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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10
votes
2answers
444 views

Intuition/idea behind a proof of the splitting principle?

The splitting principle is as follows. Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map ...
4
votes
0answers
55 views

The metric gives the optimal element in a class

In geometry there is plenty of examples in which the following happens: Some elements are considered equivalent, in some topological or algebraic sense We take the quotient The metric is usually not ...
7
votes
1answer
259 views

Intuition for Picard-Lefschetz formula

I'm trying to develop some intuition for the (local) Picard-Lefschetz formula (which I'm encountering for the first time in Deligne's paper "La Conjecture de Weil, I"). To summarize the setup, we ...
2
votes
0answers
137 views

Description of the equalizer of $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$

This is a crosspost of this MSE question. I have asked several questions in an attmept to get a general version of the Chinese remainder theorem without conditions on the ideals which will ...
16
votes
1answer
533 views

(really) basic intuition for $\mathbb A^1$-homotopy theory

Apologies in advance if this question is inappropriate for MO. I'm trying to read here and there about $\mathbb A^1$-homotopy theory in algebraic geometry. I understand some abstract machinery is ...
12
votes
1answer
435 views

Axiom of choice as zero dimensionality

In the paper Quantifiers and Sheaves by Lawvere, at the bottom of the second page, the author writes: "... the condition that every epi splits, which geometrically we would call 0-dimensionality ...
39
votes
0answers
594 views

Vector bundle $L$ admits connection if and only if degree of every direct summand of $L$ divisible by $\text{char}\,k$, intuition

Consider the following theorem of Atiyah. Let $X$ be a connected smooth projective curve over an algebraically closed field $k$. Then a vector bundle $L$ on $X$ admits a connection if and only if ...
0
votes
1answer
1k views

Can one expect the existence of a relevant approach for a proof of the Riemann hypothesis using Mochizuki's theory? [closed]

Next month at Oxford university, there will have the first workshop outside Asia on the Inter-Universal Teichmuller theory of Shinichi Mochizuki: ...
8
votes
1answer
229 views

Intuitive reasons for the existence of modular parametrizations

Whenever I encounter anything about modular parametrizations, I have a feeling it is something very unnatural: you have some kind of moduli space and all of a sudden it parametrizes an object ...
11
votes
2answers
328 views

Quotient rule, differential operator on a localization is well-defined, underlying geometry?

Using the quotient rule, we obtain that the notion of differential operator on a localization is well-defined:$$\mathcal{D}_A(B_f) \cong \mathcal{D}_A(B)_f.$$Here, $B$ is a commutative $A$-algebra, ...
18
votes
2answers
750 views

how do you visualize characteristic class?

For cohomology, there are some equivalent definitions when the object we consider is sufficiently nice. Since I mainly work with algebraic variety, I will restrict the objects I am considering to be ...
3
votes
1answer
74 views

mapping class group relations

The question I want to ask is vague in a sense. We have examples of mapping class relations, e.g. lantern relation, chain relations, etc. For instance the latern relation on a disk with three boundary ...
20
votes
2answers
1k views

Lemma 1 from Beilinson's “Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra”, intuition?

Consider Lemma 1 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows. Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, $F: ...
5
votes
0answers
137 views

Making the conceptual leap from locales to Grothendieck topologies?

I find the definition for locales and sheaves on locales to be straightforward, but I'm stumbling over the idea of a Grothendieck topology. Is there a nice way to see roughly how the latter ...
5
votes
1answer
223 views

Importance of Ornstein's isomorphism theorem

"Perhaps the most important parts of the Ornstein theory are criteria for determining whether or not a shift or flow is Bernoulli (a Bernoulli shift, $B_{ct}$ , or $B_{t}^{\infty}$) because it allows ...
1
vote
0answers
127 views

Importance and intuition of global sections in sheaf cohomology

I am trying to understand why global sections of a sheaf are "important" or interesting objects of study. Perhaps I have too weak of a background to appreciate it (and that is certainly an acceptable ...
9
votes
1answer
375 views

Grothendieck - sheaves as meter sticks

I'm trying to read parts of McLarty's Grothendieck on Simplicity and Generality. In the article, I read Grothendieck thought of sheaves over some topological space as meter sticks measuring it. ...
10
votes
3answers
530 views

Self-Similar Graphs

Many fractals can be generated using and infinite sequence of graphs. For example, Sierpinski's Gasket could be generated by the following sequence of graphs. Many definitions of fractal dimensions ...
14
votes
1answer
442 views

Intuition behind the Morse inequalities?

