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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Rigorous solution to Ricci Flow on dumbbell $S^3$
To begin a small interest in Ricci Flow and similar tools, I am starting with Hamilton's expository paper The Formation of Singularities in the Ricci Flow. This was posted in 1995, so I am wondering ...
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Intuitive pictures in characteristic p
This is a tough one, but does anyone know of any images that recall characteristic p geometry (over algebraically closed fields) in some sense? It is not enough if it is some picture that can be also ...
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1
answer
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Clarifying the link between deformation/rigidity and dual cocycles
Suppose that a type $II_{1}$ factor $M$ decomposes in two ways as a group von Neumann algebra, e.g. as $L\Gamma$ and as $L\Lambda$. The decomposition $L\Gamma$ gives rise to a comultiplication $$\...
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Examples where adding complexity made a problem simpler
I can think of a few situations in math where a problem becomes easier or an object becomes simpler when some complexity is added. Examples:
$S^n$ is never contractible, but $S^{\infty}$ is.
The ...
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1
answer
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Self-intersection and the normal bundle
Let $X/k$ be a surface nonsingular and proper over an algebraically closed field $k$. Let $C \subset X$ be a nonsingular curve. Then it is clear that the self-intersection $(C \cdot C)_X$ is $\textrm{...
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Tangent bundle and normal bundle in self-product
$\newcommand{\I}{\mathcal{I}}$ Let $X$ a variety smooth over the complex numbers. Then we know that $\Omega_{X/\mathbb{C}}$ is the (usual) pullback of the conormal sheaf $\I/\I^2$ where $\I$ the sheaf ...
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Interpreting $f^*f_*$
For a morphism of schemes $f: X\rightarrow Y$, one often considers the function $f^*f_*$ on sheaves. For example, this appears in adjunction for sheaves of $\mathcal{O}_X$-modules, the projection ...
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Schemes with isomorphic stalks
Fact: If $ X $ and $ Y $ are varieties and we have $ \mathcal{O}_{X,q} \cong \mathcal{O}_{Y,q} $ then there are neighborhoods $U$ of $p$ and $V$ of $q$ which are isomorphic.
I understand the ...
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Comparison of the classical Fourier transform and the Fourier-Mukai transform [closed]
This question has been revised. Skip to the question in bold.
Two MO questions, "Heuristic behind the Fourier-Mukai transform" and "Explaining Mukai-Fourier transforms physically," compel me to ask ...
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Clarification and intuition request for rationally equivalent algebraic cycles
I am having some difficulty lining up the definition and my intuition for rational equivalence of cycles. My intuition is based off of the idea that two cycles being rationally equivalent is analogous ...
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What do orbital integrals have to do with reciprocity?
Hi, this is my first question (of many). I am blogging for the Fields Medal Symposium and would like to get into the mathematics involved with our program.
In an attempt to sort through the articles ...
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Quantum mechanics basics [closed]
Hello. I'm thinking about where does the basic quantum mechanics things comes from. I mean the forms of operators and a Shroedinger equation. The more intuitive explanation is better.
To get forms of ...
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Philosophy behind Mochizuki's work on the ABC conjecture
Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
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What is the geometric object corresponding to a subalgebra in a polynomial ring
Many introductory texts on algebraic geometry set up some sort of algebra-geometry dictionary in which radical ideals correspond to varieties, and so on. I am wondering if there is a geometric way to ...
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Arguments against Reductio ad Absurdum [closed]
Could Reductio ad Absurdum not be consireded a valid proof method? Are there any compelling arguments against it, or at it's favor?
I feel like I am assuming some metamathematical hypothesis about my ...
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High-dimensional geometry: Top-down Vs. Bottom-up
There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of high-...
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What is the intuition for $\mathbb{Q}^{ab}$ having cohomological dimension $1$?
I frequently talk to people who think of finite fields as arithmetic analogs of punctured discs. This makes some sense: the absolute Galois group of a finite field is the profinite completion of $\...
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What is the physical difference between states and unital completely positive maps?
Mathematically, completely positive maps on C*-algebras generalize positive linear functionals in that every positive linear functional on a C*-algebra $A$ is a completely positive map of $A$ into $\...
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Why does the Section Conjecture exclude curves of genus 1?
Let $X$ be an integral proper normal curve over a (perfect) field $F$, of genus $\geq 2$. One variant of Grothendieck's "section conjecture" states that the sections $G_F \rightarrow \pi_1(X)$ of the ...
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A geometric characterization for arithmetic genus
Let $X$ be a smooth projective variety over $\mathbb{C}$. The following information is all equivalent (any of these numbers can be computed by a linear equation from any of the others):
the ...
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Other Homology Theories still Count Holes?
This may be a naive question, but since first learning homology I considered it as a tool which counts appropriate holes in your space (on top of orientation and torsion phenomena). Then I was ...
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Why are lacunary series so badly behaved?
Hi!
I just came across the Ostroski-Hadamard gap theorem, and while I can understand the proofs as well as the principle that the series $\sum_{n=0}^\infty z^{2^n}$ ought to have a singularity at ...
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Intuition behind generic points in a scheme
In a scheme, each point is a generic point of its closure. In particular each closed point is a generic point of itself (the set containing it only), but that's perhaps of little interest. A point ...
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Should the formula for the inverse of a 2x2 matrix be obvious?
As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{...
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Meaning/origin of Seiberg-Witten equations/invariants
Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from.
We take ...
