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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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
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
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 ...
13
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
387 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
7answers
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
15answers
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
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 ...
45
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
637 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 ...
26
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
729 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
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
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
8k 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
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, ...
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
810 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
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 ...
4
votes
2answers
960 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 ...
34
votes
24answers
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 ...
41
votes
46answers
14k 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" ...
23
votes
15answers
4k views

What's a magical theorem in logic?

Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less ...
38
votes
24answers
6k views

The concept of Duality

I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come. ...
12
votes
3answers
1k views

Positive definite function zoo

I've asked the following question on math.stackexchange but there has been no response so I'll ask it again here: A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a ...
9
votes
2answers
1k views

What are some other uses for Ehrenfeucht-Fraïssé games?

Let $\mathfrak{A} = \langle A, \dots \rangle$ and $\mathfrak{B} = \langle B, \dots \rangle$ be structures for a signature $\mathscr{L}$. For each ordinal $\gamma$ we define a game of perfect ...
4
votes
4answers
2k views

Which conjectures only need the Grand Riemann Hypothesis to become genuine theorems?

Hello, I've been interested in number theory for several years, and as time goes by, I read more and more articles in which theorems begin with "Assume the Riemann Hypothesis holds." But up to now, I ...
8
votes
9answers
2k views

Mathematical Advice for Interested Highschool Students

This may not be a research level math question, but I believe it is still relevant to Math Overflow. What general resources exist for students in highschool who are very interested in ...
27
votes
22answers
6k views

Titles composed entirely of math symbols

I apologize for burdening MO with such a vapid, nonresearch question, but I have been curious ever since Suvrit's popular October 2010 Most memorable titles MO question if there were any ...
45
votes
13answers
6k views

Contest problems with connections to deeper mathematics.

Hi, I already posted this on math.stackexchange, but I'm also posting it here because I think that it might get more and better answers here! Hope this is okay. We all know that problems from, for ...
52
votes
12answers
4k views

Counterexamples in PDE

Let us compile a list of counterexamples in PDE, similar in spirit to the books Counterexamples in topology and Counterexamples in analysis. Eventually I plan to type up the examples with their ...
7
votes
1answer
730 views

What makes a theorem 'good'? [closed]

I have been pondering the issue of what makes a theorem noteworthy. There are many famous examples of 'outstanding' theorems, such as Roth's theorem in Diophantine approximation, Szemeredi's Theorem, ...
9
votes
2answers
2k views

What are examples of theorems which were once “valid”, then became “invalid” as standard definitions shifted?

That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in ...
8
votes
5answers
695 views

Asymptotic Methods in Combinatorics

What are good texts to acquaint oneself with standard asymptotic techniques, particularly as they relate to probabilistic combinatorics?
12
votes
4answers
1k views

What results would follow from or imply “randomness” of the primes?

This question on random versions of deterministic problems reminded me that many conditional results in number theory hold if the primes are in some sense random, and it is common knowledge that the ...
19
votes
11answers
1k views

Random versions of deterministic problems

What are the examples of situations where "randomizing" a problem (or some part of it) and analyzing it using probabilistic techniques yields some insight into its deterministic version? An example ...
16
votes
17answers
3k views

Which book would you like to see “texified”? [closed]

Let's see if we could use MO to put some pressure on certain publishers... Although it is wonderful that it has been put online, I think it would make an even greater read if "Hodge Cycles, Motives ...
10
votes
44answers
3k views

Mathematical ideas named after places [closed]

This question is quite unimportant, so feel free to close if you think it is inappropriate. I've been thinking about how mathematicians come up with names for the ideas/objects they study, and how ...
23
votes
18answers
6k views

Interesting and Accessible Topics in Graph Theory

This summer, I will be teaching an introductory course in graph theory to talented high school seniors. The intent of the course is not to establish proficiency in graph theory, per se. Rather, I hope ...
7
votes
6answers
933 views

Seemingly emergent structures in mathematics

I rather suspect that this must have come up here on MO already, but my handful of searches didn't turn up the thread, so... I'm curious about examples of mathematical structure that seems to arise ...
4
votes
1answer
536 views

Tricks of the Trade [closed]

Can you name a mathematical theorem that is simple to state and relatively simple to prove, was essential to your research or to a work you found interesting and significant, has the potential to be ...
57
votes
26answers
4k views

What would you want on a Lie theory cheat poster?

For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, ...
31
votes
12answers
3k views

Recent Applications of Mathematics

What are the recent and new applications of Mathematics in other Sciences ? Let me try to be more precise about the question: By "recent" I mean the last 15 years. By "new" I want to exclude the ...