**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 ...

**13**

votes

**5**answers

3k 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 ...

**6**

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 ...

**3**

votes

**3**answers

1k 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

620 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 ...

**4**

votes

**1**answer

412 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" ...

**46**

votes

**5**answers

7k 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 ...

**13**

votes

**0**answers

317 views

### Why, and how badly, does the proof of “no percolation at the critical point in half-spaces” fail for full spaces?

The proof by Barsky et. al. that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples critical percolation in the half-space $\mathbb{H}^d$ with a ...

**7**

votes

**7**answers

1k views

### Intuition on finite homotopy groups

As I have been studying algebraic topology, something that I found puzzling was the existence of finite homotopy groups. For instance, $\pi_{4}(S^{2})\cong\pi_{5}(S^{4})\cong\mathbb{Z}/2\mathbb{Z}$. I ...

**8**

votes

**3**answers

1k views

### What is a twisted D-Module intuitively?

When I think about $\mathcal{D}$-Modules, I find it very often useful to envison them as vectorbundles endowed with a rule that decides whether a given section is flat. Or alternatively a notion of ...

**29**

votes

**6**answers

4k views

### How is representation theory used in modular/automorphic forms?

There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...

**2**

votes

**1**answer

622 views

### Taylor expansion to show that for Stratonovich stochastic calculus the chain rule takes the form of the classical one

As nobody seems to be able to give any kind of answer to that problem over there at math.stackexchange I post this question here:
How can I show with a heuristic argument based on a Taylor expansion ...

**1**

vote

**1**answer

716 views

### Hilbert Schmidt operators

I don't know much about the theory of Hilbert spaces but a research project has me working with them a little bit. In particular requiring an operator to be Hilbert-Schmidt is a recurring condition. ...

**41**

votes

**7**answers

4k views

### Demystifying the Caratheodory Approach to Measurability

Nowadays, the usual way to extend a measure on an algebra of sets to a measure on a $\sigma$-algebra, the Caratheodory approach, is by using the outer measure $m^* $ and then taking the family of all ...

**-8**

votes

**1**answer

1k views

### Parabolas everywhere!

There a books about the Pythagorean theorem, about the exponential function and even about the gamma constant. I haven't seen any decent book about parabolas yet...
Think about it:
they form the ...

**14**

votes

**3**answers

1k views

### What is a reasonable finitary analogue of the statement that harmonic functions are smooth?

In my answer to this question on MU, I suggested that the OP think about the difference between real-differentiable and complex-differentiable functions by using a sort of finitary analogue. One way ...

**12**

votes

**0**answers

695 views

### Seeing stacks in the Calculus of Functors

Recently I was told (by an algebraic geometer) that when algebraic geometers look at the Calculus of Functors, they think of stacks.
When I look at the Calculus of Functors, I see a categorification ...

**6**

votes

**5**answers

1k views

### The unprecedented success of the “intersection” operator

You might think that the title is an overstatement of a well-known fact but it is the best title I can come up with for the wonders the intersection operator does in some fields of math.
...

**0**

votes

**3**answers

1k views

### Intuitions/connections/examples for “eigen-*”

There are many concepts in mathematics that begin with the German word "eigen": eigenvector, eigenvalue, eigenspace, eigenstate, eigenfunction, eigensystem etc. (to name just the most important (?) ...

**5**

votes

**1**answer

638 views

### Intuitive “proof” or explanation of a result in Friedman's urn

Let $g, r, a, b$ be positive integers. In Friedman's urn model we have an urn with $r$ red and $g$ green balls in it. In each step we take one ball out of urn, register its color and return it to the ...

**41**

votes

**4**answers

7k views

### Zagier's one-sentence proof of Fermat's theorem.

Zagier has a very short proof ( MR1041893) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an involution of the set $S= \lbrace (x,y,z) \in ...

**2**

votes

**2**answers

431 views

### Definition of and intuition for regular subdivisions of a polytope

I'm doing a research project that involves subdividing a product of simplices. Specifically, I'm looking at theorem 2.4 from this paper:
math.sfsu.edu/federico/Articles/tropOMs.pdf
which references ...

**19**

votes

**5**answers

3k views

### Particle Physics and Representations of Groups

This question is asked from a point of complete ignorance of physics and the standard model.
Every so often I hear that particles correspond to representations of certain Lie groups. For a person ...

**21**

votes

**7**answers

5k views

### Intuition and/or visualisation of Ito integral/Ito's lemma

Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve.
See e.g. Wikipedia:Riemann sum
The Ito integral has due to the unbounded total variation ...

**4**

votes

**2**answers

603 views

### Intuition for the satellite of a functor

Occasionally in math I come across constructions or tools that are a bit convoluted. I can look at these constructions and see that they indeed perform the task they were made to do, but sometimes I ...

**29**

votes

**4**answers

2k views

### Does anyone know an intuitive proof of the Birkhoff ergodic theorem?

For many standard, well-understood theorems the proofs have been streamlined to the point where you just need to understand the proof once and you remember the general idea forever. At this point I ...

**8**

votes

**3**answers

2k views

### What part do arguments from authority play in mathematical reasoning?

In forming your answer you may choose to address any or all of the following aspects of the question:
Descriptive. What part do arguments from authority actually play in mathematical reasoning?
...

