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Results tagged with big-list
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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.
170
votes
Most memorable titles
The flattering lie You Could Have Invented Spectral Sequences by Timothy Y. Chow.
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 …
105
votes
Not especially famous, long-open problems which anyone can understand
Is $e+\pi $ rational?
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 …
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
70
votes
What elementary problems can you solve with schemes?
If $I,J \subset A$ are comaximal ideals in a commutative ring $A$, i.e. $I+J=A$, then
for all $n,m \in \mathbb N$ the ideals $I^n$ and $J^m$ are also comaximal.
Proof: $\emptyset= V(A)=V(I+J)=V(I)\ …
53
votes
Pseudonyms of famous mathematicians
Rainich=Rabinowitsch (of trick fame : cf. Nullstellensatz).
Here is an anecdote related by Bruce P. Palka, Editor of American Mathematical Monthly
in Vol.111 (2004) of that journal (page460).
Rai …
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!
48
votes
Errata for Atiyah–Macdonald
Dear Tim, on page 31 they consider a ring $A$ and two $A$- algebras defined by their structural ring morphisms $f:A\to B$ and $g:A\to C$. They then define the tensor product as a ring $D=B\otimes _A C …
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 $n$.
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 …
38
votes
Accepted
Justifying a theory by a seemingly unrelated example
[In front of a blackboard, in an office at Real College]
Skeptic: And why should I care about holomorphic functions?
Holomorphic enthusiast:$\;$ Can you compute $\quad$ $\sum_{n={-\infty}}^{\infty …
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 …
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' …
16
votes
Justifying a theory by a seemingly unrelated example
Let us call "division algebra over $\mathbb R$" a finite-dimensional vector space $A$ equipped with a bilinear map $A \times A \to A: (a,b) \mapsto a \bullet b$ , such that $a\bullet b=0$ implies $a= …