
834 views

elementary Ext^1 intuition

I am wondering what sort of basic basic intuitive meaning Ext1(M,N) has. As a base case: if ...
22
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 ...
20
votes
11answers
14k 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 ...
28
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 ...
22
votes
10answers
6k 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 ...