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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8
votes
14answers
7k 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
773 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 ...
3
votes
2answers
918 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
13k 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" ...
22
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 ...
37
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
1k 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 ...
30
votes
11answers
5k 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 ...
51
votes
12answers
3k 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
702 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
662 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
998 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
5k 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
909 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
516 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 ...
56
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 ...
21
votes
53answers
8k views

German mathematical terms like “Nullstellensatz”

There are quite a few german mathematical theorems or notions which usually are not translated into other languages. For example, Nullstellensatz, Hauptvermutung, Freiheitssatz, Eigenvector (the ...
4
votes
8answers
2k views

What's the difference between 2 and 3? [closed]

Here are two classical results which depend on whether a parameter is 2 or 3: It is possible to bisect an arbitrary angle with ruler and compass, but impossible to trisect it. While there are ...
56
votes
73answers
11k views

Elementary+Short+Useful

Imagine your-self in front of a class with very good undergraduates who plan to do mathematics (professionally) in the future. You have 30 minutes after that you do not see these students again. You ...
66
votes
18answers
6k views

Occurrences of (co)homology in other disciplines and/or nature

I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive ...
6
votes
10answers
821 views

Examples of “Unusual” Classifications

When one says "classification" in math, usually one of a handful of examples springs to mind: -Classification of Finite Simple Groups with 18 infinite families and 26 sporadic examples (assuming one ...
4
votes
2answers
3k views

Examples of naturally occurring Quadratic forms or quadrics.

I am always fascinated when a quadratic form (or a quadric) arises naturally. I have some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...
8
votes
4answers
1k views

Which topics/problems could you show to a bright first year mathematics student?

I am teaching a one semester course (January to June) to first year students pursuing various different degrees. Because there are students studying actuarial science, physics, other sciences, other ...
10
votes
7answers
3k views

Leibnizian calculus textbook

Where can I find a calculus textbook that emphasizes differentials? Is there such a book that I could realistically require my calculus students to use? I want a textbook that supports me when I tell ...
8
votes
9answers
1k views

“Surprising” categorical equivalences

This is inspired by this question about the equivalence between the category of finite sets and non-negative integers. Now this question was (rightly, I guess) closed, but the fact was surprising to ...
10
votes
7answers
2k views

Applications of the notion of of Gromov-Hausdorff distance

I am looking for applications of the notion of Gromov-Hausdorff convergence to prove theorems that a priori have nothing to do with it. Examples that I am aware of (thanks to wikipedia and google): ...
43
votes
29answers
9k 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 ...
51
votes
10answers
5k views

Applications of mathematics

All of us have probably been exposed to questions such as: "What are the applications of group theory...". This is not the subject of this MO question. Here is a little newspaper article that I found ...
30
votes
10answers
3k views

List of Classifying Spaces and Covers

I am looking for a list of classifying spaces $BG$ of groups $G$ (discrete and/or topological) along with associated covers $EG$; there does not seem to be such cataloging on the web. Or if not a ...
4
votes
3answers
772 views

Examples of results which were surprising but later shown to be natural. [closed]

After Ramanujan formulated his conjectures on the Tau-function, and after the importance of the function was realized, it took the development of the theory of Modular forms for the complete ...
187
votes
72answers
76k views

Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...
18
votes
3answers
2k views

Locales and Topology.

As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a ...
15
votes
12answers
2k views

Constructions unique up to non-unique isomorphism

1) Fields have algebraic closures unique up to a non-unique isomorphism. 2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism. 3) Modules have ...
91
votes
35answers
12k views

Books you would like to read (if somebody would just write them…)

I think that the title is self-explanatory but I'm thinking about mathematical subjects that have not received a full treatment in book form or if they have, they could benefit from a different ...
4
votes
10answers
1k views

Proving theorems by using functions with fixed points.

I am trying to get a better feel for solving questions where creating a function with a unique fixed point is the crux of the proof. In particular, the Inverse Function Theorem as well as the ...