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1500 questions
73
votes
10 answers

Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of x,y, assuming that x and y are sufficiently…
Terry Tao
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73
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9 answers

What is Lagrange Inversion good for?

I am planning an introductory combinatorics course (mixed grad-undergrad) and am trying to decide whether it is worth budgeting a day for Lagrange inversion. The reason I hesitate is that I know of very few applications for it -- basically just…
David E Speyer
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73
votes
3 answers

Can analysis detect torsion in cohomology?

Take, for example, the Klein bottle K. Its de Rham cohomology with coefficients in $\mathbb{R}$ is $\mathbb{R}$ in dimension 1, while its singular cohomology with coefficients in $\mathbb{Z}$ is $\mathbb{Z} \times \mathbb{Z}_2$ in dimension 1. It…
Paul Siegel
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73
votes
1 answer

Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here. $\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism (complex…
mme
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73
votes
4 answers

What's the "Yoga of Motives"?

There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new algebraico-geometric conjectures just to formulate the definition of…
Ilya Nikokoshev
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73
votes
17 answers

Mathematical research published in the form of poems

The article Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen, Math. Z. 127 (1972), no. 1, 10-16 is written in the form of a lengthy poem, in a style similar to that of the works of Wilhelm Busch. Are there any other examples of…
Stefan Kohl
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73
votes
2 answers

The inverse Galois problem and the Monster

I have a slight interest in both the inverse Galois problem and in the Monster group. I learned some time ago that all of the sporadic simple groups, with the exception of the Mathieu group $M_{23}$, have been proven to be Galois groups over…
aorq
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73
votes
2 answers

Please check my 6-line proof of Fermat's Last Theorem.

Kidding, kidding. But I do have a question about an $n$-line outline of a proof of the first case of FLT, with $n$ relatively small. Here's a result of Eichler (remark after Theorem 6.23 in Washington's Cyclotomic Fields): If $p$ is prime and the…
Cam McLeman
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73
votes
10 answers

Intuition for Group Cohomology

I'm beginning to learn cohomology for cyclic groups in preparation for use in the proofs of global class field theory (using ideal-theoretic arguments). I've seen the proof of the long exact sequence and of basic properties of the Herbrand quotient,…
David Corwin
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72
votes
5 answers

Is there an intuitive reason for Zariski's main theorem?

Zariski's main theorem has many guises, and so I will give you the freedom to pick the one that you find to be most intuitive. For the sake of completeness, I will put here one version: Zariski's main theorem: Let $f:X\rightarrow Y$ be a…
James D. Taylor
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72
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6 answers

A bestiary of topologies on Sch

The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I think. Nisnevich? pff, not a chance. There are various ideas about…
David Roberts
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72
votes
31 answers

Can infinity shorten proofs a lot?

I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein…
gowers
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72
votes
3 answers

Has the mathematical content of Grothendieck's "Récoltes et Semailles" been used?

This question is partly motivated by Never appeared forthcoming papers. Motivation Grothendieck's "Récoltes et Semailles" has been cited on various occasions on this forum. See for instance the answers to Good papers/books/essays about the thought…
Jonathan Chiche
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72
votes
7 answers

When should you, and should you not, share your mathematical ideas?

Perhaps this is a biased forum for this "downer" question, but I've never figured this out. At what stage is it safe to share mathematical ideas? Many have told me that there is serious danger of ideas being stolen if shared in a premature form, but…
Jon Bannon
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8 answers

What to do after a pure math academic path?

I don't know whether my question is in the appropriate place. I've studied physics, and then did a PhD in (pure) math and 2 postdocs. I definitely love math research, but I am not ready to apply all over the world hoping to find a position somewhere…
coco
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