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1500 questions
61
votes
10 answers
Teaching proofs in the era of Google
Dear members,
Way back in the stone age when I was an undergraduate (the mid 90's), the internet was a germinal thing and that consisted of not much more than e-mail, ftp and the unix "talk" command (as far as I can remember). HTML and web-pages…
Ryan Budney
- 43,013
61
votes
10 answers
Should I not cite an arxiv.org paper which otherwise seems to be unpublished?
I want to cite a paper which is on arxiv.org but is not published or reviewed anywhere, and no publication or review seems to be in the pipeline. Would citing this arxiv.org paper be bad? Should I wait for a paper to be peer reviewed before I cite…
ohai
- 177
61
votes
5 answers
Intuitively, what does a graph Laplacian represent?
Recently I saw an MO post Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs? that got me interested. It is about a graph parameter that is derived from the…
GraphX
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61
votes
5 answers
Bourbaki's definition of the number 1
According to a polemical article by Adrian Mathias, Robert Solovay showed that Bourbaki's definition of the number 1, written out using the formalism in the 1970 edition of Théorie des Ensembles,…
John Baez
- 21,373
61
votes
2 answers
The topological analog of flatness?
Recall that a map $f:X\to Y$ of schemes is called flat iff for any $x\in X$ the ring $O_{X,x}$ is a flat $O_{Y,f(x)}$-module.
Briefly the question is: what is the topological analog of this?
Many notions and constructions in scheme theory have…
algori
- 23,231
61
votes
3 answers
How did Lefschetz do mathematics without hands?
If people think this is the wrong forum for this question, I'll cheerfully take it elsewhere.
But: How did Solomon Lefschetz do mathematics with no hands?
Presumably there was an amanuensis to whom he dictated his papers, and then dictated his…
Steven Landsburg
- 22,477
61
votes
14 answers
What are some of the big open problems in 3-manifold theory?
From what I understand, the geometrization theorem and its proof helped to settle a lot of outstanding questions about the geometry and topology of 3-manifolds, but there still seems to be quite a lot of activity. I am not prepared to make a full…
Paul Siegel
- 28,772
61
votes
11 answers
Geometric proof of the Vandermonde determinant?
The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq i < j \leq n} (x_j-x_i).$$
There are many known…
Daniel Litt
- 22,187
61
votes
3 answers
Why is there no Cayley's Theorem for rings?
Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a technical theorem as a glorious wellspring of…
Tom Boardman
- 3,190
61
votes
7 answers
Why is the Hahn-Banach theorem so important?
Every time I hear it mentioned it is praised in the highest possible terms, and I remember one of my old lecturers saying that it is one of the 3 most important theorems in analysis. Yet the only consequences of it that I have read is that it proves…
teil
- 4,261
61
votes
4 answers
Drawing of the eight Thurston geometries?
Do you know of a picture, drawing, or other concise visual representation of the eight three-dimensional Thurston geometries?
I am imagining something akin to the standard picture (of a sphere, plane, and saddle) used to illustrate the three…
cdouglas
- 3,083
61
votes
8 answers
Physical meaning of the Lebesgue measure
Question (informal)
Is there an empirically verifiable scientific experiment that can empirically confirm that the Lebesgue measure has physical meaning beyond what can be obtained using just the Jordan measure? Specifically, is there a Jordan…
user21820
- 2,733
61
votes
3 answers
Atiyah-Singer theorem-a big picture
So far I made several attempts to really learn Atiyah-Singer theorem. In order
to really understand this result a rather broad background is required: you need
to know analysis (pseudodifferential operators), algebra (Clifford algebras,
spin…
truebaran
- 9,140
61
votes
1 answer
A dictionary of Characteristic classes and obstructions
I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians.
In an effort to get a hold of characteristic classes I had the idea…
Saal Hardali
- 7,549
61
votes
1 answer
Are there $n$ groups of order $n$ for some $n>1$?
Given a positive integer $n$, let $N(n)$ denote the number of groups of order $n$, up to isomorphism.
Question: Does $N(n)=n$ hold for some $n>1$?
I checked the OEIS-sequence https://oeis.org/A000001 as well as the squarefree numbers in the range…
Peter
- 1,203