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 ...

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14
votes
6answers
1k views

Understanding Adjointness of Sheaves in Algebraic Geometry

Pushforward and pullback are very basic operations in algebraic geometry, as is the adjointness between them. I worked out a very careful of adjointness of sheaves (below) when I was working out of ...
6
votes
2answers
374 views

Splitting of the weight filtration

All varieties are over $\mathbb{C}$. Notions related to weights etc. refer to mixed Hodge structures (say rational, but I would be grateful if the experts would point out any differences in the real ...
4
votes
1answer
346 views

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 ...
18
votes
3answers
1k views

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 ...
7
votes
1answer
269 views

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 ...
8
votes
9answers
1k views

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 ...
6
votes
1answer
1k views

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 ...
3
votes
2answers
914 views

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 ...
7
votes
2answers
559 views

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 ...
2
votes
0answers
196 views

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 ...
5
votes
1answer
280 views

What properties should a transform have to deserve the descriptor Fourier?

Two MO questions, "Heuristic behind the Fourier-Mukai transform" and "Explaining Mukai-Fourier transforms physically," compel me to ask these two related questions: 1) What properties do you feel are ...
2
votes
1answer
242 views

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 ...
3
votes
1answer
468 views

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 ...
6
votes
2answers
719 views

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 ...
191
votes
7answers
95k views

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 ...
16
votes
4answers
1k views

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 ...
1
vote
0answers
465 views

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 ...
11
votes
0answers
442 views

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 ...
11
votes
0answers
408 views

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 ...
8
votes
2answers
381 views

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 ...
17
votes
1answer
728 views

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 ...
28
votes
2answers
2k views

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 ...
13
votes
4answers
2k views

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 ...
15
votes
4answers
1k views

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 ...
16
votes
2answers
2k views

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 ...
59
votes
8answers
10k views

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 $\frac{1}{ad - bc} \begin{pmatrix} d & -b \\\ -c & a ...
32
votes
2answers
2k views

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 an ...
8
votes
1answer
422 views

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 ...
1
vote
2answers
981 views

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. ...
13
votes
1answer
858 views

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 ...
3
votes
0answers
394 views

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 ...
4
votes
4answers
727 views

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, ...
5
votes
1answer
1k views

When are technical assumptions critical? [closed]

Apart from their technical statement and proof, a usual presentation of theorems is by leading up to them with a definite motivation or intuition, for example putting the results in the wider context ...
5
votes
1answer
580 views

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 ...
13
votes
2answers
3k views

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 ...
8
votes
6answers
3k views

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 ...
26
votes
6answers
5k views

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 ...
48
votes
5answers
4k views

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 ...
29
votes
2answers
3k views

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 ...
27
votes
7answers
3k views

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 ...
3
votes
0answers
365 views

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 ...
7
votes
2answers
540 views

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 ...
23
votes
3answers
1k views

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 ...
0
votes
0answers
217 views

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 ...
24
votes
1answer
1k views

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 ...
6
votes
4answers
844 views

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 $n \times 1$ ...
24
votes
4answers
3k views

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 $D$ ...
32
votes
8answers
5k views

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 ...
28
votes
13answers
3k views

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 ...
24
votes
4answers
2k views

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 ...