Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the ...

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134
votes
79answers
21k 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 ...
65
votes
60answers
6k 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
2answers
385 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
2answers
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
1answer
625 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 ...
11
votes
11answers
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 ...
3
votes
2answers
511 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? ...
42
votes
9answers
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, ...
39
votes
35answers
6k 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
1answer
307 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
3answers
776 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
9answers
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
2answers
309 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
4answers
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 ...
13
votes
12answers
2k 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 ...
5
votes
5answers
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 ...
23
votes
3answers
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 $3X^3 + 4Y^3 + 5Z^3 = 0$. ...
41
votes
28answers
4k 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
3answers
541 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
5answers
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
0answers
514 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 - ...
32
votes
10answers
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
2answers
970 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 ...
28
votes
9answers
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
7answers
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 ...
12
votes
17answers
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
1answer
384 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 ...
42
votes
7answers
3k 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 ...
32
votes
15answers
6k 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
11answers
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 ...
44
votes
7answers
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 ...
6
votes
1answer
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 ...
18
votes
4answers
1k 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
4answers
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: ...
14
votes
1answer
592 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 ...
23
votes
4answers
2k 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
3answers
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
2answers
692 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 ...
10
votes
3answers
881 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
8answers
2k 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
14answers
6k 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 ...
28
votes
4answers
2k 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, ...
23
votes
3answers
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
4answers
739 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
3answers
988 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 ...
3
votes
2answers
886 views

Is beauty at the high school level even possible? [closed]

This question is a follow up to 74841, and follows from a suggestion by Gian-Carlo Rota that beauty as judged by the educated public differs from that experienced by mathematicians (he gives Euclidean ...
32
votes
22answers
6k views

Examples of seemingly elementary problems that are hard to solve?

I'm looking for a list of problems such that a) any undergraduate student who took multivariable calculus and linear algebra can understand the statements, (Edit: the definition of understanding here ...
14
votes
3answers
2k views

Things to keep in mind while looking for a Postdoc overseas

Hello, I would like to receive some suggestions about what you think to be the best important things that should be kept in mind while looking for a postdoc position. I'm not considering (in this ...
40
votes
44answers
12k views

An example of a beautiful proof that would be accessible at the high school level?

The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...
18
votes
7answers
2k views

Things that should be positive integers…really?

Kronecker. Nuff said. Even the numbers themselves historically started as positive integers and were subsequently generalized to hell and back. Here are some other well known concepts that "should" ...