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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Geometric interpretation of $BN$-pairs

My question is relative to a geometric interpretation of the $BN$-pairs that arise in Tits' theory of buildings. Here is a definition that comes from an article by G. Stroth (Nonspherical spheres). $[...
Thomas Connor's user avatar
86 votes
9 answers
13k views

Why should I believe the Mordell Conjecture?

It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points. I am interested to know why Mordell and ...
Barinder Banwait's user avatar
17 votes
3 answers
3k 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 ...
Michał Oszmaniec's user avatar
19 votes
1 answer
4k 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 ...
shuhalo's user avatar
  • 4,736
1 vote
2 answers
118 views

Removing a hypersurface when applying the Representation theorem to prove Positivstellensatz with uniform denominators

Let $f$ and $g$ be positive definite forms in the polynomial ring ${\mathbb{R}}[x_0,\ldots, x_n]$ such that $\deg(g)$ divides $\deg(f)$. A generalization of a theorem by Reznick is that $g^N f$ is a ...
user avatar
17 votes
1 answer
2k views

A synopsis of Adyan’s solution to the general Burnside problem?

Where can I find a high-level overview of Adyan’s original proof of the existence of finitely generated infinite groups with finite exponent? Additionally: If possible, would an expert please ...
Jon Bannon's user avatar
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56 votes
16 answers
14k 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 ...
71 votes
11 answers
8k views

How to introduce notions of flat, projective and free modules?

In the coming spring semester I will be teaching for the first time an introductory (graduate) course in Commutative Algebra. As many people know, I have been plugging away for a while at this ...
Pete L. Clark's user avatar
12 votes
1 answer
2k views

Wick rotation and the Riemann zeta function

The goal of this question is to conceptualize in some way the fact that the Riemann zeta function $\zeta(s)$, and other zeta functions like it, have analytic continuations. Background I have by now ...
Qiaochu Yuan's user avatar
67 votes
11 answers
10k 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" (sequences/...
6 votes
3 answers
1k views

Intuition for rational functions

I asked this on mathematics stack exchange and did not receive answer . I hope it is good manners to ask here. Thank you very much. Let $X$ be integral scheme and $\mathcal K$ sheaf of rationnal ...
evgeniamerkulova's user avatar
83 votes
24 answers
8k 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 ...
7 votes
2 answers
727 views

Would a supersymmetric theory of von Neumann algebras be useful?

While looking over the first chapter of 1) Quantum Fields and Strings: A Course For Mathematicians (P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and E. Witten, ...
Jon Bannon's user avatar
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12 votes
3 answers
3k views

Singularities of pairs

In the next days I have to give a talk in which I need to explain some of the usual singularities of pairs that one meets when dealing with the minimal model program: KLT, DLT and LC pairs. In ...
Gianni Bello's user avatar
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34 votes
5 answers
9k views

Intuition behind Alexander duality

I was wondering if anyone could offer some intuition for why Alexander duality holds. Of course, the proof is easy enough to check, and it is also easy to work out many examples by hand. However, I ...
Aaron S's user avatar
  • 351
12 votes
3 answers
3k views

The localization of a regular local ring is regular

I've heard, as I'm sure many have, that the theorem that the localization of a regular local ring at any prime ideal is regular is one of the first major applications of homological methods to pure ...
James D. Taylor's user avatar
5 votes
3 answers
3k views

Intuition behind the notion of distance between curves

Let $(M,g)$ be a Riemannian manifold and let $p$ and $q$ be two points on it and define $d(p,q)$ as the length of the minimizing geodesic between them. Now given two rectifiable paths $\gamma_1$ and $\...
Anirbit's user avatar
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10 votes
2 answers
1k views

Canonical geometric examples

The proofs without words post has some great entries. I'm interested in a similar concept: examples where a problem in math or physics is accompanied by a geometric figure that illuminates some key ...
7 votes
1 answer
736 views

Intuition on Log-Concave Sequences

A sequence $(a_n)$ is said to be log-concave provided $a_i^2 \geq a_{i-1}a_{i+1}$ for all $i$. What sorts of intuition can one have about log-concave sequences? In particular, what kind of "picture" ...
86 votes
5 answers
18k views

What is sheaf cohomology intuitively?

