Questions tagged [intuition]

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 rough description of ones understanding of the subject at hand (on a technical level).

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Dimension of the moduli stack of vector bundles over a curve

Let $Vect_{n}(C)$ the moduli stack of vector bundles $V$ of rank $n$ over a smooth curve $C$ of genus $g$. It is well known that $Vect_{n}(C)$ is a smooth stack of dimension $\dim(H^{0}(C,End(V)))-\...
Martin Hurtado's user avatar
19 votes
1 answer
2k views

Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?

Wikipedia calls resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use ...
William Bell's user avatar
2 votes
0 answers
315 views

Why can one compute the sum of divisors of $n$ without factoring $n$?

Question links to paper which states: $$ \sigma(n)= \frac{6}{n^2(n-1)}\sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\sigma(n-k) \qquad (1) $$ where $\sigma(n)$ is the sum of divisors of $n$. Another similar ...
joro's user avatar
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10 votes
1 answer
391 views

Why are root data a natural candidate for classifying connected reductive groups?

For the purpose of this question, you may assume that we are working over the complex numbers. Given a connected reductive group $G$, one can choose a maximal torus $T$, and then let $T$ act on the ...
Tim Phalange's user avatar
12 votes
1 answer
688 views

Intuition behind choosing a specific test function

I am learning about elliptic PDEs using the book by Chen & Wu, especially on the maximum principle. The author uses the De Giorgi iteration technique to establish the weak maximum principle for ...
Yongmin Park's user avatar
12 votes
1 answer
795 views

Intuition for categorical fibrations?

I think I have a pretty good intuitive understanding of most types of fibrations of quasicategories: a (trivial) Kan fibration is a bundle of (contractible) spaces with equivalent fibers, a left/...
JoeyBF's user avatar
  • 223
4 votes
0 answers
411 views

Brouwer's fixed point theorem and the one-point topology [closed]

I posted this question last week on Math SE and got upvotes and helpful comments that allowed me to make the question more precise https://math.stackexchange.com/q/3765546/810513. As I did not get an ...
R. Srivastava's user avatar
2 votes
1 answer
690 views

Interpretation around conjugacy classes in group theory [closed]

this is Rajeev Srivastava and his colleague Ravinder Padmanabha, specializing in computational geometry algorithms for application in science and research. We would like to include methods from ...
R. Srivastava's user avatar
17 votes
3 answers
1k views

Intuition behind stability and instability in model theory

In A survey of homogeneous structures by Macpherson (Discrete Mathematics, vol. 311, 2011), a stable or unstable theory is defined as (Definition 3.3.1): A complete theory $T$ is unstable if there is ...
Eugene Zhang's user avatar
13 votes
1 answer
1k views

Intuition about ordinal fixed points $\alpha = \aleph_\alpha$

I wanted to ask for your intuition about ordinal fixed points $\alpha = \aleph_\alpha$, where $\aleph_\alpha$ stands for the $\alpha$-th Aleph number in the Aleph sequence of cardinalities. For ...
Claus's user avatar
  • 6,777
-2 votes
1 answer
62 views

Interpretation of the word Random [closed]

I have previous knowledge of what a random experiment is, but sometimes I get confused by the use of the word Random. I can express my doubts as the following questions: if something is random them it ...
Walquer X. Valles's user avatar
44 votes
3 answers
2k views

Why is the Vandermonde determinant harmonic?

It can be checked that the Vandermonde determinant defined as $$V(\alpha_1, \cdots, \alpha_n) = \prod_{1 \le i < j \le n}(\alpha_i-\alpha_j) $$ is a harmonic function, that is $\Delta V = 0$ where ...
Sandeep Silwal's user avatar
10 votes
0 answers
364 views

Grothendieck categories and their morphisms

I am not an algebraic geometer in the first place, and I am mainly familiar with topology and category theory. Recently I am studying Grothendieck categories and I am struggling with getting ...
Ivan Di Liberti's user avatar
39 votes
9 answers
4k views