Forgive me if this is sort of a vague question, but can someone supply me with their intuition behind the Morse inequalities?
15
votes
1answer
1k views

Why is there a $\sqrt{5}$ in Hurwitz's Theorem?

Hurwitz's theorem is an extension of Minkowski's Theorem and deals with rational approximations to irrational numbers. The theorem states: For every irrational number $\alpha$, there are infinitely ...
4
votes
4answers
460 views

What is the motivation and purpose of the Floretion group?

When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...
3
votes
0answers
89 views

Lie group cohomology with coefficients in Lie algebra

I'm looking for a reference, and basic results, about Lie algebra as modules over a Lie group (with the adjoint representation), from the point of view of cohomology. Links with the Lie algebra ...
17
votes
5answers
1k views

How to explain the concentration-of-measure phenomenon intuitively?

One way to phrase the "concentration-of-measure" phenomenon is that, for a Euclidean sphere $S^d$ in $d$ dimensions, for large $d$, "most of the mass is close to the equator, for any equator."1 ...
11
votes
3answers
722 views

Why does the bitxor function appear in Nim?

I am conducting research in Combinatorial Game Theory (CGT). Although I have done a considerable amount of reading, I do not completely understand why the bit-xor function also known as the nim-sum ...
17
votes
1answer
2k views

Why do people use “formal calculation” to describe informal calculations?

Many times, I see the word formal being used to describe a calculation that is not rigorous. I would think that such calculations should rather be termed informal than formal. What is the explanation ...
1
vote
0answers
299 views

What does the Riemann–Stieltjes integral measure? [closed]

The Riemann–Stieltjes integral is a generalization of the Riemann integral, and has a definition based on a sum analogous to the Riemann sum: $$ S(P,f,g) =\sum_{k=1}^{n} f(x_k)\Delta g(x_k) $$ where ...
12
votes
1answer
835 views

How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices?

I asked this question in Math Stack Exchange earlier here: http://math.stackexchange.com/questions/1199380/what-is-the-intuition-behind-how-can-we-interpret-the-eigenvalues-and-eigenvec and since I ...
5
votes
2answers
321 views

Choice of fibrations is like a choice of a basis of a module

In some notes on derived stacks, in describing categories of fibrant objects, the author drops this parenthetical: (Grothendieck said in his famous letter to Quillen that the choice of $\mathscr ...
2
votes
1answer
295 views

Yang-Mills Functional and Energy

I have a question about the meaning of Yang-Mills Functional. It is stated everywhere that the Yang-Mills Functional is a measure of energy. But the formal definition of the Yang-Mills Functional is: ...
1
vote
0answers
255 views

Intuitive Approach to Sheaf and Cech Cohomology [closed]

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
1
vote
0answers
149 views

Intuition for hereditary torsion theories

I'm looking for intuition and references for the definition of a hereditary torsion theory and two facts found here. First, the definition and facts: Definition. A torsion theory $(\mathcal ...
7
votes
1answer
418 views

Intuitive Aproach to Dolbeault Cohomology [closed]

(Duplicated from math.stackexchange) I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. ...
0
votes
1answer
440 views

What is the meaning of non-Hausdorff spaces in algebraic geometry [closed]

At the beginning I should warn everybody reading this post: I don't know much about algebraic geometry so specialists in this subject may see my question as ignorant. As far I understood one on the ...
1
vote
0answers
114 views

Role of determinant of the matrix corresponding to $i$-th Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
27
votes
4answers
2k views

Why the Dold-Thom theorem?

Dold-Thom Theorem: $$\pi_i(SP(X))\cong\tilde{H}_i(X)$$ It's pretty miraculous, no? I've seen its proof, where you show that the composition of the functors on the left-side satisfies the axioms of a ...
13
votes
2answers
921 views

What justification can you give for the fact that “most ODEs do not have an explicit solution”?

What justification can you give for the fact that "most ODEs do not have an explicit solution"?
5
votes
1answer
279 views

Compositional inversion and generating functions in algebraic geometry

The exponential generating function of the graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus zero satisfying the associativity equations of physics (the WDVV ...
7
votes
2answers
397 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 ...
6
votes
2answers
578 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 ...
7
votes
6answers
451 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 ...
0
votes
0answers
53 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^2( \mathbb{R} ^d ) $ ...
6
votes
5answers
628 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 ...
4
votes
1answer
214 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
494 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 ...
11
votes
3answers
675 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 ...
4
votes
2answers
365 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 ?
22
votes
3answers
1k 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 ...
6
votes
1answer
549 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 ...
20
votes
1answer
685 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 ...
30
votes
5answers
1k 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 ...