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1500 questions
168
votes
1 answer
Ultrafilters and automorphisms of the complex field
It is well-known that it is consistent with $ZF$ that the only automorphisms of the complex field $\mathbb{C}$ are the identity map and complex conjugation. For example, we have that $\vert\operatorname{Aut}(\mathbb{C})| = 2$ in $L(\mathbb{R})$. But…
Simon Thomas
- 8,348
167
votes
37 answers
Too old for advanced mathematics?
Kind of an odd question, perhaps, so I apologize in advance if it is inappropriate for this forum. I've never taken a mathematics course since high school, and didn't complete college. However, several years ago I was affected by a serious illness…
MattyZ
- 309
167
votes
50 answers
17 camels trick
The following popular mathematical parable is well known:
A father left 17 camels to his three sons and, according to the will, the eldest son should be given a half of all camels, the middle son the one-third part and the youngest son the…
Fedor Petrov
- 102,548
166
votes
3 answers
Convergence of $\sum(n^3\sin^2n)^{-1}$
I saw a while ago in a book by Clifford Pickover, that whether the Flint Hills series $\displaystyle \sum_{n=1}^\infty\frac1{n^3\sin^2 n}$ converges is open.
I would think that the question of its convergence is really about the density in $\mathbb…
Andrés E. Caicedo
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164
votes
21 answers
When should a supervisor be a co-author?
What are people's views on this? To be specific: suppose a PhD student has produced a piece of original mathematical research. Suppose that student's supervisor suggested the problem, and gave a few helpful comments, but otherwise did not…
MrB
- 399
164
votes
9 answers
Endless controversy about the correctness of significant papers
In principle, a mathematical paper should be complete and correct. New statements should be supported by appropriate proofs. But this is only theory. Because we often cannot enter into the smallest details, we "prove" wrong statements here and then.…
Denis Serre
- 51,599
163
votes
38 answers
Short exact sequences every mathematician should know
I'd like to have a big-list of "great" short exact sequences that capture some vital phenomena. I'm learning module theory, so I'd like to get a good stock of examples to think about. An elementary example I have in mind is the SES:
$$
0 \rightarrow…
Siddharth Bhat
- 1,191
163
votes
46 answers
Every mathematician has only a few tricks
In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert have only a few tricks which they use over and over again.
Assuming…
Ivan Meir
- 4,782
163
votes
11 answers
Why is the Gamma function shifted from the factorial by 1?
I've asked this question in every math class where the teacher has introduced the Gamma function, and never gotten a satisfactory answer. Not only does it seem more natural to extend the factorial directly, but the integral definition $\Gamma(z) =…
Kevin Casto
- 3,071
163
votes
28 answers
Cool problems to impress students with group theory
Since this forum is densely populated with algebraists, I think I'll ask it here.
I'm teaching intermediate level algebra this semester and I'd like to entertain my students with some clever applications of group theory. So, I'm looking for problems…
fedja
- 59,730
161
votes
23 answers
Do you read the masters?
I often hear the advice, "Read the masters" (i.e., read old, classic texts by great mathematicians). But frankly, I have hardly ever followed it. What I am wondering is, is this a principle that people give lip service to because it sounds good,…
Timothy Chow
- 78,129
160
votes
37 answers
Conceptual reason why the sign of a permutation is well-defined?
Teaching group theory this semester, I found myself laboring through a proof that the sign of a permutation is a well-defined homomorphism $\operatorname{sgn} : \Sigma_n \to \Sigma_2$. An insightful student has pressed me for a more illuminating…
Tim Campion
- 60,951
159
votes
18 answers
Ideas on how to prevent a department from being shut down.
[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 employer, is planning to shut down its pure math section and fire four tenured…
Tilman
- 6,042
158
votes
14 answers
What is an integrable system?
What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" and "chaotic"? (There is an interesting Wikipedia…
Gil Kalai
- 24,218
158
votes
8 answers
Resources for mathematics advising.
This question is possibly ill-advised. (If it is not right for this site I will delete it.)
I, suddenly, have students.
It is very clear to me that there is nothing in my education that has prepared me for the task of training graduate…
anonymous
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