Interpretation of the action in classical mechanics

In classical mechanics the dynamics on a manifold $M$ are characterised by the minimisation of a functional $$ \min_{q \in C^\infty(\mathbb{R},M)} \int_{\mathbb{R}}L(q(t),\dot{q}(t))dt, $$ where $L:TM\...
Jannik Pitt's user avatar
  • 1,163
5 votes
0 answers
262 views

Intuition for the McGerty-Nevins compactification of quiver varieties

In Section 4 of the paper Kirwan surjectivity for quiver varieties (Inventiones Math. 2018) McGerty and Nevins define a compactification of the moduli space of representations of the preprojective ...
Yellow Pig's user avatar
  • 2,490
14 votes
0 answers
516 views

Seeing what gets Harvey Friedman's "tangible incompleteness" principles into large cardinal territory

I'm trying to wrap my head around some of Harvey Friedman's recent, unpublished work on his tangible incompleteness project, and I'm trying to see the link between his "tangible statements" (...
Malice Vidrine's user avatar
5 votes
0 answers
101 views

Intuition behind the local limit theorem in hyperbolic groups

Let $\Gamma$ be a finitely generated group and let $\mu$ be a probability measure on $\Gamma$. Denote by $X_n$ the induced random walk. Finally, let $p_n=\mu^{*n}(e)=P_e(X_n=e)$. The local limit ...
M. Dus's user avatar
  • 1,900
8 votes
1 answer
1k views

Geometric intuition behind this chain homotopy

My question has to do with the chain homotopy that appears in Lee's Introduction to Topological Manifols and Rotman's Introduction to Algebraic Topology proofs that the inclusion $$C_\bullet^\mathcal{...
Akerbeltz's user avatar
  • 484
5 votes
1 answer
405 views

Geometric intuition for Mather's cube theorem

Mather's cube theorem for the category of topological spaces says that given a homotopy-commutative cube: If one pair of opposite faces are homotopy pushouts and the two remaining faces adjecent ...
Arrow's user avatar
  • 10.3k
4 votes
1 answer
854 views

Geometric interpretation of sections $H^0(\Theta_X, X)$ of the Tangent sheaf over curve

I'm reading Mumford's & Oda's Algebraic Geometry II and I'm confused about explanations on geometric intuition of sections $H^0(\Theta_X, X)$ of the tangent sheaf on page 287: Let $X$ a ...
user267839's user avatar
  • 5,948
1 vote
0 answers
47 views

Geometric interpretation for uniformly elliptic pde of 2 second order

Let $\Omega \subset \mathbb{R}^{2}$ a domain,let $u \in C^{2}(\Omega)$, the operator $Lu= tr(A.D^{2}u) + <\nabla u,b> +cu$ where $A$ is a symmetric matrix, $b$ is a vector field continuous ...
Cézar Bezerr's user avatar
-2 votes
1 answer
408 views

Building intuition in algebraic number theory [closed]

How do you build your intuition in algebraic number theory? Generally my intuition in elementary number theory came from just numerically fiddling with python (and also elementary number theory is ...
Lelouch's user avatar
  • 105
75 votes
3 answers
3k views

Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?

Here is a couple of examples of the similarity from Wikipedia, in which the expressions differ only in signs. I encountered other analogies as well. $${\begin{aligned}\gamma &=\int _{0}^{1}\int _{...
Anixx's user avatar
  • 9,306
2 votes
2 answers
359 views

How should I think about concrete functors and in particular about concrete isomorphism?

All the definitions that follow is taken from The Joy of Cats. Definition 1. Let $\bf{X}$ be a category. A concrete category over $\bf{X}$ is a pair $({\bf{A}},U)$, where $\bf{A}$ is a category ...
user avatar
3 votes
0 answers
659 views

Deligne's Mixed Hodge Theory

Deligne constructs Mixed Hodge Structures (MHS) on the cohomology, $H^{*}(X)$, of an algebraic variety $X$, in his papers Hodge II and Hodge III. Really the question below is rather vague, and is ...
user avatar
5 votes
2 answers
538 views

Tricks for getting a creative idea [closed]

Caveat: I fear that people will criticize me for asking this potentially inappropriate question here, but I guess that the community here is quite unique in the ability of potentially answering my ...
user avatar
18 votes
3 answers
4k views

What does the torsion-free condition for a connection mean in terms of its horizontal bundle?

