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1500 questions
45
votes
2 answers
Why do we care whether a PID admits some crazy Euclidean norm?
An integral domain $R$ is said to be Euclidean if it admits some Euclidean norm: i.e., a function $N: R \rightarrow \mathbb{N} = \mathbb{Z}^{\geq 0}$ such that: for all $x, y \in R$ with $N(y) > 0$, either $y$ divides $x$ or there exists $q \in R$…
Pete L. Clark
- 64,763
45
votes
1 answer
Is there a nullstellensatz for trigonometric polynomials?
Let
$$ f(x) = \sum_{j=1}^n c_j e^{2\pi i\alpha_j x}, g(x) = \sum_{k=1}^m d_k e^{2\pi i\beta_k x}$$
be two (quasi-periodic) trigonometric polynomials, where the coefficients $c_j, d_k$ are complex and the frequencies $\alpha_j,\beta_k$ are real (but…
Terry Tao
- 108,865
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- 517
45
votes
3 answers
"Cute" applications of the étale fundamental group
When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the circle that induces the identity on the circle.…
Libli
- 7,210
45
votes
4 answers
How to invoke constants badly
In a nice and witty lecture titled "how to write mathematics badly" (available on YouTube at https://www.youtube.com/watch?v=ECQyFzzBHlo&t=23s), Jean-Pierre Serre describes various ways in which a paper can be poorly/confusingly/inaccurately…
Alessandro Della Corte
- 4,263
45
votes
4 answers
How to write computer-assisted mathematics well?
Much has been said about writting good papers in mathematics. A short google query yields countless sources of advice. This skill also appears to be quite transferrable between basic branches of mathematics: a well-written paper in analysis follows…
Jakub Konieczny
- 1,582
45
votes
1 answer
Has gnu(2048) been found?
The gnu (or Group NUmber) function describes how many groups there are of a given order. The number of groups of each order are known up to 2047, see https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf
Has any progress been made on the number…
Thomas
- 2,691
45
votes
8 answers
What is Realistic Mathematics?
This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers will understand this post and the nature of the…
Andreas Thom
- 25,252
45
votes
3 answers
What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory?
This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields is essentially unfounded, and that Kummer was…
Emerton
- 56,762
45
votes
2 answers
On proof-verification using Coq
So i recently learnt that there is now a certain software called ''Coq'' by which one can check the validity of mathematical proofs. My questions are:
Are there limitations on the kinds of proofs that Coq can verify?
How long on average does Coq…
45
votes
1 answer
Exotic $R^4$ as the universal covering space
Is there a smooth compact 4-manifold whose universal covering is an exotic $R^4$, i.e. is homeomorphic but not diffeomorphic to $R^4$?
Remark. I am aware of examples (due to Mike Davis) of compact $n$-manifolds whose universal covering spaces are…
Moishe Kohan
- 9,664
45
votes
3 answers
A second Ph.D. in mathematics?
I have now some problems about my research Career, I would like to tell my stories. I am a Chinese guy, but now a Ph.D. candidate in Germany, in the field of so called 'Geometric Analysis', but I do not feel happy when I work in such a field.…
Zhiqiang Sun
- 111
45
votes
18 answers
Mathematical research interrupted by a war
I am not sure that this is appropriate at MO, so if not, please, delete this.
This is inspired by David Hansen's question where he asks about mathematics done during the WWII. I would like to ask the opposite question:
what are some examples of…
Sergei Akbarov
- 7,274
45
votes
1 answer
Hilbert's alleged proof of the Continuum Hypothesis in "On the Infinite"
As is known, Hilbert attempted a proof sketch of the Continuum Hypothesis in the latter part of his paper, "On the Infinite". It is also known that it is false.
Has there ever been a published analysis of this alleged proof showing where the…
Thomas Benjamin
- 6,024
45
votes
3 answers
Citing exercises in an article
I'm writing a paper in which I cite a lot of results that appear in Schikhof's Ultrametric Calculus. Some of these results are exercises in Schikhof's book. These exercises are not difficult, but are laborious. Thus, if I write the proofs, the…
efs
- 3,099
45
votes
5 answers
Fibonacci series captures Euler $e=2.718\dots$
The Fibonacci recurrence $F_n=F_{n-1}+F_{n-2}$ allows values for all indices $n\in\mathbb{Z}$. There is an almost endless list of properties of these numbers in all sorts of ways. The below question might even be known. Yet, if true, I like to ask…
T. Amdeberhan
- 41,802