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1500 questions
90
votes
5 answers
Does this property characterize straight lines in the plane?
Take a plane curve $\gamma$ and a disk of fixed radius whose center moves along $\gamma$. Suppose that $\gamma$ always cuts the disk in two simply connected regions of equal area. Is it true that $\gamma$ must be a straight line?
Alessandro Della Corte
- 4,263
90
votes
3 answers
What is homology anyway?
Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid mistakes please treat them as such and try to…
Saal Hardali
- 7,549
90
votes
8 answers
Books on music theory intended for mathematicians
Some time ago I attended a colloquium given by Princeton music theorist Dmitri Tymoczko, where he gave a fascinating talk on the connection between music composition and certain geometric objects (as I recall, the work of Chopin can naturally be…
Stanley Yao Xiao
- 25,509
90
votes
24 answers
Examples of major theorems with very hard proofs that have not dramatically improved over time
This question complement a previous MO question: Examples of theorems with proofs that have dramatically improved over time.
I am looking for a list of
Major theorems in mathematics whose proofs are very hard but was not dramatically improved over…
Gil Kalai
- 24,218
89
votes
10 answers
Is there any formal foundation to ultrafinitism?
Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to Wikipedia, it has been primarily studied by Alexander Esenin-Volpin. On his…
Michael O'Connor
- 1,001
89
votes
13 answers
If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?
There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What is the probability that the three segments…
Michael Lugo
- 13,858
89
votes
27 answers
Modern Mathematical Achievements Accessible to Undergraduates
While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even most research topologists would not be able to…
Lennart Meier
- 11,877
88
votes
5 answers
Algorithm or theory of diagram chasing
One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ideally also an algorithm?
To be precise about what I…
Greg Kuperberg
- 56,146
88
votes
7 answers
If I exchange infinitely many digits of $\pi$ and $e$, are the two resulting numbers transcendental?
If I swap the digits of $\pi$ and $e$ in infinitely many places, I get two new numbers. Are these two numbers transcendental?
user10290
88
votes
9 answers
Work of plenary speakers at ICM 2014
The next International Congress of Mathematicians (ICM) will take place in 2014 in Seoul, Korea. The present question is meant to gather brief overviews of the work of the plenary speakers for the ICM 2014.
More precisely, anybody who feels…
Koushik
- 2,076
88
votes
5 answers
When is $A$ isomorphic to $A^3$?
This is totally elementary, but I have no idea how to solve it: let $A$ be an abelian group such that $A$ is isomorphic to $A^3$. is then $A$ isomorphic to $A^2$? probably no, but how construct a counterexample? you can also ask this in other…
Martin Brandenburg
- 61,443
87
votes
38 answers
Where is number theory used in the rest of mathematics?
Where is number theory used in the rest of mathematics?
To put it another way: what interesting questions are there that don't appear to be about number theory, but need number theory in order to answer them?
To put it another way still: imagine a…
Tom Leinster
- 27,167
87
votes
12 answers
Why do we make such big deal about the 'unsolvability' of the quintic?
The unsolvability of a general quintic equation in terms of the basic arithmetic operations and $n$th roots (i.e. the Abel–Ruffini theorem) is considered a major result in the mathematical canon. I have recently become confused as to why this is the…
Arthur
- 1,379
87
votes
14 answers
LaTeX tricks that save time in typesetting
In ${\rm\LaTeX}$ typesetting, when we repeat a long and complex formula in long documents, it is appropriate to create a new command that just by calling this new command we get the desired output. For example, I have used the following math…
C.F.G
- 4,165
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87
votes
15 answers
The importance of EGA and SGA for "students of today"
That fact that EGA and SGA have played mayor roles is uncontroversial. But they contain many volumes/chapters and going through them would take a lot of time, especially if you do not speak French.
This raises the question if a student, like me,…
user1161
- 193