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1500 questions
53
votes
12 answers
Looking for an introduction to orbifolds
Is there any source where the basic facts about orbifolds are written and proved in full detail?
I found the article by Satake "The Gauss-Bonnet Theorem for V-manifolds", but I'd like to have a more complete and modern source.
Themaninthebox
- 539
- 5
- 3
53
votes
3 answers
Sheaves and bundles in differential geometry
Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as the "main theory" used to describe geometric data. All sheaf data…
Harry Gindi
- 19,374
53
votes
3 answers
Everywhere differentiable function that is nowhere monotonic
It is well known that there are functions $f \colon \mathbb{R} \to \mathbb{R}$ that are everywhere continuous but nowhere monotonic (i.e. the restriction of $f$ to any non-trivial interval $[a,b]$ is not monotonic), for example the Weierstrass…
Ricky
- 3,674
53
votes
2 answers
Why is it hard to prove that the Euler Mascheroni constant is irrational?
Philosophically why should proving that $\gamma$ is irrational (let alone transcendental) be so much harder than proving $\pi$ or $e$ are irrational?
user16557
- 1,513
- 2
- 14
- 14
53
votes
3 answers
Is it true that, as $\Bbb Z$-modules, the polynomial ring and the power series ring over integers are dual to each other?
Is it true that, in the category of $\mathbb{Z}$-modules, $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}[x],\mathbb{Z})\cong\mathbb{Z}[[x]]$ and $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}[[x]],\mathbb{Z})\cong\mathbb{Z}[x]$?
The first isomorphism is…
Maharana
- 1,742
53
votes
30 answers
Fundamental problems whose solution seems completely out of reach
In many areas of mathematics there are fundamental problems that are embarrasingly natural or simple to state, but whose solution seem so out of reach that they are barely mentioned in the literature even though most practitioners know about them.…
alvarezpaiva
- 13,238
52
votes
10 answers
Where to publish a math textbook in Creative Commons
The question I am asking concerns publishing and not mathematics in a strict sense, the argument might be off-topic for this site... but I will try anyway.
Suppose that you are writing a textbook in advanced mathematics, say a book for mature…
Bruno Martelli
- 10,164
- 2
- 37
- 69
52
votes
5 answers
Beautiful descriptions of exceptional groups
I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need not so many words to describe this).
For $G_2$…
zroslav
- 1,412
52
votes
2 answers
a categorical Nakayama lemma?
There are the following Nakayama style lemmata:
(the classical Nakayama lemma) Let $R$ be a commutative ring with $1$ and $M$ a finitely generated $R$-module. If $m_1, \ldots, m_n$ generate $M$ modulo $I$, where $I \subset \mathrm{Jac}(R)$, then…
user19475
52
votes
4 answers
Changing field of study post-PhD
I am doing my PhD in algebraic graph theory, for not much more reason than that was what was available. However, I love deep structure and theory in mathematics, and I do not particularly want to be a graph theorist for the rest of my life.
I…
Adam
- 1,327
52
votes
6 answers
Why is the standard definition of cocycle the one that _always_ comes up??
This question might not have a good answer. It was something that occurred to me yesterday when I found myself in a pub, needing to do an explicit calculation with 2-cocycles but with no references handy (!).
Review of group cohomology.
Let $G$ be a…
Kevin Buzzard
- 40,559
52
votes
3 answers
What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?
On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis holds. He doesn't give any clue as to the proof of…
David White
- 29,779
52
votes
8 answers
What are the external triumphs of matroid theory?
As a relatively new abstraction, matroids clearly enjoy a rich theory unto themselves and also offer a viewpoint that suggests interesting analogies and clarifies aspects of the foundations of venerable subjects.
All that said, a very harsh metric…
David Feldman
- 17,466
52
votes
1 answer
Open map D⁴ → S²
Is it possible to construct an embedding $D^4\hookrightarrow S^2\times
\mathbb R^2$
such that the projection $D^4\to S^2$ is an open map?
Here $D^n$ denotes closed $n$-ball.
An open map D⁴ → S².
It is easy to construct an embedding…
ε-δ
- 1,785
52
votes
9 answers
Is there a preferable convention for defining the wedge product?
There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)!}{k!l!}Alt(\alpha\otimes\beta)$,
where $\alpha$…
agt
- 4,246