**26**

votes

**1**answer

1k views

### Does the Pfaffian have a geometric meaning?

While reviewing the proof of Gauss-Bonnet in John Lee's book, I noticed the following paragraph:
"
...In a certain sense, this might be considered a very satisfactory generalization of Gauss-Bonnet. ...

**5**

votes

**1**answer

293 views

### Compositional inversion and generating functions in algebraic geometry

The exponential generating function of the graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus zero satisfying the associativity equations of physics (the WDVV ...

**7**

votes

**2**answers

412 views

### Twist in identification with singular cohomology

Let $X$ be a smooth projective variety over $\mathbb{Q}$ and $$V = H^m(X(\mathbb{C}), \mathbb{Q} \cdot (2\pi i)^r)$$ Then I've seen people write the comparison with complex cohomology (an isomorphism ...

**6**

votes

**2**answers

599 views

### Can we define an “empirically generic” real number?

Summary: My question, in a nutshell, is how we should intuitively imagine a generic real number (as opposed to a random one), and whether we can construct numbers which empirically behave like generic ...

**7**

votes

**6**answers

465 views

### Do you have examples of such “transitive” elements?

(I've asked the same question at the MSE, so far with no answers, so I thought I'd try it here as well. If there's some clash with any site rules, please let me know and I'll abide.)
Let $A$ be a set ...

**0**

votes

**0**answers

56 views

### Why Does a quadratic phase term in BNLS causes collapse?

I've heard a couple of times that in the Biharmonic Nonlinear Schrodinger Equation,
$i\psi_z + \Delta ^2 \psi + |\psi | ^{2\sigma } \psi =0 $, $\psi (x, 0) = \psi _0 (x) \in H^2( \mathbb{R} ^d ) $
...

**6**

votes

**5**answers

636 views

### What makes a set random?

There are many results in number theory, where the existence
of some $B \subseteq \mathbb{N}$ with certain properties is proved by
a probabilistic argument employing "random sets". One such example
...

**4**

votes

**1**answer

228 views

### what characterizes a characteristic function of a probability measure in separable Hilbert spaces?

As we all know on real line $\mathbb{R}$, the following is valid
A $\mathbb{C}$-valued function $\varphi$ is a characteristic function of a probability measure on $\mathbb{R}$ if and only if $\...

**11**

votes

**2**answers

511 views

### References for particular topics related to Langlands

I have never really concentrated on Langlands, which explains my poor level of understanding of it. But I have read quite a few introductory papers related to Langlands, and to the circle of ideas ...

**11**

votes

**3**answers

733 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 $\mathbb{G}(...

**4**

votes

**2**answers

397 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 ?

**23**

votes

**3**answers

1k 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

**1**answer

601 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 ...

**20**

votes

**1**answer

724 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 ...

**30**

votes

**5**answers

1k 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 ...

**4**

votes

**2**answers

400 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.
...

**26**

votes

**3**answers

2k views

### Intuition behind the ricci flow

I hope you don't shoot me for this question.
I try to understand among other things the Ricci flow. However I have no idea of the intuition behind the definition. So my questions is:
What is the ...

**4**

votes

**1**answer

651 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
$h(\mathcal{A},...

**3**

votes

**3**answers

756 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 ...

**6**

votes

**0**answers

143 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) = \...

**2**

votes

**1**answer

145 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

**2**answers

1k 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 ...

**24**

votes

**7**answers

2k 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 ...

**19**

votes

**2**answers

3k views

### Intuition of law of iterated logarithm?

Let $X_i$ be iid random variables with $EX_i = 0$ and $Var X_i=1$ and $S_n=X_1+\cdots+X_n$. Then the law of the iterated logarithm says almost everywhere we have
$$\limsup_{n\to\infty}\frac{S_n}{\...

**4**

votes

**0**answers

209 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

**1**answer

597 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

**2**answers

736 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 ...

**9**

votes

**2**answers

1k 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 ...

**9**

votes

**4**answers

1k 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 ...

**15**

votes

**6**answers

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 ...

**7**

votes

**2**answers

438 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

**1**answer

370 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 ...

**23**

votes

**3**answers

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

**1**answer

287 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

**9**answers

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 ...

**7**

votes

**1**answer

2k 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 $\textrm{...

**3**

votes

**2**answers

1k 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

**2**answers

645 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

**0**answers

215 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 the ...

**5**

votes

**1**answer

286 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 ...

**3**

votes

**1**answer

257 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

**1**answer

485 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

**2**answers

752 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 ...

**213**

votes

**7**answers

109k 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 ...

**17**

votes

**4**answers

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

**0**answers

475 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

**0**answers

473 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 high-...

**11**

votes

**0**answers

467 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

**2**answers

401 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 $\...

**19**

votes

**1**answer

808 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 ...