**211**

votes

**99**answers

34k 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 ...

**532**

votes

**207**answers

135k views

### Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...

**168**

votes

**36**answers

46k views

### Widely accepted mathematical results that were later shown wrong?

I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time ...

**109**

votes

**39**answers

32k views

### Most interesting mathematics mistake?

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove ...

**95**

votes

**42**answers

18k views

### Examples of eventual counterexamples

Define an "eventual counterexample" to be
$P(a) = T $ for $a < n$
$P(n) = F$
$n$ is sufficiently large for $P(n) = T\ \ \forall n \in \mathbb{N}$ to be a 'reasonable' conjecture to make.
where ...

**126**

votes

**76**answers

42k views

### Best online mathematics videos?

I know of two good mathematics videos available online, namely:
Sphere inside out (part I and part II)
Moebius transformation revealed
Do you know of any other good math videos? Share.

**116**

votes

**94**answers

63k views

### Famous mathematical quotes [closed]

Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?
Standard community wiki rules apply: one ...

**147**

votes

**67**answers

48k views

### Awfully sophisticated proof for simple facts [closed]

It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ...

**113**

votes

**46**answers

36k views

### Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?
Please give a new way in each answer, and if possible give reference. I start by giving two:
...

**140**

votes

**30**answers

50k views

### Real-world applications of mathematics, by arxiv subject area?

What are the most important applications outside of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to arxiv mathematics categories, ...

**121**

votes

**33**answers

69k views

### Best Algebraic Geometry text book? (other than Hartshorne)

I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.
Then what might be the 2nd best?
It can be a book, preprint, online lecture note, webpage, etc.
One suggestion ...

**120**

votes

**58**answers

24k views

### What are your favorite instructional counterexamples?

Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical.
In many branches of mathematics, it seems to me that a good counterexample can be ...

**177**

votes

**30**answers

18k views

### Which journals publish expository work?

I wonder if anyone else has noticed that the market for expository papers in mathematics is very narrow (more so than it used to be, perhaps).
Are there any journals which publish expository work, ...

**116**

votes

**69**answers

23k views

### Which math paper maximizes the ratio (importance)/(length)?

My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are ...

**72**

votes

**33**answers

66k views

### Mathematicians who were late learners?-list [closed]

It is well-known that many great mathematicians were prodigies.
Were there any great mathematicians who started off later in life?

**31**

votes

**9**answers

5k 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 ...

**19**

votes

**63**answers

13k views

### What's your favorite equation, formula, identity or inequality? [closed]

Certain formulas I really enjoy looking at like the Euler-Maclaurin formula or the Leibniz integral rule. What's your favorite equation, formula, identity or inequality?

**58**

votes

**53**answers

14k views

### Pseudonyms of famous mathematicians

Many mathematicians know that Lewis Carroll was quite a good mathematician, who wrote about logic (paradoxes) and determinants. He found an expansion formula, which bears his real name (Charles ...

**222**

votes

**68**answers

104k views

### Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
(One could ask if this is of interest to mathematicians, and I ...

**144**

votes

**77**answers

24k views

### Your favorite surprising connections in Mathematics

There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the ...

**92**

votes

**55**answers

16k views

### Counterexamples in Algebra?

This is certainly related to "What are your favorite instructional counterexamples?", but I thought I would ask a more focused question. We've all seen Counterexamples in Analysis and Counterexamples ...

**87**

votes

**19**answers

13k views

### Do you read the masters?

I often hear the advice, "Read the masters" (i.e., read old, classic texts by great mathematicians). But frankly, I have hardly ever followed it. What I am wondering is, is this a principle that ...

**72**

votes

**25**answers

28k views

### What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...

**91**

votes

**42**answers

20k views

### What are the most attractive Turing undecidable problems in mathematics?

What are the most attractive Turing undecidable problems in mathematics?
There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on ...

**66**

votes

**31**answers

17k views

### Quick proofs of hard theorems

Mathematics is rife with the fruit of abstraction. Many problems which first are solved via "direct" methods (long and difficult calculations, tricky estimates, and gritty technical theorems) later ...

