Most Popular
1500 questions
207
votes
14 answers
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 mathematics because errors were found?, etc...
My…
J. Doe
- 141
202
votes
72 answers
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 worth just as much as a good theorem or lemma. The…
Qiaochu Yuan
- 114,941
199
votes
89 answers
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 name as many examples of "great writing" as…
Anton Geraschenko
- 23,718
194
votes
18 answers
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 Groups, and Modular forms! (and an undergraduate…
aglearner
- 13,995
193
votes
43 answers
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 rigorous formalization. I'm wondering if anybody…
Marco Disce
- 303
- 3
- 4
- 8
193
votes
30 answers
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. math.AT, math.QA, math.CO, etc.
This is a…
Scott Morrison
- 7,540
192
votes
94 answers
Famous mathematical quotes
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 quote per post.
Sam Derbyshire
- 5,440
191
votes
34 answers
What is convolution intuitively?
If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability distribution of $X+Y$. This is the only intuition I…
Kim Greene
- 3,583
- 10
- 42
- 41
191
votes
47 answers
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 approach.
(I do hope this is not inappropriate for…
Gonçalo Marques
- 496
191
votes
12 answers
Do you know important theorems that remain unknown?
Do you know of any very important theorems that remain unknown? I mean results that could easily make into textbooks or research monographs, but almost
nobody knows about them. If you provide an answer, please:
State only one theorem per answer.…
Piotr Hajlasz
- 27,279
189
votes
79 answers
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 homeomorphic but not diffeomorphic to the…
David Hansen
- 13,018
187
votes
81 answers
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 true and 0 otherwise; so for example Σ1≤n
Richard Borcherds
- 20,442
186
votes
3 answers
Issue UPDATE: in graph theory, different definitions of edge crossing numbers - impact on applications?
QUICK FINAL UPDATE: Just wanted to thank you MO users for all your support. Special thanks for the fast answers, I've accepted first one, appreciated the clarity it gave me. I've updated my torus algorithm with ${\rm cr}(G)$. Works fine on my full…
user161819
- 243
186
votes
8 answers
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 point $x$ such that $f(x)=g(x)$?
If one of the mappings is invertible,…
fedja
- 59,730
185
votes
47 answers
Magic trick based on deep mathematics
I am interested in magic tricks whose explanation requires deep mathematics. The trick should be one that would actually appeal to a layman. An example is the following: the magician asks Alice to choose two integers between 1 and 50 and add them.…
Richard Stanley
- 49,238