What is sheaf cohomology intuitively? For local systems it is ordinary cohomology with twisted coefficients. But what if the sheaf in question is far from being constant? Can one still understand ...
Jan Weidner's user avatar
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14 votes
0 answers
561 views

Why, and how badly, does the proof of "no percolation at the critical point in half-spaces" fail for full spaces?

The proof by Barsky et. al. that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples critical percolation in the half-space $\mathbb{H}^d$ with a ...
Louigi Addario-Berry's user avatar
17 votes
7 answers
5k views

Intuition on finite homotopy groups

As I have been studying algebraic topology, something that I found puzzling was the existence of finite homotopy groups. For instance, $\pi_{4}(S^{2})\cong\pi_{5}(S^{4})\cong\mathbb{Z}/2\mathbb{Z}$. I ...
Daniel Miller's user avatar
20 votes
3 answers
4k views

What is a twisted D-Module intuitively?

When I think about $\mathcal{D}$-Modules, I find it very often useful to envison them as vectorbundles endowed with a rule that decides whether a given section is flat. Or alternatively a notion of ...
Jan Weidner's user avatar
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51 votes
6 answers
10k views

How is representation theory used in modular/automorphic forms?

There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...
David Corwin's user avatar
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2 votes
2 answers
1k views

Taylor expansion to show that for Stratonovich stochastic calculus the chain rule takes the form of the classical one

As nobody seems to be able to give any kind of answer to that problem over there at math.stackexchange I post this question here: How can I show with a heuristic argument based on a Taylor expansion ...
vonjd's user avatar
  • 5,855
2 votes
1 answer
1k views

Hilbert Schmidt operators

I don't know much about the theory of Hilbert spaces but a research project has me working with them a little bit. In particular requiring an operator to be Hilbert-Schmidt is a recurring condition. ...
solbap's user avatar
  • 3,938
85 votes
9 answers
16k 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 ...
Michael Greinecker's user avatar
17 votes
3 answers
2k views

What is a reasonable finitary analogue of the statement that harmonic functions are smooth?

In my answer to this question on MU, I suggested that the OP think about the difference between real-differentiable and complex-differentiable functions by using a sort of finitary analogue. One way ...
Qiaochu Yuan's user avatar
28 votes
2 answers
2k views

Seeing stacks in the Calculus of Functors

Recently I was told (by an algebraic geometer) that when algebraic geometers look at the Calculus of Functors, they think of stacks. When I look at the Calculus of Functors, I see a categorification ...
B. Bischof's user avatar
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7 votes
5 answers
2k views

The unprecedented success of the “intersection” operator

You might think that the title is an overstatement of a well-known fact but it is the best title I can come up with for the wonders the intersection operator does in some fields of math. Recently,(on ...
Unknown's user avatar
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0 votes
3 answers
1k views

Intuitions/connections/examples for "eigen-*"

There are many concepts in mathematics that begin with the German word "eigen": eigenvector, eigenvalue, eigenspace, eigenstate, eigenfunction, eigensystem etc. (to name just the most important (?) ...
vonjd's user avatar
  • 5,855
5 votes
1 answer
1k views

Intuitive "proof" or explanation of a result in Friedman's urn

Let $g, r, a, b$ be positive integers. In Friedman's urn model we have an urn with $r$ red and $g$ green balls in it. In each step we take one ball out of urn, register its color and return it to the ...
Jankir Dezmin's user avatar
116 votes
8 answers
32k views

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

Zagier has a very short proof (MR1041893, JSTOR) 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) \...
Keivan Karai's user avatar
  • 6,064
2 votes
2 answers
2k views