I must have read and re-read introductory differential geometry texts ten times over the past few years, but the "torsion free" condition remains completely unintuitive to me. The aim of this ...
Andrew NC's user avatar
  • 2,011
1 vote
0 answers
164 views

Imagining linear maps between finite fields

I can't imagine a right picture of a linear transformation $\mathbb{F}_{p} \mapsto \mathbb{F}_p$ or $\mathbb{F}_{p^2} \rightarrow \mathbb{F}_{p^2}$ etc (over the field $\mathbb{F}_p$) although they ...
katana_0's user avatar
  • 353
12 votes
0 answers
524 views

Additive and multiplicative convolution deeply related in modular forms

From the fact spaces of modular forms are finite dimensional, from the decomposition in Hecke eigenforms, and from the duplication formula for $\Gamma(s)$ there are a lot of identities mixing additive ...
reuns's user avatar
  • 3,405
33 votes
4 answers
4k views

How to visualize the Riemann-Roch theorem from complex analysis or geometric topology considerations?

As the question title asks for, how do others visualize the Riemann-Roch theorem with complex analysis or geometric topology considerations? That is all Riemann would have had back in the day, and he ...
Squid with Black Bean Sauce's user avatar
5 votes
1 answer
715 views

Meaningful interpretation for fixed point of a probability generating function

Suppose $f$ is the probability generating function for the Galton-Watson branching process. What intuition makes the fact that $f(s) = s$ is the probability of extinction obvious? Moreover, can one ...
libby's user avatar
  • 151
2 votes
1 answer
236 views

volume of parallelotope in $L^2(\mathbb R).$ [closed]

Let $L^2(\mathbb R)$ is complex Hilbert space with standard inner product. Does it make sense to talk of volume of parallelotope formed by following vectors in $L^2(\mathbb R):$ say, e.g., $$\{ f(...
Math Learner 's user avatar
10 votes
0 answers
245 views

Best proof of Artin approximation?

I'm trying to learn deformation theory, where the algebraic Artin approximation theorem is crucial. However, the proofs I've seen* seem to go like: Keep reducing the theorem until one is in a ...
Pulcinella's user avatar
  • 5,506
4 votes
2 answers
577 views

Orthogonal Polynomials and Sturm Liouville operators

Classical Orthogonal polynomials (e.g., Hermite, Legendre) are eigenfunctions of Sturm Liouville operators. For example, define $L[u]=u''-xu'$, then the $n$-th order Hermite polynomial satisfies $...
Amir Sagiv's user avatar
  • 3,544
3 votes
1 answer
265 views

If $A$ is a (shifted) Poisson algebra, what does $A[\varepsilon]$ represent?

I have a question which is not really precise, unfortunately. Let $A$ be a Poisson $n$-algebra, i.e. a graded commutative algebra with a Lie bracket of degree $n-1$ s.t. the bracket is a biderivation ...
Najib Idrissi's user avatar
6 votes
0 answers
209 views

"A typical pair of two-dimensional surfaces in four dimensions will intersect at finitely many points"

The title quotes C.H. Taubes in the Princeton Companion to Mathematics (p.404). I found this surprising despite the natural lower-dimensional analog (a typical pair of loops in $\mathbb{R}^2$ will ...
Joseph O'Rourke's user avatar
9 votes
0 answers
613 views

Why are the open and closed adic discs defined the way that they are?

The closed adic disc is defined as $Spa(\mathbb{Q}_p\langle T\rangle,\mathbb{Z}_p\langle T\rangle)$, and the open adic disc is defined to be the fiber $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ ...
Andrew NC's user avatar
  • 2,011
2 votes
1 answer
1k views

Intuition for coercive functions

I have been working with $\Gamma$-convergence for some time now; it has lead me to wonder: What is the intuition behind coercive functions?
ABIM's user avatar
  • 5,019
17 votes
1 answer
1k views

How to visualize a Witt vector?

