**29**

votes

**2**answers

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

**4**answers

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

**4**answers

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 ...

**18**

votes

**2**answers

3k 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 ...

**67**

votes

**8**answers

12k 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 \end{...

**34**

votes

**2**answers

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

**1**answer

453 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 (vague)...

**1**

vote

**2**answers

1k 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

**1**answer

890 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

**0**answers

445 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 ...

**6**

votes

**4**answers

916 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

**1**answer

2k 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

**1**answer

623 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

**2**answers

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 $K$...

**14**

votes

**7**answers

4k 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 ...

**29**

votes

**6**answers

6k 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 ...

**50**

votes

**5**answers

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 ...

**32**

votes

**2**answers

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 c'}F(c,c'...

**34**

votes

**7**answers

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 ...

**4**

votes

**0**answers

447 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

**2**answers

588 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 ...

**25**

votes

**5**answers

6k views

### 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 ...

**24**

votes

**3**answers

2k 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 stack-...

**0**

votes

**0**answers

226 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 ...

**25**

votes

**1**answer

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 claim?...

**6**

votes

**4**answers

890 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

**4**answers

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 $D$ ...

**36**

votes

**8**answers

6k 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 ...

**29**

votes

**13**answers

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

**4**answers

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 ...

**66**

votes

**34**answers

11k views

### 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 ...

**5**

votes

**2**answers

694 views

### 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).
$[...

**61**

votes

**9**answers

8k 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 ...

**13**

votes

**3**answers

2k 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 ...

**16**

votes

**1**answer

2k 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 ...

**0**

votes

**2**answers

105 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 ...

**15**

votes

**1**answer

1k 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 ...

**32**

votes

**14**answers

8k 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 ...

**50**

votes

**11**answers

4k 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 ...

**9**

votes

**1**answer

1k 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 ...

**53**

votes

**11**answers

6k 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/...

**5**

votes

**3**answers

774 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 ...

**58**

votes

**25**answers

5k 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 ...

**6**

votes

**2**answers

589 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, ...

**8**

votes

**3**answers

1k 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 ...

**19**

votes

**5**answers

4k 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 ...

**9**

votes

**3**answers

1k 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 ...

**4**

votes

**3**answers

2k 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 $\...

**5**

votes

**2**answers

708 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 ...

**5**

votes

**1**answer

468 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" ...