**75**

votes

**33**answers

18k views

### Computer Algebra Errors

In the course of doing mathematics, I make extensive use of computer-based calculations. There's one CAS that I use mostly, even though I occasionally come across out-and-out wrong answers.
After ...

**52**

votes

**25**answers

9k views

### Cocktail party math [closed]

Ok, hotshots. You're at a party, and you're chatting with some non-mathematicians. You tell them that you're a mathematician, and then they ask you to elaborate a bit on what you study, or they ask ...

**57**

votes

**41**answers

9k 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 ...

**17**

votes

**21**answers

24k views

### Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?

**29**

votes

**18**answers

10k 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 ...

**28**

votes

**14**answers

7k views

### Open problems in Euclidean geometry?

Which are some (research level) open problems in Euclidean geometry ?
(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a ...

**246**

votes

**72**answers

92k 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 ...

**144**

votes

**136**answers

30k views

### Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)
I'd love to learn about ...

**152**

votes

**64**answers

26k views

### Proofs that require fundamentally new ways of thinking [closed]

I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is ...

**180**

votes

**109**answers

46k views

### What are some examples of colorful language in serious mathematics papers? [closed]

The popular MO question "Famous mathematical quotes" has turned
up many examples of witty, insightful, and humorous writing by
mathematicians. Yet, with a few exceptions such as Weyl's "angel of
...

**100**

votes

**67**answers

33k views

### Math puzzles for dinner [closed]

You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a ...

**112**

votes

**81**answers

88k views

### Do good math jokes exist? [closed]

Have a good joke? Share.
I know this is subjective, but the principle "should be of interest to mathematicians" trumps. (I hope.)

**67**

votes

**54**answers

14k views

### Free, high quality mathematical writing online? [closed]

I often use the internet to find resources for learning new mathematics and due to an explosion in online activity, there is always plenty to find. Many of these turn out to be somewhat unreadable ...

**147**

votes

**27**answers

18k views

### What are some reasonable-sounding statements that are independent of ZFC?

Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC."
For example, suppose A is an abelian group such that every ...

**61**

votes

**22**answers

17k views

### What's the “best” proof of quadratic reciprocity?

For my purposes, you may want to interpret "best" as "clearest and easiest to understand for undergrads in a first number theory course," but don't feel too constrained.

**104**

votes

**27**answers

14k views

### How To Present Mathematics To Non-Mathematicians?

(Added an epilogue)
I started a job as a TA, and it requires me to take a five sessions workshop about better teaching in which we have to present a 10 minutes lecture (micro-teaching).
In the last ...

**119**

votes

**37**answers

15k 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 ...

**103**

votes

**40**answers

11k views

### Generalizing a problem to make it easier

One of the many articles on the Tricki that was planned but has never been written was about making it easier to solve a problem by generalizing it (which initially seems paradoxical because if you ...

**82**

votes

**18**answers

8k 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 ...

**122**

votes

**64**answers

16k views

### Suggestions for good notation

I occasionally come across a new piece of notation so good that it makes life easier by giving a better way to look at something. Some examples:
Iverson introduced the notation [X] to mean 1 if X is ...

**61**

votes

**27**answers

7k views

### Good papers/books/essays about the thought process behind mathematical research

Papers in mathematics are generally written as if the major insights suddenly appeared, unbidden, in a notebook on the researcher's desk and then were fleshed out into the final paper.
While this is ...

**36**

votes

**23**answers

10k views

### Open source mathematical software

I want some recommendation on which software I should install on my computer. I'm looking for an open source program for general abstract mathematical purposes (as opposed to applied mathematics).
I ...

**107**

votes

**27**answers

14k views

### Extremely messy proofs

Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what ...

**51**

votes

**51**answers

17k views

### Colloquial catchy statements encoding serious mathematics

As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as ...

**43**

votes

**24**answers

8k 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.
...