As the question title asks for, how do others "visualize" Witt vectors? I just think of them as algebraic creatures. Bonus points for pictures.
Squid with Black Bean Sauce's user avatar
1 vote
2 answers
1k views

Understanding reduced suspension of $S^1$ [closed]

I know this is just $S^2$. To see it, I use the CW structure of $S^1$ x $S^1$ , consisting of one 0-cell, two 1-cells and a 2-cell. Then since the reduced suspension is the cartesian product ...
vanmeri's user avatar
  • 61
8 votes
3 answers
1k views

Intuition behind the canonical projective resolution of a quiver representation

Let $Q$ be a finite acyclic quiver, and $X$ some representation of $Q$. For $i \in Q_0$ define the $kQ$-modules $P_i = kQe_i$, and $X(i) = e_i X$. The representation $X$ has a canonical projective ...
Mike Pierce's user avatar
  • 1,149
1 vote
0 answers
155 views

Should the power series solution to $y' = y, y(0) = 1$ be obvious? [closed]

My Understanding: I would derive the Poisson random variable as follows: I consider an experiment which consists of a continuum of trials on an interval $[0,t)$. The result of the experiment takes the ...
jaslibra's user avatar
  • 111
5 votes
1 answer
519 views

Few questions regarding Heath-Brown's identity

Heath-Brown's identity states: Let $K \geq 1, z \geq 1.$ Then for any $n < 2 z^K$ we have $$ \Lambda(n) = - \sum_{1 \leq k \leq K} (-1)^k {{K}\choose{k}} \sum_{ \substack{ m_1 \cdots m_k n_1 \cdots ...
Johnny T.'s user avatar
  • 3,547
7 votes
5 answers
2k views

What are Lie groupoids intuitively?

I am trying to understand about Lie groupoids but not able to get feeling for what it actually is. So, question here is, What are Lie groupoids? How similar are they to Lie groups, Groupoids and ...
Praphulla Koushik's user avatar
2 votes
0 answers
62 views

What is the intuition of lower global bases?

In the paper: Crystallizing the Q-analogue of Universal Enveloping Algebras, Kashiwara introduced the upper global bases. In the paper: https://projecteuclid.org/euclid.dmj/1077295931, Kashiwara ...
Jianrong Li's user avatar
  • 6,101
7 votes
1 answer
204 views

The Giraud-Benabou construction for splitting fibrations

I'm currently reading "Revisiting the categorical interpretation of dependent type theory" and they give a very terse description of the Giraud-Benabou construction: For a fibration $p : \mathbb E \...
Daniel Satanove's user avatar
22 votes
1 answer
2k views

Reasons behind assuming the existence of Siegel zeros can be used to prove something stronger than assuming GRH?

There are few results that I am aware of where one can prove something stronger by assuming the existence of Siegel zeros than by assuming the GRH. For example Heath-Brown proved the existence of ...
Johnny T.'s user avatar
  • 3,547
21 votes
2 answers
4k views

intuition for hochschild homology

According to this post Intuition for group homology, I wonder what is the intuition for Hochschild homology. The Hochschild homology is defined as the homology of this complex chain. Given a ...
Zbigniew's user avatar
  • 416
16 votes
2 answers
932 views

Is there a simple system that has $\text{SU}(3)$ symmetry?

The buckle at the end of a belt has $\text{SU}(2)$ symmetry, if the rotations around the three coordinate axes are taken as generators. See, for example, the paper by Hart, Francis and Kauffman, ...
Francesco's user avatar
  • 161
23 votes
1 answer
3k views

Geometric intuition for Fontaine-Wintenberger?

I asked my advisor the question in the title. He told me it was a stupid question and that I should focus on my research. Thus we're asking here. The statement of Fontaine-Winterberger, per their ...
Student's user avatar
  • 273

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