302
votes
169answers
75k views
Examples of common false beliefs in mathematics. [closed]
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 p …
233
votes
4answers
17k views
Polynomial bijection from QxQ to Q?
Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}\ $ such that $f:\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?
157
votes
21answers
16k 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 tho …
156
votes
18answers
11k views
Ideas on how to prevent a department from being shut down. [closed]
[Please stop upvoting. I don't want to get a gold badge out of this...]
[Too late... is there a way to give a medal back?]
Dear community,
the VU University Amsterdam, my employ …
147
votes
7answers
64k views
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 explai …
147
votes
8answers
9k 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 polynomi …
144
votes
11answers
9k 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 …
128
votes
70answers
44k views
Mathematical “urban legends” [closed]
When I was a young and impressionable graduate student at Princeton, we scared each other with the story of a Final Public Oral, where Jack Milnor was dragged in against his will t …
126
votes
108answers
33k 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 W …
126
votes
59answers
64k views
Proofs without words [closed]
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 mathe …
125
votes
8answers
5k views
Two commuting mappings in the disk
Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk (or, if you prefer, the closed unit ball in $R^n$) to itself. Does there always exist a poi …
124
votes
28answers
11k 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 publi …
117
votes
41answers
43k 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 y …
115
votes
65answers
44k 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. Fortun …
113
votes
31answers
13k views
What should be learned in a first serious schemes course?
I've just finished teaching a year-long "foundations of algebraic
geometry" class. It
was my third time teaching it, and my notes are gradually converging.
I've enjoyed it for a n …
