Tagged Questions

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 ...

412 views

Effects of “weak” vs. “strict” categories in Eckmann-Hilton arguments

A standard example for demonstrating the need for genuinely weak n-categories is that a weak 3-category with unique 0- and 1-cells amounts to the same thing as a braided monoidal category (by an ...
3k views

Intuition for Primitive Cohomology

In complex projective geometry, we have a specified Kähler class $\omega$ and we have a Lefschetz operator $L:H^i(X,\mathbb{C})\to H^{i+2}(X,\mathbb{C})$ given by $L(\eta)=\omega\wedge \eta$. We then ...
7k views

Truth of the Poisson summation formula

The Poisson summation says, roughly, that summing a smooth $L^1$-function of a real variable at integral points is the same as summing its Fourier transform at integral points(after suitable ...
805 views

Geometric Intuition for Big Monodromy

In various contexts, I have come across results referred to as "big monodromy." A standard arithmetic example is the open image theorem for the image of Galois action on non-CM elliptic curves. A ...
9k views

Visualization of Riemann–Stieltjes Integrals

The Riemann–Stieltjes integral $\int_a^b f(x)\,dg(x)$ is a generalization of the Riemann integral. It is e.g. heavily used as a starting point for stochastic integration. The approximating Riemann–...
2k views

Understanding moment maps and Lie brackets

I'm trying to learn about moment maps in symplectic topology (suppose our Lie group is $G$ with Lie algebra $\mathfrak g$, acting on the symplectic manifold $(M,\omega)$ by symplectomorphisms). I'm ...
3k 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 ...
3k 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 ...
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 ...
2k 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 ...
8k 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 ...
18k 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 ...
11k 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 ...
110k 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 ...
441 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. ...
3k 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 ...
9k 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 ...
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 ...
2k 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 ...
2k 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 ...
3k 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 ...
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 ...
6k 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, ...
8k 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 ...
4k 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 ...
34k 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 ...
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 ...
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 ...
3k 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 i,...
3k 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 \$C_c^\infty(...
803 views

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} \... 3answers 5k 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 ... 12answers 6k 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 ... 51answers 18k 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 ... 5answers 7k 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 ... 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. 6answers 4k 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 (... 11answers 14k 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 ... 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 ... 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 ... 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 ... 3answers 919 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 '... 4answers 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 @...
5k views

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 ...
957 views

elementary Ext^1 intuition

I am wondering what sort of basic basic intuitive meaning Ext1(M,N) has. As a base case: if <...
3k 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 ...
30k views

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 ...
5k 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 ...