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1500 questions
64
votes
7 answers
Is Thompson's Group F amenable?
Last year a paper on the arXiv (Akhmedov) claimed that Thompson's group $F$ is not amenable, while another paper, published in the journal "Infinite dimensional analysis, quantum probability, and related topics" (vol. 12, p173-191) by Shavgulidze…
ADL
- 2,762
64
votes
8 answers
Example of a good Zero Knowledge Proof
I am working on my zero knowledge proofs and I am looking for a good example of a real world proof of this type. An even better answer would be a Zero Knowledge Proof that shows the statement isn't true.
George
- 659
64
votes
2 answers
What is descent theory?
I read the article in wikipedia, but I didn't find it totally illuminating. As far as I've understood, essentially you have a morphism (in some probably geometrical category) $Y \rightarrow X$, where you interpret $Y$ as being the "disjoint union"…
Qfwfq
- 22,715
64
votes
6 answers
Are dagger categories truly evil?
Recall that a dagger category is a category equipped with an involution $*:Hom(x,y)\to Hom(y,x)$ that satisfies $f^{**}=f$ and $f^* g^*=(gf)^*$. A prominent example of a dagger category is the category of Hilbert spaces and continuous linear…
André Henriques
- 42,480
64
votes
16 answers
How helpful is non-standard analysis?
So, I can understand how non-standard analysis is better than standard analysis in that some proofs become simplified, and infinitesimals are somehow more intuitive to grasp than epsilon-delta arguments (both these points are debatable).
However,…
Tony Huynh
- 31,500
64
votes
12 answers
Why don't more mathematicians improve Wikipedia articles?
Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. Here is a list of the 500 most popular math articles. The number of regular Wikipedia readers is…
Mark M
- 111
64
votes
5 answers
The unreasonable effectiveness of Padé approximation
I am trying to get an intuitive feel for why Padé approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence.
But what I can't grasp is how it manages to approximate the original…
Felix Goldberg
- 6,872
64
votes
2 answers
Where are the second- (and third-)generation proofs of the classification of finite simple groups up to?
According the the Wikipedia page, the second generation proof is up to at least nine volumes: six by Gorenstein, Lyons and Solomon dated 1994-2005, two covering the quasithin business by Aschbacher and Smith in 2004, and one by Aschbacher, Lyons,…
David Roberts
- 33,851
64
votes
3 answers
Forcing as a new chapter of Galois Theory?
There is a (very) long essay by Grothendieck with the ominous title La Longue Marche à travers la théorie de Galois (The Long March through Galois Theory). As usual, Grothendieck knew what he was talking about: Galois Theory, far from being confined…
Mirco A. Mannucci
- 7,603
63
votes
1 answer
Smooth proper scheme over Z
Does every smooth proper morphism $X \to \operatorname{Spec} \mathbf{Z}$ with $X$ nonempty have a section?
EDIT [Bjorn gave additional information in a comment below, which I am recopying here. -- Pete L. Clark]
Here are some special cases,…
Bjorn Poonen
- 23,617
63
votes
3 answers
Is there a "Basic Number Theory" for elliptic curves?
Tate's thesis showed how to profitably analyze $\zeta$ functions of number fields in terms of adelic points on the multiplicative group. In particular, combining Fourier analysis and topology, Tate gave new and cleaner proofs of the finiteness of…
David E Speyer
- 150,821
63
votes
16 answers
What is the high-concept explanation on why real numbers are useful in number theory?
The utopian situation in mathematics would be that the statement and the proof of every result would live "in the same world", at the same level of mathematical complexity (in a broad sense), unless there were a good conceptual reason for the…
Daniel Moskovich
- 21,887
63
votes
4 answers
When is the product of two ideals equal to their intersection?
Consider a ring $A$ and an affine scheme $X=\operatorname{Spec}A$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds to the union $V(I\cap J)=V(I)\cup V(J)$. But a…
evgeniamerkulova
- 894
63
votes
8 answers
Fair but irregular polyhedral dice
I am interested in determining a collection of geometric conditions that will guarantee that a convex polyhedron
of $n$ faces is a fair die in the sense that, upon random rolling, it has an equal $1/n$ probability of
landing on each of its…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
63
votes
14 answers
Unnecessary uses of the axiom of choice
What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?
I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and…
Tom Leinster
- 27,167