**42**

votes

**30**answers

5k views

### Fundamental problems whose solution seems completely out of reach [closed]

In many areas of mathematics there are fundamental problems that are embarrasingly natural or simple to state, but whose solution seem so out of reach that they are barely mentioned in the literature ...

**24**

votes

**2**answers

850 views

### Properties of functors and their adjoints

I am interested in collecting in this question a list of properties a functor $F$ may have and what those properties imply for left and right adjoints, $F^L$ and $F^R$, assuming they exist. There are ...

**181**

votes

**96**answers

30k views

### Not especially famous, long-open problems which anyone can understand

Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list ...

**69**

votes

**60**answers

7k views

### Blackbox Theorems [closed]

By a blackbox theorem I mean a theorem that is often applied but whose proof is understood in detail by relatively few of those who use it. A prototypical example is the Classification of Finite ...

**2**

votes

**2**answers

498 views

### Resources for special functions, integral identities

In the past weeks, I have struggled with finding suitable tables for integral indentities for Beta functions, Chebyshev polynomials and their like.
I would like to ask for online/offline resources ...

**3**

votes

**2**answers

1k views

### Papers whose title defines a new terminology [duplicate]

To explain a new signal processing technique based on Fourier Transform, Bogert et al went on to define a new vocabulary. The new terminology was published in a paper with the title:
The Quefrency ...

**6**

votes

**1**answer

683 views

### On the concept of point in category theory

In principle, many different abstractions of the set-theoretic notion of point / element are available in the framework of categories, but they are not equally effective and, what is more interesting ...

**12**

votes

**11**answers

2k views

### Approachable French Masters

It has been my general desire for a few years to acquire the basics in other European languages for the purpose of reading some of the classics in their original language, in a similar vein to this ...

**4**

votes

**2**answers

556 views

### Neat results from algebraic graph theory

Studying algebraic graph theory, I've stumbled across a wide range of results that I found pretty stunning and also useful. I'd like to share them here and ask for your favorite result from this area?
...

**43**

votes

**8**answers

5k views

### Have we ever lost any mathematics?

The history of mathematics over the last 200 years has many occasions when the fundamental assumptions of an area have been shown to be flawed, or even wrong. Yet I cannot think of any examples where, ...

**31**

votes

**2**answers

3k views

### Open problems/questions in representation theory and around?

What are open problems in representation theory?
What are the sources (books/papers/sites) discussing this?
Any kinds of problems/questions are welcome - big/small, vague/concrete.
Some estimation ...

**39**

votes

**35**answers

7k views

### Examples of theorems with proofs that have dramatically improved over time

I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider ...

**6**

votes

**1**answer

313 views

### Polygons that are hard to guard

Given an $n$-vertex polygonal 'art gallery' $P$ in the plane, it is possible to cover the interior of $P$ by placing 'guards' at (at worst) $\lfloor n/3\rfloor$ of the vertices of $P$. That this is ...

**16**

votes

**3**answers

840 views

### Proofs that inspire and teach

I was just listening to the show "A Splendid Table" (which I'd recommend, if you're interested in food) in which they were discussing how to spot a good recipe: one which you can follow successfully ...

**11**

votes

**9**answers

1k views

### math circles video lectures for school children?

Hello,
I am from India. I find the mathoverflow amazing.
I have a question: Are there any good quality video lectures on school math topics?
There are a lot of high quality lectures available on ...

**5**

votes

**2**answers

326 views

### Do (Banach) ultrapowers carry some sort of 'elementary equivalence'?

The (model-theoretic) ultrapowers had been used for studying elementary equivalnce of first-order structures. Then, they have been adapted to Banach spaces, which are, let me say, second-order ...

**6**

votes

**4**answers

2k views

### Math blog directory [closed]

Does anyone have a list of high quality mathematics (or related) blogs. I am of course aware of Terry Tao's most excellent blog, and also of ldtopology.wordpress.com, but I am sure the complete list ...

**14**

votes

**12**answers

3k views

### Excellent uses of induction and recursion

Can you make an example of a great proof by induction or construction by recursion?
Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen ...

**6**

votes

**5**answers

1k views

### Application of polynomials with non-negative coefficients

Question 1: Are there any deeper applications (in any field of mathematics) of polynomials (with possibly more than one variable) over the real numbers whose coefficients are non-negative? So far I ...

**24**

votes

**3**answers

2k views

### Arithmetic geometry examples

(This is inspired by Algebraic geometry examples.)
I want to collect here (counter)examples in arithmetic geometry.
Curves violating the Hasse principle: The Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$. ...

**43**

votes

**28**answers

5k views

### What are some examples of ingenious, unexpected constructions?

Many famous problems in mathematics can be phrased as the quest for a specific construction. Often such constructions were sought after for centuries or even millennia and later proved impossible by ...

**17**

votes

**3**answers

777 views

### Sperner Lemma Applications

I was always fascinated with this result. Sperner's lemma is a combinatorial result which can prove some pretty strong facts, as Brouwer fixed point theorem. I know at least another application of ...

**4**

votes

**5**answers

2k views

### Easy and Hard problems in Mathematics [closed]

Modified question:
I would like to know some examples of problems in Mathematics, for pedagogical purposes, which do not involve difficult techiques to solve the problem but with a change of context ...

**5**

votes

**0**answers

544 views

### Open problems with practical outcome in a visible future ? [closed]

I believe that any non-trivial idea will sooner or later find application in real life.
However "sooner" is better than "later":)
If we look at famous open problems - e.g. Millennium Prize problems - ...

**15**

votes

**12**answers

3k views

### Mathematics and cancer research?

What are applications of mathematics in cancer research?
My answer.
Unfortunately I heard quite small about math, but I heard something about
applications of physics. And let me put this story here, ...