Definition of and intuition for regular subdivisions of a polytope

I'm doing a research project that involves subdividing a product of simplices. Specifically, I'm looking at theorem 2.4 from this paper: math.sfsu.edu/federico/Articles/tropOMs.pdf which references ...
Chris G's user avatar
  • 21
28 votes
5 answers
6k views

Particle Physics and Representations of Groups

This question is asked from a point of complete ignorance of physics and the standard model. Every so often I hear that particles correspond to representations of certain Lie groups. For a person ...
Makhalan Duff's user avatar
51 votes
11 answers
20k views

Intuition and/or visualisation of Itô integral/Itô'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 Itô integral has due to the unbounded total variation ...
vonjd's user avatar
  • 5,855
4 votes
2 answers
1k views

Intuition for the satellite of a functor

Occasionally in math I come across constructions or tools that are a bit convoluted. I can look at these constructions and see that they indeed perform the task they were made to do, but sometimes I ...
Eric's user avatar
  • 855
51 votes
4 answers
5k views

Does anyone know an intuitive proof of the Birkhoff ergodic theorem?

For many standard, well-understood theorems the proofs have been streamlined to the point where you just need to understand the proof once and you remember the general idea forever. At this point I ...
Paul Siegel's user avatar
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8 votes
3 answers
3k views

What part do arguments from authority play in mathematical reasoning?

In forming your answer you may choose to address any or all of the following aspects of the question: Descriptive. What part do arguments from authority actually play in mathematical reasoning? ...
61 votes
3 answers
6k views

Why is there no Cayley's Theorem for rings?

Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a ...
Tom Boardman's user avatar
  • 3,190
7 votes
4 answers
3k views

Geometric interpretation of the fundamental groupoid

Motivation The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy ...
Josh's user avatar
  • 1,402
20 votes
2 answers
2k views

Geometric interpretation of group rings?

For a group $G$, is there an interpretation of $\mathbb C[G]$ as functions over some noncommutative space? If so, what does this space "look like"? What are its properties? How are they related to ...
Jan Weidner's user avatar
  • 12.8k
8 votes
1 answer
4k views

Skellam distribution: Deep connection between Poisson distributions and Bessel function?

The probability mass function for the Skellam distribution for a count difference $k=n_1-n_2$ from two Poisson-distributed variables with means $\mu_1$ and $\mu_2$ is given by: $$ f(k;\mu_1,\mu_2)= ...
vonjd's user avatar
  • 5,855
37 votes
15 answers
12k views

Geometric imagination of differential forms

In order to explain to non-experts what a vector field is, one usually describes an assignment of an arrow to each point of space. And this works quite well also when moving to manifolds, where a ...
Mircea's user avatar
  • 2,031
249 votes
29 answers
163k 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 ...
8 votes
3 answers
2k views

Intuition for a formula that expresses the class number of an imaginary quadratic field by counting quadratic residues

If $p$ is a prime of the form $4n+3$, the class number $h$ of $Q[\sqrt{-p}]$ can be expressed using the number $V$ of quadratic residues and $N$ nonresidues in the interval $[1,\frac{p-1}{2}]$: If $...
danseetea's user avatar
  • 954
5 votes
2 answers
1k views

Gaining intuition for how submodules behave

I'm studying elementary commutative algebra this semester, largely following Atiyah-MacDonald. I often find myself in a situation where I'm interested in whether some property of an R-module M is ...
Nick Salter's user avatar
  • 2,790
39 votes
7 answers
8k 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 ...
Hans-Peter Stricker's user avatar
17 votes
2 answers
4k views

What is the physical meaning of a Lie algebra symmetry?

The physical meaning of a Lie group symmetry is clear: for example, if you have a quantum system whose states have values in some Hilbert space $H$, then a Lie group symmetry of the system means that $...
Qiaochu Yuan's user avatar
20 votes
2 answers
3k views

Why are normal crossing divisors nice?

This question is going to be extremely vague. It seems that wherever I go (especially about Grothendieck's circle of ideas) the higher-dimensional analogue of a curve minus a finite number of points ...

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