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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6
votes
2answers
310 views

Twist in identification with singular cohomology

Let $X$ be a smooth projective variety over $\mathbb{Q}$ and $$V = H^m(X(\mathbb{C}), \mathbb{Q} \cdot (2\pi i)^r)$$ Then I've seen people write the comparison with complex cohomology (an isomorphism ...
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 ...
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 ...
10
votes
4answers
6k 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 ...
50
votes
26answers
4k views

Proof synopsis collection

I hate to keep going with the big lists, but the question about one-sentence summaries of topics/areas spurred this question...and I just can't help myself! Definition (Fraleigh): A proof synopsis ...
6
votes
2answers
402 views

Can we define an “empirically generic” real number?

Summary: My question, in a nutshell, is how we should intuitively imagine a generic real number (as opposed to a random one), and whether we can construct numbers which empirically behave like generic ...
13
votes
1answer
2k views

Intuition for the Hardy space $H^1$ on $R^n$

the standard intuition for Lebesgue spaces $L^p(\mathbb R^n)$ for $p \in [1,\infty]$ are measurable functions with certain decay properties at infinity or at the singularities. In particular, a ...
80
votes
37answers
14k views

Examples of eventual counterexamples

Define an "eventual counterexample" to be $P(a) = T $ for $a < n$ $P(n) = F$ $n$ is sufficiently large for $P(n) = T\ \ \forall n \in \mathbb{N}$ to be a 'reasonable' conjecture to make. where ...
21
votes
7answers
3k views

Spectral graph theory: Interpretability of eigenvalues and -vectors

I thought "Wow!" when I learned that the eigenvector of the adjacency matrix of a cycle graph $C_n$ corresponding to the second largest eigenvalue gives the coordinates of the vertices when equally ...
162
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 ...
6
votes
5answers
579 views

What makes a set random?

There are many results in number theory, where the existence of some $B \subseteq \mathbb{N}$ with certain properties is proved by a probabilistic argument employing "random sets". One such example ...
15
votes
4answers
4k views

Intuition and/or visualisation of Ito integral/Ito's lemma

Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve. See e.g. Wikipedia:Riemann sum The Ito integral has due to the unbounded total variation ...
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 ...
7
votes
6answers
383 views

Do you have examples of such “transitive” elements?

(I've asked the same question at the MSE, so far with no answers, so I thought I'd try it here as well. If there's some clash with any site rules, please let me know and I'll abide.) Let $A$ be a set ...
26
votes
2answers
2k views

A geometric characterization for arithmetic genus

Let $X$ be a smooth projective variety over $\mathbb{C}$. The following information is all equivalent (any of these numbers can be computed by a linear equation from any of the others): the ...
0
votes
0answers
24 views

Why Does a quadratic phase term in BNLS causes collapse?

I've heard a couple of times that in the Biharmonic Nonlinear Schrodinger Equation, $i\psi_z + \Delta ^2 \psi + |\psi | ^{2\sigma } \psi =0 $, $\psi (x, 0) = \psi _0 (x) \in H^1( \mathbb{R} ^d ) $ ...
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 ...
167
votes
7answers
82k views

Philosophy behind Mochizuki's work on the ABC conjecture

Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
4
votes
1answer
138 views

what characterizes a characteristic function of a probability measure in separable Hilbert spaces?

As we all know on real line $\mathbb{R}$, the following is valid A $\mathbb{C}$-valued function $\varphi$ is a characteristic function of a probability measure on $\mathbb{R}$ if and only if ...
11
votes
2answers
408 views

References for particular topics related to Langlands

I have never really concentrated on Langlands, which explains my poor level of understanding of it. But I have read quite a few introductory papers related to Langlands, and to the circle of ideas ...
13
votes
1answer
848 views

Intuition of law of iterated logarithm?

Let $X_i$ be iid random variables with $EX_i = 0$ and $Var X_i=1$ and $S_n=X_1+\cdots+X_n$. Then the law of the iterated logarithm says almost everywhere we have ...
31
votes
2answers
2k views

Meaning/Origin of Seiberg-Witten Equations/Invariants

Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from. We take an ...
10
votes
3answers
519 views

What is the intuition behind the definition of cuspidal representations?

Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the ...
2
votes
2answers
192 views

Intuition on Lindeberg condition

I want to know how Lindeberg came up with the condition which is sufficient for CLT to hold ? What is the intuition behind such an expression ?
26
votes
2answers
2k views

Intuition for coends

Let $D$ be a co-complete category and $C$ be a small category. For a functor $F:C^{op}\times C \to D$ one defines the co-end $$ \int^{c\in C} F(c,c) $$ as the co-equalizer of $$ \coprod_{c\to ...
20
votes
3answers
812 views

“Paradoxes” in $\mathbb{R}^n$

One may think of this question as a duplicate of this one. I see it more like an extension. The "inscribed sphere paradox" discussed in the aforementioned question states that if you inscribe a ...
4
votes
3answers
1k views

Is there a way to graphically imagine smash product of two topological spaces?

