**201**

votes

**22**answers

23k views

### Thinking and Explaining

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words ...

**82**

votes

**38**answers

15k views

### Examples of eventual counterexamples

Define an "eventual counterexample" to be
$P(a) = T $ for $a < n$
$P(n) = F$
$n$ is sufficiently large for $P(n) = T\ \ \forall n \in \mathbb{N}$ to be a 'reasonable' conjecture to make.
where ...

**116**

votes

**130**answers

27k views

### Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)
I'd love to learn about ...

**71**

votes

**16**answers

17k views

### What recent discoveries have amateur mathematicians made?

E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what ...

**50**

votes

**11**answers

3k views

### Why is Set, and not Rel, so ubiquitous in mathematics?

The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations.
Why was there the necessity of singling ...

**26**

votes

**2**answers

2k views

### How to unify various reconstruction theorems (Gabriel-Rosenberg, Tannaka,Balmers)

What I am talking about are reconstruction theorems for commutative scheme and group from category. Let me elaborate a bit. (I am not an expert, if I made mistake, feel free to correct me)
...

**28**

votes

**3**answers

1k views

### “Softness” vs “rigidity” in Geometry

According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually, geometries${}^*$ ...

**19**

votes

**6**answers

3k views

### Heuristic behind the Fourier-Mukai transform

What is the heuristic idea behind the Fourier-Mukai transform? What is the connection to the classical Fourier transform?
Moreover, could someone recommend a concise introduction to the subject?

**45**

votes

**14**answers

4k views

### What is the high-concept explanation on why real numbers are useful in number theory?

The utopian situation in mathematics would be that the statement and the proof of every result would live "in the same world", at the same level of mathematical complexity (in a broad sense), unless ...

**16**

votes

**5**answers

1k views

### Why are finiteness conditions important (and how to recognize them)?

I think everybody here has met lots of finiteness conditions, like those requiring a vector space to be finite dimensional, an abelian group to be finitely generated, a ring to be Noetherian, a ...

**2**

votes

**1**answer

418 views

### Is there a well defined subset of the integers that cannot be defined as a property of a recursive process or Turing Machine?

Is there a well defined subset of the integers that cannot be defined as a property of a recursive process or Turing Machine?
I have long been intrigued by the observation that much of mathematics ...

**102**

votes

**61**answers

19k views

### Your favorite surprising connections in Mathematics

There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the ...

**50**

votes

**16**answers

8k views

### Why is it a good idea to study a ring by studying its modules?

This is related to another question of mine. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules ...

**66**

votes

**11**answers

12k views

### What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...

**65**

votes

**6**answers

8k views

### How to find ICM talks?

I am very interested in reading some and skimming through the list of invited talks at the International Congress of Mathematicians. Since the proceedings contain talks supposedly by top experts in ...

**44**

votes

**11**answers

5k views

### What precisely Is “Categorification”?

(And what's it good for.)
Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.

**35**

votes

**8**answers

5k views

### Sheaf cohomology and injective resolutions

In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...

**49**

votes

**12**answers

9k views

### Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic.
The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...

**53**

votes

**16**answers

5k views

### What makes four dimensions special?

Do you know properties which distinguish four-dimensional spaces among the others?
What makes four-dimensional topological manifolds special?
What makes four-dimensional differentiable manifolds ...

**43**

votes

**12**answers

5k views

### Is there a high-concept explanation for why characteristic 2 is special?

The structure of the multiplicative groups of Z/pZ or of Zp is the same for odd primes, but not for 2. Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for 2. So ...

**29**

votes

**21**answers

6k views

### What are some slogans that express mathematical tricks?

Many "tricks" that we use to solve mathematical problems don't correspond nicely to theorems or lemmas, or anything close to that rigorous. Instead they take the form of analogies, or general methods ...

**30**

votes

**9**answers

4k views

### Collection of equivalent forms of Riemann Hypothesis

This forum brings together a broad enough base of mathematicians to collect a "big list" of equivalent forms of the Riemann Hypothesis...just for fun. Also, perhaps, this collection could include ...

**24**

votes

**1**answer

1k views

### Are there any “homotopical spaces” ?

This is a somewhat vague question; I don't know how "soft" it is, and even if it makes sense.
[Edit: in the light of the comments, we can state my question in a formally precise way, that is: "Is ...

**9**

votes

**2**answers

1k views

### Why does homotopy behave well with respect to fibrations and homology with respect to cofibrations?

(I apologize that this is a vague question).
I seems to me somehow that homotopy groups behave well with respect to (Serre)-fibrations. For example you get a long exact sequence of homotopy groups ...

**19**

votes

**6**answers

3k views

### Why the triangle inequality?

[Maybe this is asking to be closed; but I thought I'd risk it.]
A metric satisfies the axioms:
$d(x,y)=0$ if and only if $x=y$.
$d(x,y) = d(y,x)$.
$d(x,y) \leq d(x,z) + d(z,y)$.
Similarly (and ...

**12**

votes

**2**answers

1k views

### Using higher-order Bring radicals to solve arbitrary polynomials

It is well known that there is no general formula for the solution of the quintic. Of course, what this really means is that there is no general formula that only involves addition, subtraction, ...

**21**

votes

**2**answers

2k views

### Elevator pitch for the Virtual Fibering Theorem

There has been a great deal of excitement among topologists about the proof of the Virtual Haken Theorem, and in fact of the Virtual Fibering Theorem (for closed hyperbolic 3-manifolds, but I'm ...

**12**

votes

**2**answers

757 views

### What is the status of (universal) algebra in type theory?

With the recent interest in homotopy type theory as a foundation for mathematics, it seems natural to develop algebra within the framework of type theory. So far, I can't find much literature ...

**10**

votes

**4**answers

4k views

### Is the ABC conjecture known to imply the Riemann hypothesis?

I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...

**7**

votes

**2**answers

1k views

### Duality of eta product identities: a new idea?

Looking at the collection of Eta Function Product Identities by Michael Somos, it seems like generally those identities come in pairs:
let's call two eta product identities $\sum\limits_{i=1}^r ...

**6**

votes

**1**answer

242 views

### Under which constraints are there only finite numbers of irreducible eta product identities?

For the Dedekind eta function, defined as usual by $\eta(q) = q^{\frac1{24}} \prod\limits_{n=1}^{\infty} (1-q^{n})$, let for brevity $e_k:=\eta(q^k)$.
An eta product identity (or eta identity for ...