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1500 questions
69
votes
7 answers
Is Grothendieck a computer?
I can't resist asking this companion question to the one of Gowers. There, Tim Dokchitser suggested the idea of Grothendieck topologies as a fundamentally new insight. But Gowers' original motivation is to probe the boundary between a human's way…
Minhyong Kim
- 13,471
69
votes
28 answers
Results from abstract algebra which look wrong (but are true)
There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup of $G$? As an experienced mathematician, we see…
Martin Brandenburg
- 61,443
69
votes
2 answers
Identity of J. L. Rabinowitsch (of Rabinowitsch Trick)
For some time, it seemed widely accepted that G. Y. Rainich was the author of the note Rabinowitsch, J. L., Zum Hilbertschen Nullstellensatz., Math. Ann. 102, 520 (1929). JFM 55.0103.04., which describes a short proof of Hilbert's Nullstellensatz by…
Olaf Teschke
- 1,114
69
votes
3 answers
Can you solve the listed smallest open Diophantine equations?
In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and evaluate. For example, the size $H$ of the…
Bogdan Grechuk
- 5,947
- 1
- 25
- 42
69
votes
4 answers
$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$
I read (in a paper by Emil Saucan) that the flat torus may be isometrically embedded
in $\mathbb{R}^3$ with a $C^1$ map by the Kuiper extension of the Nash Embedding Theorem,
a claim repeated in this Wikipedia entry. I have been unsuccessful in…
Joseph O'Rourke
- 149,182
- 34
- 342
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69
votes
24 answers
PhD dissertations that solve an established open problem
I search for a big list of open problems which have been solved in a PhD thesis by the Author of the thesis (or with collaboration of her/his supervisor).
In my question I search for every possible open problem but I prefer (but not limited) to…
Ali Taghavi
- 312
69
votes
5 answers
What was Hilbert's view of Gödel's Incompleteness Theorems?
According to Solomon Feferman, in his slide presentation "Three Problems for Mathematics", Hilbert wrote (in regards to Gödel's second incompleteness theorem):
...the end goal [is] to establish as consistent all our usual methods
of mathematics. …
Thomas Benjamin
- 6,024
69
votes
4 answers
Is a "non-analytic" proof of Dirichlet's theorem on primes known or possible?
It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) | n \in \mathbb{Z} \}$ must lie in certain arithmetic progressions,…
Qiaochu Yuan
- 114,941
69
votes
6 answers
What is a cumulant really?
A cumulant is defined via the cumulant generating function
$$ g(t)\stackrel{\tiny def}{=} \sum_{n=1}^\infty \kappa_n \frac{t^n}{n!},$$
where
$$
g(t)\stackrel{\tiny def}{=} \log E(e^{tX}).
$$
Cumulants have some nice properties, including additivity-…
Daniel Moskovich
- 21,887
69
votes
2 answers
Galoisian sets of prime numbers
The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved by Khare-Wintenberger), or certain cases of the…
Chandan Singh Dalawat
- 18,449
68
votes
9 answers
When have we lost a body of mathematics because errors were found?
The history of mathematics over the last 200 years has many occasions when the fundamental assumptions of an area have been shown to be flawed, or even wrong. Yet I cannot think of any examples where, as the result the mathematics itself had to be…
Edmund Harriss
- 1,314
68
votes
12 answers
Life after Hartshorne (the book, not the person)...
I was wondering what material in algebraic geometry is crucial and is a logical step for a serious graduate student in algebraic geometry once they've finished Hartshorne. Good answers could include a list of areas of algebraic geometry or…
anonymous
- 191
68
votes
30 answers
A book you would like to write
Writing a book from the beginning to the end is (so I heard) a very hard process. Planning a book is easier. This question is dual in a sense to the question "Books you would like to read (if somebody would just write them)". It is about a book that…
Gil Kalai
- 24,218
68
votes
2 answers
Is there a category structure one can place on measure spaces so that category-theoretic products exist?
The usual category of measure spaces consists of objects $(X, \mathcal{B}_X, \mu_X)$, where $X$ is a space, $\mathcal{B}_X$ is a $\sigma$-algebra on $X$, and $\mu_X$ is a measure on $X$, and measure preserving morphisms $\phi \colon (X,…
Damek Davis
- 783
68
votes
4 answers
Nelson's program to show inconsistency of ZF
At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say:
Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel axioms for set theory are necessarily even…
Andreas Thom
- 25,252