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Results tagged with big-list
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user 450
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
15
votes
Consolidation: Aftermathematics of fads
In the seventies and eighties of the preceding century, existence and classification of vector bundles on projective space $\mathbb P^n$ were all the rage, with contributions from such luminaries as A …
159
votes
What are some examples of colorful language in serious mathematics papers?
Andre Weil (Oeuvres, vol. 2, page 558) purporting to be R.Lipschitz writing from Hades:
"Unfortunately, it appears that there is now in your world a race of
vampires, called referees, who clamp down m …
2
votes
Interesting examples of vacuous / void entities
A The empty set is a covering map of any topological space. More generally, a covering map needn't be surjective (although many books claim just that). For example the inclusion of a closed and open …
34
votes
Theorems for nothing (and the proofs for free)
Wedderburn's theorem: "Every finite division ring is a field."
This is really astonishing if you think of quaternions: nothing analogous in the finite case.
Then of course the classification of finit …
73
votes
Accepted
Cardinalities larger than the continuum in areas besides set theory
The Zariski tangent space at any point of a positive dimensional $C^1$-manifold $X$ has dimension $2^{2^{\aleph_0}}= 2^{\frak c}$.
Let me explain in the case when $X=\mathbb R$.
Consider the ring $C^ …
- 47.5k
52
votes
Fundamental problems whose solution seems completely out of reach
Is every algebraic curve in $\mathbb P^3$ the set-theoretic intersection of two algebraic surfaces ? Not known!
3
votes
Individual mathematical objects whose study amounts to a (sub)discipline?
$SL_2\mathbb R$ and its evil universal covering.
20
votes
What should be learned in a first serious schemes course?
Dear Ravi, here is a small suggestion. I think one might emphasize as soon as possible that the subschemes of an affine scheme $Spec A$ exactly correspond to the set of ideals of the ring $A$.( I don' …
43
votes
What should be learned in a first serious schemes course?
Since in 2007-2008 you evoked [ Class 24, §1.8, The problem with locally free sheaves] the equivalence between locally free sheaves and vector bundles on a scheme, the following point, potentially co …
88
votes
Widely accepted mathematical results that were later shown to be wrong?
In 1882 Kronecker proved that every algebraic subset in $\mathbb P^n$ can be cut out by $n+1$ polynomial equations.
In 1891 Vahlen asserted that the result was best possible by exhibiting
a curve in …
10
votes
Any reference on multilinear algebra
Here are three excellent books.
Tensor Spaces and Exterior Algebra by Takeo Yokonuma.
Translations of Mathematical Monographs, volume 108, AMS 1992
You can browse it in Google books here
Laurent Sc …
16
votes
Proof synopsis collection
Fermat's little theorem: $n^p\equiv n \; (mod \;p)$ for $p$ prime and all integers $n$.
Synopsis of proof: Reduce to nontrivial case where $p$ doesn't divide $n$, interpret as equality in field of $p …
44
votes
Theorems that are 'obvious' but hard to prove
That $\mathbb R^n$ has topological dimension $n$. In a similar vein that affine space $\mathbb A^n_k$ over a field $k$ has Zariski dimension $n$.
13
votes
What should be learned in a first serious schemes course?
If you decide to teach a more arithmetically flavoured algebraic geometry, students should be made aware that schemes over a ring $A$ are stranger than they might think.
For example $A$-rational poin …
105
votes
Not especially famous, long-open problems which anyone can understand
Is $e+\pi $ rational?