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).
$[...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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/...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 ...
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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" ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 (?) ...
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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 ...
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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) \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
...
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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 ...
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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 ...
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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 ...
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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)= ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 $...
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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 ...