**60**

votes

**32**answers

10k 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

684 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).
...

**60**

votes

**9**answers

7k 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 ...

**6**

votes

**3**answers

1k 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 ...

**15**

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

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 ...

**15**

votes

**1**answer

850 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 ...

**27**

votes

**13**answers

6k 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 ...

**47**

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 ...

**50**

votes

**11**answers

5k 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" ...

**5**

votes

**3**answers

747 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 ...

**54**

votes

**26**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

576 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 ...

**18**

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

662 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

434 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" ...

**53**

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

336 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 ...

**8**

votes

**7**answers

2k 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 ...

**31**

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

636 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

734 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. ...

**40**

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 ...

**15**

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

714 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

663 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 ...

**43**

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

489 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 ...

**20**

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 ...

**26**

votes

**9**answers

6k 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

632 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 ...

**30**

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?
...

**32**

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 ...

**5**

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)= ...

**25**

votes

**13**answers

5k views

### Geometric imagination of differential forms

In order to explain to non-experts what a vector field is, one usually describes an assignment of an arrow to each point of space. And this works quite well also when moving to manifolds, where a ...

**92**

votes

**25**answers

38k 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

564 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 ...