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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6
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0answers
133 views

Geometric meaning of Todd classes for non-tangent bundles

While Todd classes are defined for arbitrary vector bundles, the only cases I actually see in practical use are Todd classes of tangent bundles (as in Hirzebruch-Riemann-Roch or Grothendieck-Riemann-...
10
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0answers
268 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 ...
36
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7answers
3k 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\...
5
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0answers
169 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 ...
10
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0answers
253 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I'm trying to wrap my head around some of Harvey Friedman's recent, unpublished work on his tangible incompleteness project, and I'm trying to see the link between his "tangible statements" (...
5
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0answers
78 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 ...
7
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1answer
548 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{...
4
votes
1answer
259 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 ...
2
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1answer
278 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 ...
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0answers
34 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 ...
-2
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1answer
248 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 ...
43
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1answer
932 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 _{...
2
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2answers
220 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 ...
3
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0answers
374 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 ...
5
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2answers
390 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 ...
15
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3answers
811 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 ...
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0answers
65 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 ...
9
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0answers
294 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 ...
30
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4answers
2k 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 ...
2
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1answer
212 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 ...
2
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1answer
117 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(...
10
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0answers
181 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 ...
4
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2answers
265 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 $...
3
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1answer
205 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 ...
6
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0answers
199 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 ...
8
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0answers
322 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 $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ ...
1
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1answer
456 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?
14
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1answer
799 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.
1
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2answers
422 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 ...
6
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3answers
623 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 ...
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0answers
153 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 ...
3
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1answer
242 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 ...
4
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4answers
847 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 ...
6
votes
1answer
98 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 \...
18
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1answer
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 ...
18
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2answers
2k 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 ...
15
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2answers
618 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, ...
20
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1answer
2k 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 ...
2
votes
3answers
167 views

The independence of all sub-paths between adjacent returns to origin in random walks

In random walks, a path may return to origin for the $r$-th time in $n$-th step ($r$ is given but $n$ is not given), and under this condition, these $n$ steps can be split into $r$ sections where each ...
2
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0answers
142 views

Right adjoint completions

Forgive me if this question is not well thought out. I don't know how else to ask it. The nlab page on completion gives some examples of completions which are left adjoints. These completions are "...
6
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0answers
404 views

Grothendieck construction and coends

In category theory, both the Grothendieck construction and coends are represented by a sort of "integral sign", respectively: $$ \int F $$ for a functor $F:C\to\mathbf{Cat}$, and: $$ \int^x G(x,x) $$ ...
24
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2answers
1k views

Intuition for symplectic groups

My question essentially breaks down to How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you ...
10
votes
3answers
1k views

Why linearization leads to arithmetization?

Sorry for this question, but I think it is really important the intuition here. Motives can be seen as the 'best' way of linearizing the study of schemes, des-composing them into "cohomological atoms"...
1
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2answers
326 views

Non-trivial examples of taking the exponential of an integral

This question is inspired by a recent course I did on random matrix theory and also from common mistakes high-schoolers make in algebra :). In random matrix theory, one often encounters somewhat ...
10
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0answers
480 views

Mumford's intuition for flatness

In Mumford's book Algebraic Geometry II, he writes on page 179..."In order to get at what I consider the intuitive content of "flat" we need first a deeper fact..." After the deeper fact is proven he ...
14
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3answers
2k views

Understanding Vaughan's Identity

Vaughan's identity https://proofwiki.org/wiki/Vaughan%27s_Identity is a very useful identity in analytic number theory. The identity expresses the von-Mangoldt function $\Lambda(n)$ as a sum of ...
10
votes
1answer
1k views

Intuition behind the Dehn Invariant

EDIT: as pointed-out below, this has been posted on math.stackexchange. I'll leave it up to the community whether or not to delete this question, but I do think there is room for a more technical ...
3
votes
0answers
224 views

Intuition behind results in Mumford's “Lectures on curves on an algebraic surface”, II

NOTE: This is a followup to my question here. These are some questions concerning Mumford's "Lectures on curves on an algebraic surface". We concern ourselves with questions of the Picard variety $P$...
6
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1answer
756 views

Intuition behind results in Mumford's “Lectures on curves on an algebraic surface”, I

These are some questions concerning Mumford's "Lectures on curves on an algebraic surface". We concern ourselves with questions of the Picard variety $P$, and its dimension, of a complete nonsingular ...
3
votes
0answers
289 views

Intuition behind Artin/Grothendieck's Riemann existence theorem

The following result is due to Grothendieck and Artin and can be found in SGA 1. Let $X$ be a normal scheme of finite type over $\mathbb{C}$. Let $\mathfrak{X}'$ be a normal complex analytic space, ...

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