**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: ...

**14**

votes

**1**answer

618 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

**4**answers

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

**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

707 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

**3**answers

934 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

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

**14**answers

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

**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, ...

**23**

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

780 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 ...

**3**

votes

**2**answers

923 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

**22**answers

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

**3**answers

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

**44**answers

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

**7**answers

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

**15**answers

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

**24**answers

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

**3**answers

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

**2**answers

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

**4**answers

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

**9**answers

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

**22**answers

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 ...

**40**

votes

**13**answers

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

**12**answers

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

**1**answer

707 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

**2**answers

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

**5**answers

668 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

**4**answers

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

**11**answers

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

**17**answers

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

**44**answers

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

**18**answers

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

**6**answers

910 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

**1**answer

518 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

**26**answers

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

**12**answers

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 ...

**20**

votes

**10**answers

2k views

### What math institutes offer research in pairs/research in teams?

Some math institutes offer programs in which a small number of researchers are enabled to meet at the institute for a week or more. A list seemed as if it could be useful.

**21**

votes

**53**answers

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

**8**answers

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

**73**answers

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 ...

**67**

votes

**17**answers

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

**10**answers

826 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

**2**answers

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

**4**answers

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

**7**answers

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

**9**answers

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

**7**answers

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

**29**answers

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 ...