# Tagged Questions

**11**

votes

**2**answers

778 views

### What justification can you give for the fact that “most ODEs do not have an explicit solution”?

What justification can you give for the fact that "most ODEs do not have an explicit solution"?

**21**

votes

**3**answers

840 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 ...

**7**

votes

**4**answers

827 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 ...

**17**

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

**9**answers

998 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 ...

**1**

vote

**2**answers

776 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. ...

**6**

votes

**1**answer

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 ...

**28**

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 ...

**43**

votes

**29**answers

9k 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 ...

**0**

votes

**2**answers

104 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 ...

**6**

votes

**2**answers

562 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, ...

**3**

votes

**1**answer

398 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" ...

**43**

votes

**5**answers

6k views

### What is sheaf cohomology intuitively?

What is sheaf cohomology intuitively?
For local systems it is ordinary cohomology with twisted coefficients. But what
if the sheaf in question is far from being constant?
Can one still understand ...

**70**

votes

**25**answers

28k views

### Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-
An engineer, a physicist, and a mathematician are discussing how to visualise four ...

**13**

votes

**2**answers

2k views

### Why are normal crossing divisors nice?

This question is going to be extremely vague.
It seems that wherever I go (especially about Grothendieck's circle of ideas) the higher-dimensional analogue of a curve minus a finite number of points ...

**14**

votes

**4**answers

4k views

### What is a symplectic form intuitively?

Hi,
to completely describe a classical mechanical system, you need to do three things:
-Specify a manifold $X$, the phase space. Intuitively this is the space of all possible states of your system.
...

**15**

votes

**1**answer

689 views

### Symmetric groups which are not quotients of Z/2Z*Z/3Z

Somehow this question made me think of instances of small exceptions in general, and I remembered the statement I heard once that $S_5,A_6,S_6,A_7,A_8,S_8$ are the only instances of ...

**50**

votes

**15**answers

6k views

### What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?

Before you close for "homework problem", please note the tags.
Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about ...

**69**

votes

**8**answers

5k views

### Why are flat morphisms “flat?”

Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason!
Is ...

**36**

votes

**12**answers

5k views

### Cures for mathematician's block (as in writer's block) [closed]

What kind of things do you find that help you get the "creative juices flowing," to use a tired cliche, when you're stuck or burnt out on a problem? I've read about some studies that suggest listening ...

**47**

votes

**52**answers

16k views

### Colloquial catchy statements encoding serious mathematics

As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as ...

**12**

votes

**15**answers

1k views

### Most helpful heuristic?

What's the most useful piece of mathematical "folk wisdom" you've encountered? I'm talking here about things that aren't theorems, or even conjectures, or even shadows of conjectures -- just broad ...