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Basic examples of induction on scales arguments
An important ingredient in recent progress on Euclidean harmonic analysis has been that of "inductions on scales". A few examples are the papers of Wolff, Tao, and Bourgain and Guth.
Here is a (vague)...
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Intrinsic vs. Extrinsic [closed]
Undoubtedly, these terms play essential roles in (pure) mathematics. My problem is that I have feelings what they mean in different fields, such as, differential geometry (abstract manifolds vs. ...
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Lagrange's theorem for Hopf algebras
Under what conditions is a Hopf algebra free over any of its sub-Hopf algebras?
I am reading "Hopf algebras and their actions on rings" by Susan Montgomery, specifically chapter 3. Lagrange's theorem ...
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Geometric picture behind tilting sheaves
I am trying to read "Tilting exercises" and have trouble to see any geometric pictures behind the formulas.
So my questions are, how to think about tilting perverse sheaves?
Are they just formal ...
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What is the intuition of connections for cubical sets?
I am beggining to do some work with cubical sets and thought that I should have an understanding of various extra structures that one may put on cubical sets (for purposes of this question, ...
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What is the intuition behind the proof of the algebraic version of Cartan's theorem A?
I am trying to understand the idea behind the proof of GAGA. A crucial step is the following:
Theorem: Let $X=\mathbb{P}^r_{\mathbb{C}}$ (either as a variety or as an analytic space), and let $\...
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Why should the anabelian geometry conjectures be true?
I had probed friends of mine about Grothendieck's motivation for making the anabelian geometry conjectures, and they gave me the following explanation:
If $X$ is a hyperbolic curve over some field $K$...
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What is a complex inner product space "really"?
This is an extended re-post of a question that I have asked on MSE not a long time ago. But anyway, it seems more appropriate for MO.
To begin with, in a real inner product space we have a geometric ...
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What does Mellin inversion "really mean"?
Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...
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Is there an intuitive reason for Zariski's main theorem?
Zariski's main theorem has many guises, and so I will give you the freedom to pick the one that you find to be most intuitive. For the sake of completeness, I will put here one version:
Zariski's main ...
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Intuition for coends
Let $D$ be a co-complete category and $C$ be a small category. For a functor $F:C^{op}\times C \to D$ one defines the co-end
$$
\int^{c\in C} F(c,c)
$$
as the co-equalizer of
$$
\coprod_{c\to c'}F(c,c'...
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What is a coalgebra intuitively?
How to think about coalgebras? Are there geometric interpretations of coalgebras?
If I think of algebras and modules as spaces and vectorbundles, what are coalgebras and comodules? What basic ...
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Relationship between R-transform and free convolution of random matrices?
I've been using the R-transform to calculate the free convolution of the eigenvalue spectra of two random matrices and I am trying to understand how it works, and in particular how it relates to ...
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Intuition behind the age grading in quantum cohomology of orbifolds
Let $\mathscr{X}$ be a smooth DM-stack with projective coarse moduli space. I am interested in the orbifold cohomology ring $H^\mathrm{orb}(\mathscr{X})$, as defined by Chen-Ruan (for orbifolds) and ...
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Physicist's request for intuition on covariant derivatives and Lie derivatives
A friend of mine is studying physics, and asks the following question which, I am sure, others could respond to better:
What is the difference between the covariant derivative of $X$ along the curve ...
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Help motivating log-structures
I'm currently reading a thesis that uses log-structures. I should mention that this is my first encounter with them, and the thesis (as well as my expertise) is scheme-theoretic (in fact stack-...
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Weight filtration of MHSs
This is probably a very stupid question, but could someone explain to me where the weight filtration of mixed Hodge structures come from and why we actually need it?
If the Hodge-to-de Rham spectral ...
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Tomita-Takesaki versus Frobenius: where is the similarity?
I've often heard Alain Connes say that the modular flow of Tomita-Takesaki theory should be thought of as a characteristic zero analog of the Frobenius endomorphism. ... can anyone justify this claim?...
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How to solve Ax=b incrementally ?
Hi, everyone.
What I am struggling is the following problem. I have a linear matrix equation $Ax=b$, where $A$ is a known $n \times n$ large sparse real matrix, $x$ and $b$ are known $n \times 1$ ...
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Overview of the interplay of Harmonic Analysis and Number Theory
I'm kind of disappointed that the question here was never sharpened.
The Laplacian $\Delta$ on the upper half-plane is $-y^{2}(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}))$. Suppose $D$ ...
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What is a Lagrangian submanifold intuitively?
What are good ways to think about Lagrangian submanifolds?
Why should one care about them?
More generally: same questions about (co)isotropic ones.
Answers from a classical mechanics point of view ...
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Surprising and Useful Physical Intuition for Mathematical Objects
I believe I.M. Gelfand said that when beginning to learn a new subject, one should learn it like a physicist.
In this spirit, what are some helpful and surprising physical intuitions accompanying ...
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Algebraic P vs. NP
I recently attended a lecture where the speaker mentioned that what he was talking about was connected to the algebraic version of the $P$ vs. $NP$ problem. Could someone explain what that means in a ...
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What notions are used but not clearly defined in modern mathematics?
"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
Felix Klein
What notions are used but not ...
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What is the intuition behind the Freudenthal suspension theorem?
The Freudenthal suspension theorem states in particular that the map
$$
\pi_{n+k}(S^n)\to\pi_{n+k+1}(S^{n+1})
$$
is an isomorphism for $n\geq k+2$.
My question is: What is the intuition behind the ...