**32**

votes

**10**answers

2k views

### papers archives? (especially not indexed by google)

http://www.digizeitschriften.de/index.php?id=239&L=2
has many papers with free access (e.g. Inventiones Mathematicae)
but when you search with scholar.google.com it does not index this site!
Are ...

**1**

vote

**2**answers

1k views

### An undergraduate's guide to the foundational theorems of logic [closed]

How would you explain one of these theorems in the foundations of mathematics to a fellow colleague outside the field of logic (or rather to an undergraduate mathematics student) handwaving over the ...

**29**

votes

**9**answers

4k views

### Non-computational software useful to mathematicians

The MathOverflow question Open source mathematical software contains a list of programs that are useful to perform various computational tasks, such as computer algebra systems.
However, evaluating ...

**15**

votes

**7**answers

1k views

### Unexpected applications of the fact that nth degree polynomials are determined by n+1 points

I had a funny idea for proving an identity in Euclidean geometry. While it didn't end up being a very nice proof strategy in my case, I would still like to collect nice examples of where the proof ...

**13**

votes

**17**answers

2k views

### Short Course Suggestions For High School Students

I am planning to teach a course for talented high school students at a summer camp and I need suggestions for possible topics. The students usually have different backgrounds but most of them are ...

**0**

votes

**1**answer

395 views

### Weak algebraic structures

The following question can be thought as a sequel of this one.
Here I'm looking for a big list of example of weak algebraic structures: here weak means that the structure (i.e. operations) need not ...

**43**

votes

**7**answers

4k views

### The main theorems of category theory and their applications

This question first arose as I wrote an answer for the question: Is there a nice application of category theory to functional/complex/harmonic analysis?; it can also be regarded as a (hopefully) more ...

**36**

votes

**15**answers

7k views

### Is there a nice application of category theory to functional/complex/harmonic analysis?

[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.]
I've read looked at the examples in ...

**20**

votes

**11**answers

7k views

### Noteworthy achievements in and around 2010?

The goal of this question is to compile a list of noteworthy mathematical achievements from about 2010 (so somewhat but not too far in the past).
In particular, this is meant to include (but not ...

**45**

votes

**7**answers

3k views

### Are higher categories useful?

Of course, personally, I think the answer is a big Yes!
However once, a while ago, while giving a talk about higher category theory, I was asked a question about whether higher category theory was ...

**5**

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 ...

**19**

votes

**4**answers

2k views

### What information is contained in the Kazhdan-Lusztig polynomials?

The Kazhdan-Lusztig polynomials contain all kinds of representation theoretic (and other kinds of) informations.
For example the character of a simple module over a Lie algebra with Weyl group $W$ ...

**11**

votes

**4**answers

1k views

### Casual tours around proofs

(this is basically the same question, only in math, as Alessandro Cossentino asked about theoretical computer science at TCS.SE: ...

**15**

votes

**1**answer

682 views

### What are “good” examples of string manifolds?

Based on this mathoverlow question, I would like to have a similar list for the case of string manifolds. An $n$-dim. Riemannian manifold $M$ is said to be string, if the classifying map of its bundle ...

**27**

votes

**4**answers

3k views

### What are “good” examples of spin manifolds?

I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly:
What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin ...

**14**

votes

**3**answers

1k views

### Rhombus tilings with more than three directions

The point of this question is to construct a list of references on the following subject: Fix vectors $v_1$, $v_2$, ..., $v_g$ in $\mathbb{R}^2$, all lying in a half plane in that cyclic order. I am ...

**7**

votes

**2**answers

744 views

### Virtual algebraic calculation within proofs

It seems to me that the undergraduates I teach have particular difficulty with proofs that involve reasoning about algebraic calculations that arise only theoretically. Since I have in mind doing ...

**11**

votes

**3**answers

1k views

### Proofs of parity results via the Handshaking lemma

I particularly like the following strategy to prove that the number of some combinatorial objects is even: to construct a graph, in which they correspond to vertices of odd degree (=valency).
Let me ...

**12**

votes

**8**answers

3k views

### Failures that lead eventually to new mathematics [duplicate]

Possible Duplicate:
Most interesting mathematics mistake?
In the 25-centuries old history of Mathematics, there have been landmark points when a famous mathematician claimed to have proven a ...

**8**

votes

**14**answers

9k views

### Movies about mathematics/mathematicians [closed]

I would like to watch a movie about mathematics/mathematicians (english/french language is OK, italian would be the best! Both real and invented stories are OK, maybe I would prefer something based on ...

**45**

votes

**13**answers

8k views

### How has modern algebraic geometry affected other areas of math?

I have a friend who is very biased against algebraic geometry altogether. He says it's because it's about polynomials and he hates polynomials. I try to tell him about modern algebraic geometry, ...

**29**

votes

**4**answers

3k views

### What motivates modern algebraic geometry for a combinatorial/constructive algebraist?

This is, basically, me trying to generalize "Why should I care for sheaves and schemes?" into a reasonable question. Whether successfully, time will tell, but let me hope that if not the question, ...

**25**

votes

**3**answers

2k views

### In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?

In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra.
He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie ...

**8**

votes

**4**answers

876 views

### Applications of Hilbert's metric

Among the fascinating constructions in mathematics is the Hilbert metric on a bounded convex subset of ${\mathbb R}^n$.
Where, within mathematics, is it used ? I know at least a proof of the ...

**8**

votes

**3**answers

1k views

### Undecidable problems in geometry

Are there any (many) algorithmically undecidable problems in computational (combinatorial/discrete) geometry?
Update: the Wang tiles answer the question with "any". (I have somewhat overlooked to ...