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1500 questions
118
votes
5 answers
What do epimorphisms of (commutative) rings look like?
(Background: In any category, an epimorphism is a morphism $f:X\to Y$ which is "surjective" in the following sense: for any two morphisms $g,h:Y\to Z$, if $g\circ f=h\circ f$, then $g=h$. Roughly, "any two functions on $Y$ that agree on the image of…
Anton Geraschenko
- 23,718
117
votes
13 answers
How do I fix someone's published error?
Paper A is in the literature, and has been for more than a decade.
An error is discovered in paper A and is substantial in that many
details are affected, although certain fundamental properties
claimed by the theorems are not. (As a poor…
Gerhard Paseman
- 12,968
117
votes
29 answers
Papers that debunk common myths in the history of mathematics
What are some good papers that debunk common myths in the history of mathematics?
To give you an idea of what I'm looking for, here are some examples.
Tony Rothman, "Genius and biographers: The fictionalization of Evariste Galois," Amer. Math.…
Timothy Chow
- 78,129
116
votes
4 answers
Is the analysis as taught in universities in fact the analysis of definable numbers?
Ten years ago, when I studied in university, I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those which can be unambiguously defined by a limited set…
Anixx
- 9,302
116
votes
29 answers
Where does a math person go to learn statistical mechanics?
The more math I read, the more I see concepts from statistical mechanics popping up -- all over the place in combinatorics and dynamical systems, but also in geometric situations. So naturally I've been trying to get a grasp on statistical mechanics…
Harrison Brown
- 12,543
116
votes
8 answers
Zagier's one-sentence proof of a theorem of Fermat
Zagier has a very short proof (MR1041893, JSTOR) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $ which is easily seen…
Keivan Karai
- 6,074
116
votes
24 answers
Tools for collaborative paper-writing
I personally use a revision control system (git) to manage my own paper writing, back things up, and synchronize between different machines. However, I've found most programmer's revision control systems to require a bit too much training to try…
Tyler Lawson
- 51,259
116
votes
5 answers
How did Cole factor $2^{67}-1$ in 1903?
I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193{,}707{,}721\times 761{,}838{,}257{,}287$. There doesn't seem to be a historical record of how Cole achieved this;…
David E Speyer
- 150,821
115
votes
2 answers
Why is the Hodge Conjecture so important?
The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm able to understand what it says, and at first…
Fitzcarraldo
- 1,253
114
votes
6 answers
What properties make $[0,1]$ a good candidate for defining fundamental groups?
The title essentially says it all. Consider the category $\mathfrak{Top}_2$ of triples $(J,e_0,e_1)$ where $J$ is a topological space, and $e_i \in J$. There is an obvious generalization of the definition of homotopic maps. Suppose we have selected…
Daniel Miller
- 5,609
114
votes
32 answers
What notions are used but not clearly defined in modern mathematics?
"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
Felix Klein
What notions are used but not clearly defined in modern mathematics?
To clarify further…
kakaz
- 1,596
114
votes
96 answers
What would you want to see at the Museum of Mathematics?
EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no longer relevant".
As some of you may already know,…
Timothy Chow
- 78,129
114
votes
4 answers
Is the series $\sum_n|\sin n|^n/n$ convergent?
Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent?
(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for solution is "butelka miodu pitnego", see page…
Lviv Scottish Book
- 4,435
114
votes
3 answers
The number $\pi$ and summation by $SL(2,\mathbb Z)$
Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$. (it is the defect in the triangle inequality)
Then, we discovered by heuristic arguments and then verified by computer that
$$\sum f(a,b,c,d)^n = 2-\pi/2$$
where the sum runs…
Nikita Kalinin
- 4,761
114
votes
36 answers
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 turn out to follow beautifully from basic…
Paul Siegel
- 28,772