# All Questions

155,596
questions

1052
votes

292
answers

345k
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### 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 ...

522
votes

3
answers

56k
views

### Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?

428
votes

16
answers

65k
views

### Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...

423
votes

92
answers

148k
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 ...

401
votes

84
answers

187k
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### 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 ...

395
votes

23
answers

67k
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, ...

386
votes

115
answers

108k
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### 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 in ...

386
votes

53
answers

146k
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### Widely accepted mathematical results that were later shown to be wrong?

Are there 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 later, possibly ...

364
votes

31
answers

77k
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### Geometric interpretation of trace

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

335
votes

16
answers

156k
views

### What's a mathematician to do?

I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of ...

328
votes

34
answers

93k
views

### Why is a topology made up of 'open' sets? [closed]

I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...

318
votes

22
answers

103k
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### Why do so many textbooks have so much technical detail and so little enlightenment? [closed]

I think/hope this is okay for MO.
I often find that textbooks provide very little in the way of motivation or context. As a simple example, consider group theory. Every textbook I have seen that ...

295
votes

8
answers

142k
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### Philosophy behind Mochizuki's work on the ABC conjecture

Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...

294
votes

34
answers

51k
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 ...

293
votes

7
answers

22k
views

### Polynomial representing all nonnegative integers

Lagrange proved that every nonnegative integer is a sum of 4 squares.
Gauss proved that every nonnegative integer is a sum of 3 triangular numbers.
Is there a 2-variable polynomial $f(x,y) \in \...

292
votes

125
answers

91k
views

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

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
...

292
votes

34
answers

39k
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, ...

281
votes

47
answers

109k
views

### Examples of unexpected mathematical images

I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...

270
votes

67
answers

139k
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 ...

263
votes

29
answers

89k
views

### Mathematical games interesting to both you and a 5+-year-old child

Background: My daughter is 6 years old now, once I wanted to think on some math (about some Young diagrams), but she wanted to play with me...
How to make both of us to do what they want ? I guess ...

254
votes

41
answers

98k
views

### A single paper everyone should read? [closed]

Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues.
Do ...

250
votes

16
answers

69k
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### Why worry about the axiom of choice?

As I understand it, it has been proven that the axiom of choice is independent of the other axioms of set theory. Yet I still see people fuss about whether or not theorem X depends on it, and I don't ...

250
votes

29
answers

165k
views

### Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-
An engineer, a physicist, and a mathematician are discussing how to visualise four ...

249
votes

37
answers

173k
views

### Best algebraic geometry textbook? (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 ...

244
votes

8
answers

30k
views

### Need advice or assistance for son who is in prison. His interest is scattering theory

The letter below is written by my son. I have been sending him text books and looking for answers on the internet to keep his interest up. He has progressed so far on his own and now he needs ...

236
votes

14
answers

75k
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 ...

235
votes

36
answers

35k
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### Conway's lesser-known results

John Horton Conway is known for many achievements:
Life, the three sporadic groups in the "Conway constellation," surreal numbers, his "Look-and-Say" sequence analysis, the Conway-Schneeberger $15$-...

235
votes

10
answers

42k
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### If $f$ is infinitely differentiable then $f$ coincides with a polynomial

Let $f$ be an infinitely differentiable function on $[0,1]$ and suppose that for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{(n)}(x)=0$. Then does $f$ coincide on $[0,1]$ ...

233
votes

16
answers

56k
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### What elementary problems can you solve with schemes?

I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary self-...

231
votes

46
answers

88k
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 ...

231
votes

13
answers

41k
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### Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain.
I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...

230
votes

89
answers

44k
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 ...

230
votes

4
answers

15k
views

### Is $\mathbb R^3$ the square of some topological space?

The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X \...

228
votes

9
answers

24k
views

### John Nash's Mathematical Legacy

It would seem that John Nash and his wife Alicia died tragically in a car accident on May 23, 2015 (reference). My condolences to his family and friends.
Maybe this is an appropriate time to ask a ...

226
votes

20
answers

38k
views

### How can a mathematician handle the pressure to discover something new?

Suppose I'm an aspiring mathematician-to-be, who started doing research. Although this is really what I love doing, I found that one disturbing point is that there's always the pressure of discovering ...

225
votes

11
answers

26k
views

### Refereeing a Paper [closed]

I've refereed at least a dozen papers in my (short) career so far and I still find the process completely baffling. I'm wondering what is actually expected and what people tend to do...
Some things ...

223
votes

4
answers

16k
views

### A game on Noetherian rings

A friend suggested the following combinatorial game. At any time, the state of the game is a (commutative) Noetherian ring $\neq 0$. On a player's turn, that player chooses a nonzero non-unit element ...

220
votes

140
answers

49k
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 ...

218
votes

8
answers

34k
views

### How to memorise (understand) Nakayama's lemma and its corollaries?

Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...

216
votes

67
answers

46k
views

### Proofs that require fundamentally new ways of thinking

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 ...

215
votes

24
answers

47k
views

### What is torsion in differential geometry intuitively?

Hi,
given a connection on the tangent space of a manifold, one can define its torsion:
$$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$
What is the geometric picture behind this definition&...

214
votes

0
answers

17k
views

### Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?

Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...

213
votes

40
answers

38k
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### Demonstrating that rigour is important

Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with ...

212
votes

26
answers

51k
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### The most outrageous (or ridiculous) conjectures in mathematics

The purpose of this question is to collect the most outrageous (or ridiculous) conjectures in mathematics.
An outrageous conjecture is qualified ONLY if:
1) It is most likely false
(Being hopeless is ...

209
votes

51
answers

81k
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:
...

208
votes

14
answers

59k
views

### Why doesn't mathematics collapse even though humans quite often make mistakes in their proofs?

To begin with, I am aware of these questions, which seems to be related:
How do I fix someone's published error?, Examples of common false beliefs in mathematics, When have we lost a body of ...

204
votes

72
answers

50k
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### 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 ...

199
votes

89
answers

53k
views

### Examples of great mathematical writing

This question is basically from Ravi Vakil's web page, but modified for Math Overflow.
How do I write mathematics well? Learning by example is more helpful than being told what to do, so let's try to ...

194
votes

43
answers

43k
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### Are there other nice math books close to the style of Tristan Needham?

I've been very positively impressed by Tristan Needham's book "Visual Complex Analysis", a very original and atypical mathematics book which is more oriented to helping intuition and insight than to ...

194
votes

18
answers

16k
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### Great graduate courses that went online recently

In 09.2020 by pure chance I discovered the YouTube channel of Richard Borcherds where he gives graduate courses in Group Theory, Algebraic Geometry, Schemes, Commutative Algebra, Galois Theory, Lie ...