**31**

votes

**3**answers

3k views

### Why is there no Cayley's Theorem for rings?

Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a ...

**5**

votes

**4**answers

2k views

### Geometric interpretation of the fundamental groupoid

Motivation
The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy ...

**19**

votes

**2**answers

1k views

### Geometric interpretation of group rings?

For a group $G$, is there an interpretation of $\mathbb C[G]$ as functions over some noncommutative space?
If so, what does this space "look like"? What are its properties? How are they related to ...

**4**

votes

**1**answer

2k views

### Skellam distribution: Deep connection between Poisson distributions and Bessel function?

The probability mass function for the Skellam distribution for a count difference $k=n_1-n_2$ from two Poisson-distributed variables with means $\mu_1$ and $\mu_2$ is given by:
$$
f(k;\mu_1,\mu_2)= ...

**15**

votes

**10**answers

5k views

### Geometric imagination of differential forms

In order to explain to non-experts what is a vectorfield, one usually describes an assignemnt of an arrow to each point of space, and this works quite well, also when moving to manifolds (where a ...

**83**

votes

**25**answers

32k 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 ...

**7**

votes

**3**answers

1k views

### Intuition for a formula that expresses the class number of an imaginary quadratic field by counting quadratic residues

If $p$ is a prime of the form $4n+3$, the class number $h$ of $Q[\sqrt{-p}]$ can be expressed using the number $V$ of quadratic residues and $N$ nonresidues in the interval $[1,\frac{p-1}{2}]$:
If ...

**5**

votes

**2**answers

526 views

### Gaining intuition for how submodules behave

I'm studying elementary commutative algebra this semester, largely following Atiyah-MacDonald. I often find myself in a situation where I'm interested in whether some property of an R-module M is ...

**25**

votes

**7**answers

3k views

### Spectral graph theory: Interpretability of eigenvalues and -vectors

I thought "Wow!" when I learned that the eigenvector of the adjacency matrix of a cycle graph $C_n$ corresponding to the second largest eigenvalue gives the coordinates of the vertices when equally ...

**13**

votes

**2**answers

2k views

### What is the physical meaning of a Lie algebra symmetry?

The physical meaning of a Lie group symmetry is clear: for example, if you have a quantum system whose states have values in some Hilbert space $H$, then a Lie group symmetry of the system means that ...

**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 ...

**7**

votes

**2**answers

582 views

### A split short exact sequence of algebraic fundamental groups

If we have a variety, $X$, over a field, $k$, and $x$ is a geometric point of $X$, and let $\bar x$ be a geometric point of $X_{k^s} := X \times_k k^s$ above $x$ then we have the following short exact ...

**6**

votes

**4**answers

672 views

### What is the intuitive meaning of star and box in a pure type system?

The systems of the λ-cube have the axiom $\star:\square$.
I've listed a few meanings that the Curry-Howard isomorphism gives to $t : T$ below. What are the intuitive meanings of $\star$ and ...

**8**

votes

**1**answer

1k views

### why isn't the mobius band an algebraic line bundle?

When I hear the phrase "line bundle" the first thing that pops into my head is a mobius band. But this is a bad picture from an algebraic point of view since any line bundle on an affine variety is ...

**18**

votes

**6**answers

1k views

### What can you do with a compact moduli space?

So sometime ago in my math education I discovered that many mathematicians were interested in moduli problems. Not long after I got the sense that when mathematicians ran across a non compact moduli ...

**8**

votes

**4**answers

702 views

### Intuition behind existence of moduli space of stable curves

I'm not entirely sure that the title is what I'm looking for. What I'm really asking is for intuition as to why $\bar{\mathcal{M}_g}$ is the compactification of $\mathcal{M}_g$. I'm sure this is ...

**6**

votes

**1**answer

2k views

### Is there good intution of the trace map?

I have never understood the trace map,not even after reading Geometric Interpretation of Trace. The problem with many answers in the above discussion is the geometric intuition does not apply to other ...

**26**

votes

**7**answers

8k views

### What is the exterior derivative intuitively?

Hi,
actually I have several related questions, not worth opening different threads:
What is the of the exterior derivative intuitively? What is its geometric meaning?
A possible answer I know is, ...

**46**

votes

**15**answers

12k views

### Most 'unintuitive' application of the Axiom of Choice?

It is well-known that the axiom of choice is equivalent to many other assumptions, such as the well-ordering principle, Tychonoff's theorem, and the fact that every vector space has a basis. Even ...

**22**

votes

**6**answers

5k views

### What is the difference between homology and cohomology?

In intuitive terms, what is the main difference? We know that homology is essentially the number of $n$-cycles that are not $n$-boundaries in some simplicial complex $X$. This is, more or less, the ...

**83**

votes

**16**answers

10k views

### What is torsion in differential geometry intuitively?

Hi,
given a connection on the tangent space of a manifold, one can define its torsion:
$$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$
What is the geometric picture behind this ...

**18**

votes

**3**answers

2k views

### Surprising Analogue of Q

I was describing Manish Kumar's work a few weeks ago to a fellow graduate student, and she stumped me with a big-picture question I couldn't answer.
Manish Kumar proved that the commutator subgroup ...

**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.
...