Recently I've been reading "Topology" by Klaus Janich. I find this book very entertaining as it contains lots of graphical illustrations that appeal to my "geometrical" imagination. In paragraph 3.6 ...
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 ...
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 ...
76
votes
15answers
9k views

What is torsion in differential geometry intuitively?

Hi, given a connection on the tangent space of a manifold, one can define its torsion: $$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$ What is the geometric picture behind this ...
6
votes
1answer
386 views

Getting the story of Dynkin and Satake diagrams straight

I've been trying to teach myself the theory of Lie groups. The sources I've been reading reference Lie algebras in the context of Dynkin and Satake diagrams, but not Lie groups (which I am more ...
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 ...
28
votes
5answers
874 views

are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?

I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary ...
46
votes
5answers
3k views

Is there an intuitive reason for Zariski's main theorem?

Zariski's main theorem has many guises, and so I will give you the freedom to pick the one that you find to be most intuitive. For the sake of completeness, I will put here one version: Zariski's ...
17
votes
1answer
479 views

What motivated Rademacher's contour along the Ford circles?

Apologies if this question isn't suitable for MathOverflow; I posted it on MSE here but it didn't get a response and it felt like it was on the cusp of being suitable for here. After Ramanujan and ...
16
votes
11answers
5k views

Examples of using physical intuition to solve math problems

For the purposes of this question let a "physical intuition" be an intuition that is derived from your everyday experience of physical reality. Your intuitions about how the spin of a ball affects ...
40
votes
7answers
3k views

Demystifying the Caratheodory Approach to Measurability

Nowadays, the usual way to extend a measure on an algebra of sets to a measure on a $\sigma$-algebra, the Caratheodory approach, is by using the outer measure $m^* $ and then taking the family of all ...
41
votes
4answers
6k views

Zagier's one-sentence proof of Fermat's theorem.

Zagier has a very short proof ( MR1041893) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares. The proof defines an involution of the set $S= \lbrace (x,y,z) \in ...
11
votes
2answers
4k 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 ...
5
votes
1answer
985 views

Self-intersection and the normal bundle

Let $X/k$ be a surface nonsingular and proper over an algebraically closed field $k$. Let $C \subset X$ be a nonsingular curve. Then it is clear that the self-intersection $(C \cdot C)_X$ is ...
4
votes
2answers
279 views

How does Tate cohomology fit into a derived categories framework?

I've read through one class field theory text after another, but there's something very non-intuitive for me about cohomology that makes it hard for me to understand why Tate cohomology was invented. ...
3
votes
1answer
436 views

What do orbital integrals have to do with reciprocity?

Hi, this is my first question (of many). I am blogging for the Fields Medal Symposium and would like to get into the mathematics involved with our program. In an attempt to sort through the articles ...
6
votes
4answers
777 views

How to solve Ax=b incrementally ?

Hi, everyone. What I am struggling is the following problem. I have a linear matrix equation $Ax=b$, where $A$ is a known $n \times n$ large sparse real matrix, $x$ and $b$ are known $n \times 1$ ...
4
votes
1answer
319 views

Intuition of Kolmogorov-Sinai entropy

For a measurable entropy of measurable transformation $T$ from $(X,\mathcal{B},m)$ to itself. For each finite measurable partition $\mathcal{A}=\{A_i\}_{i=1}^{m}$ of $X$, we can define ...
65
votes
24answers
27k views

Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows- An engineer, a physicist, and a mathematician are discussing how to visualise four ...
6
votes
6answers
3k views

What is a complex inner product space “really”?

This is an extended re-post of a question that I have asked on MSE not a long time ago. But anyway, it seems more appropriate for MO. To begin with, in a real inner product space we have a geometric ...
3
votes
3answers
564 views

Mock Theta Functions

I am studying about Mock modular forms and Mock theta functions. I wonder how Zwegers connected mock theta functions with Harmonic Maass Forms? I mean, what was the philosophy/idea of Mock Theta ...
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 ...
48
votes
11answers
4k views

How should one think about non-Hausdorff topologies?

In most basic courses on general topology, one studies mainly Hausdorff spaces and finds that they fit quite well with our geometric intuition and generally, things work "as they should" ...
5
votes
0answers
121 views

What is the analogy between the Hilbert function and L-functions?

In his book Projective Varieties and Modular Forms, M. Eichler uses the notation $L(\lambda, M)$ for the Hilbert function of a finite graded $R=k[x_0, \dots, x_n]$-module $M$. So, $L(\lambda, M) = ...