**573**

votes

**218**answers

145k 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 ...

**225**

votes

**99**answers

37k 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 ...

**186**

votes

**36**answers

50k 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 ...

**120**

votes

**46**answers

40k 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:
...

**117**

votes

**39**answers

35k 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 ...

**129**

votes

**33**answers

77k 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 ...

**118**

votes

**94**answers

66k 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 ...

**155**

votes

**67**answers

53k 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 ...

**235**

votes

**68**answers

110k 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 ...

**133**

votes

**76**answers

45k 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.

**241**

votes

**21**answers

30k views

### Thinking and Explaining

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...

**143**

votes

**30**answers

55k 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, e.g....

**181**

votes

**30**answers

19k 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, ...

**75**

votes

**33**answers

71k 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?

**75**

votes

**33**answers

19k 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 ...

**24**

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?

**64**

votes

**59**answers

15k 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 ...

**122**

votes

**58**answers

25k 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 ...

**91**

votes

**19**answers

14k 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 ...

**93**

votes

**55**answers

17k 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 ...

**112**

votes

**45**answers

21k 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 '...

**93**

votes

**42**answers

21k 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 ...

**88**

votes

**19**answers

9k 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 ...

**64**

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 ...

**62**

votes

**16**answers

22k views

### What is the definition of “canonical”?

I just received a referee report criticizing that I would too often use the word "canonical". I have a certain understanding of what "canonical" should stand for, but the report shows me that other ...

**37**

votes

**25**answers

8k views

### Theorems for nothing (and the proofs for free) [closed]

Some theorems give far more than you feel they ought to: a weak hypothesis is enough to prove a strong result. Of course, there's almost always a lot of machinery hidden below the waterline. Such ...

**34**

votes

**14**answers

8k 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 bit ...

**261**

votes

**72**answers

97k 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

31k 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 ...

**155**

votes

**64**answers

27k 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 ...

**184**

votes

**109**answers

49k 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
...

**121**

votes

**81**answers

96k 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.)

**171**

votes

**22**answers

25k views

### Geometric Interpretation of Trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandry;
Is there a geometric interpretation of the trace of a matrix?
This question ...

**156**

votes

**27**answers

19k 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 ...

**120**

votes

**68**answers

24k 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 ...

**61**

votes

**22**answers

19k 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

15k 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 ...

**113**

votes

**40**answers

12k 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 ...

**122**

votes

**37**answers

16k 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 ...

**59**

votes

**16**answers

15k views

### Most 'unintuitive' application of the Axiom of Choice?

It is well-known that the axiom of choice is equivalent to many other assumptions, such as the well-ordering principle, Tychonoff's theorem, and the fact that every vector space has a basis. Even ...

**64**

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 ...

**154**

votes

**11**answers

46k views

### Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...

**75**

votes

**51**answers

12k views

### Important formulas in Combinatorics

Motivation:
The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered ...

**111**

votes

**27**answers

15k 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 ...

**72**

votes

**33**answers

12k views

### Theorems with unexpected conclusions [closed]

I am interested in theorems with unexpected conclusions. I don't mean
an unintuitive result (like the existence of a space-filling curve), but
rather a result whose conclusion seems disconnected from ...

**65**

votes

**10**answers

9k views

### Why do Bernoulli numbers arise everywhere?

I have seen Bernoulli numbers many times, and sometimes very surprisingly. They appear in my textbook on complex analysis, in algebraic topology, and of course, number theory. Things like the criteria ...

**54**

votes

**51**answers

18k 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 ...

**45**

votes

**35**answers

13k views

### Most intricate and most beautiful structures in mathematics

In the December 2010 issue of Scientific American, an article "A Geometric Theory of
Everything" by A. G. Lisi and J. O. Weatherall states "... what is arguably the most
intricate structure known to ...

**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 ...

**44**

votes

**11**answers

6k views

### Interesting results in algebraic geometry accessible to 3rd year undergraduates

On another thread I asked how I could encourage my final year undergraduate colleagues to take an algebraic geometry or complex analysis courses during their graduate studies.
Willie Wong proposed me ...