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1500 questions
83
votes
4 answers
Parallelizability of the Milnor's exotic spheres in dimension 7
Are the Milnor's seven dimensional exotic spheres parallelizable?
Hamed
- 1,226
83
votes
24 answers
Proof synopsis collection
I hate to keep going with the big lists, but the question about one-sentence summaries of topics/areas spurred this question...and I just can't help myself!
Definition (Fraleigh): A proof synopsis is a one or two sentence synopsis of a proof,…
Jon Bannon
- 6,987
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- 110
83
votes
0 answers
Which finite abelian groups aren't homotopy groups of spheres?
Someone asked me if all finite abelian groups arise as homotopy groups of spheres. I strongly doubted it, and I bet ten bucks that $\mathbb{Z}_5$ is not $\pi_k(S^n)$ for any $n,k$. But I don't know how to prove it's not.
Which finite abelian…
John Baez
- 21,373
83
votes
6 answers
What’s the etiquette on using diagrams that need color to be understood?
I’m working on a paper that makes heavy use of colorful diagrams to supplement the text. For most of these it would probably not be possible to create grayscale versions that convey the same information as effectively. I’m a bit worried about this…
GMB
- 1,379
83
votes
2 answers
Why is differential Galois theory not widely used?
E.R. Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. Whether it is of closed form or not. see)
My question…
Henry.L
- 7,951
83
votes
4 answers
How to find ICM talks?
I am very interested in reading some and skimming through the list of invited talks at the International Congress of Mathematicians. Since the proceedings contain talks supposedly by top experts in each area, even the list of invited talks would…
Hailong Dao
- 30,261
83
votes
10 answers
Why can't there be a general theory of nonlinear PDE?
Lawrence Evans wrote in discussing the work of Lions fils that
there is in truth no central core
theory of nonlinear partial
differential equations, nor can there
be. The sources of partial
differential equations are so many -
physical,…
Steve Huntsman
- 15,258
82
votes
18 answers
Contest problems with connections to deeper mathematics
I already posted this on math.stackexchange, but I'm also posting it here because I think that it might get more and better answers here! Hope this is okay.
We all know that problems from, for example, the IMO and the Putnam competition can…
Dedalus
- 1,071
82
votes
5 answers
how does one understand GRR? (Grothendieck Riemann Roch)
I tried to answer an earlier question as to uses of GRR, just from my reading, although i do not understand GRR. Today i tried to understand the possible idea behind GRR. After editing my answer accordingly, it occurred to me i was asking a…
roy smith
- 12,063
82
votes
6 answers
What is a cohomology theory (seriously)?
This question has bugged me for a long time. Is there a unifying concept behind everything that is called a "cohomology theory"?
I know that there exist generalized cohomology theories, Weil cohomology theories and perhaps one might include…
user717
- 5,153
82
votes
8 answers
What is the motivation for a vertex algebra?
The mathematical definition of a vertex algebra can be found here:
http://en.wikipedia.org/wiki/Vertex_operator_algebra
Historically, this object arose as an axiomatization of "vertex operators" in "conformal field theory" from physics; I don't know…
user332
- 3,878
82
votes
16 answers
Tools for long-distance collaboration
Background
In general, I am aware of four and a half methods of long-distance collaboration:
Telephone (including voice-chat, VOIP, etc.; anything that is voice based)
Text chat (chat room, IM, gchat, things like that)
E-mail (or other…
Willie Wong
- 37,551
82
votes
15 answers
Theorems that impeded progress
It may be that certain theorems, when proved true, counterintuitively retard
progress in certain domains. Lloyd Trefethen provides two examples:
Faber's Theorem on polynomial interpolation: Interpreted as saying that polynomial interpolants are…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
82
votes
17 answers
Examples of algorithms requiring deep mathematics to prove correctness
I am looking for examples of algorithms for which the proof of correctness requires deep mathematics ( far beyond what is covered in a normal computer science course).
I hope this is not too broad.
Gorka
- 1,825
82
votes
30 answers
Applications of the Chinese remainder theorem
As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel numbering for sequences...)
Do you know some other…
JoeCamel
- 121