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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9
votes
3answers
421 views

What is the intuition behind the definition of cuspidal representations?

Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the ...
2
votes
2answers
159 views

Intuition on Lindeberg condition

I want to know how Lindeberg came up with the condition which is sufficient for CLT to hold ? What is the intuition behind such an expression ?
20
votes
3answers
720 views

“Paradoxes” in $\mathbb{R}^n$

One may think of this question as a duplicate of this one. I see it more like an extension. The "inscribed sphere paradox" discussed in the aforementioned question states that if you inscribe a ...
6
votes
1answer
358 views

Getting the story of Dynkin and Satake diagrams straight

I've been trying to teach myself the theory of Lie groups. The sources I've been reading reference Lie algebras in the context of Dynkin and Satake diagrams, but not Lie groups (which I am more ...
16
votes
1answer
387 views

What motivated Rademacher's contour along the Ford circles?

Apologies if this question isn't suitable for MathOverflow; I posted it on MSE here but it didn't get a response and it felt like it was on the cusp of being suitable for here. After Ramanujan and ...
28
votes
5answers
728 views

are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?

I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary ...
5
votes
2answers
257 views

How does Tate cohomology fit into a derived categories framework?

I've read through one class field theory text after another, but there's something very non-intuitive for me about cohomology that makes it hard for me to understand why Tate cohomology was invented. ...
4
votes
1answer
277 views

Intuition of Kolmogorov-Sinai entropy

For a measurable entropy of measurable transformation $T$ from $(X,\mathcal{B},m)$ to itself. For each finite measurable partition $\mathcal{A}=\{A_i\}_{i=1}^{m}$ of $X$, we can define ...
3
votes
3answers
528 views

Mock Theta Functions

I am studying about Mock modular forms and Mock theta functions. I wonder how Zwegers connected mock theta functions with Harmonic Maass Forms? I mean, what was the philosophy/idea of Mock Theta ...
5
votes
0answers
119 views

What is the analogy between the Hilbert function and L-functions?

In his book Projective Varieties and Modular Forms, M. Eichler uses the notation $L(\lambda, M)$ for the Hilbert function of a finite graded $R=k[x_0, \dots, x_n]$-module $M$. So, $L(\lambda, M) = ...
1
vote
1answer
119 views

Explaining a comment: Difference between a transformation of points and a transformation of coordinates

In a comment to the top answer of this question Darij Grinberg says that the problem with the dynamical perspective is that it is way harder to grasp for algebraic/combinatorial-minded people ...
3
votes
2answers
466 views

Visualization of the real projective plane [closed]

Consider a closed (compact and without boundary) and non-orientable 2-manifold $M$. By Whitney embedding theorem, one can embed $M$ in $\mathbb{R}^4$. $M$ cannot be embeded in $\mathbb{R}^3$ and just ...
21
votes
7answers
838 views

Intuition for failure of Implicit Function theorem on Frechet Manifolds

When dealing with moduli spaces of, say connections or metrics, I am using the notions of Frechet spaces/manifolds/groups. I have become familiar with Banach manifolds (I think), but Frechet manifolds ...
4
votes
0answers
135 views

What is the meaning of the cospecialization map?

This question comes from the same place as my other one. In reading SGA 4 1/2, but not SGA4 itself (at least, not the obvious sections xv + xvi), one can learn about the "cospecialization morphisms" ...
0
votes
1answer
197 views

Generalization of join of simplicial complexes

The join of two abstract simplicial complexes $K$ and $L$, denoted $K\star L$ is defined as a simplicial complex on the base set $V(K)\dot{\cup} V(L)$ whose simplices are disjoint union of simplices ...
2
votes
2answers
301 views

Schönhage–Strassen algorithm

After brief intro to Fourier series, CFT, DFT and their basic properties I enjoyed implementing forward and backward FFT algorithm in complex numbers. I was happy to, at least, have an idea how is it ...
8
votes
2answers
622 views

Intuition behind the spectral density of random matrices

Hi, I have read that the spectral density of an NxN random matrix consisting of iid random variables with zero mean and unit variance converges as N goes to infinity to the uniform distribution on ...
7
votes
4answers
749 views

Coboundaries and Gluing in Cech Cohomology - Intuition?

I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...
13
votes
6answers
799 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
270 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
288 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 ...
16
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
253 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 ...
7
votes
9answers
981 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 ...
5
votes
1answer
842 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
572 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
468 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
176 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
265 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
212 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
420 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
661 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 ...
159
votes
7answers
78k 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
439 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 ...
8
votes
0answers
371 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
336 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
320 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 ...
16
votes
1answer
616 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 ...
22
votes
2answers
1k 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 ...
12
votes
4answers
1k 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
829 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 ...
47
votes
8answers
7k 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 ...
28
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 ...
6
votes
0answers
281 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
699 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
795 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
323 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
512 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, ...