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1500 questions
66
votes
5 answers
Heuristic argument that finite simple groups _ought_ to be "classifiable"?
Obviously there exists a list of the finite simple groups, but why should it be a nice list, one that you can write down?
Solomon's AMS article goes some way toward a historical / technical explanation of how work on the proof proceeded. But,…
Tim Campion
- 60,951
66
votes
6 answers
Why is the Fourier transform so ubiquitous?
Many operations and equivalences in mathematics arise as some sort of Fourier transform. By Fourier transform I mean the following:
Let $X$ and $Y$ be two objects of some category with products, and consider the correspondence $X \leftarrow X \times…
Exit path
- 2,969
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66
votes
1 answer
Why is the Eisenstein ideal paper so great?
I am currently trying to decipher Mazur's Eisenstein ideal paper (not a comment about his clarity, rather about my current abilities). One of the reasons I am doing that is that many people told me that the paper was somehow revolutionary and…
user140761
66
votes
4 answers
Why did Voevodsky consider categories "posets in the next dimension", and groupoids the correct generalisation of sets?
Earlier today, I stumbled upon this article written by V. Voevodsky about the "philosophy" behind the Univalent Foundations program. I had read it before around the time of his passing, and one passage that I remember vividly is this, for which I…
Soham Chowdhury
- 755
66
votes
4 answers
What's there to do in category theory?
Disclaimer: I posted this question on MSE only a few days ago; and received very few comments. I know that the etiquette is to wait a bit more than that before moving a post from MSE to MO, but I figured that posting it on MO would be an actual…
Maxime Ramzi
- 13,453
66
votes
9 answers
Axiom of choice, Banach-Tarski and reality
The following is not a proper mathematical question but more of a metamathematical one. I hope it is nonetheless appropriate for this site.
One of the non-obvious consequences of the axiom of choice is the Banach-Tarski paradox and thus the…
ThiKu
- 10,266
66
votes
17 answers
Shortest/Most elegant proof for $L(1,\chi)\neq 0$
Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$?
Background: The non-vanishing of $L(1,\chi)$ plays an essential role in the…
M.G.
- 6,683
66
votes
41 answers
Major mathematical advances past age fifty
From A Mathematician’s Apology, G. H. Hardy, 1940:
"I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than…
Halfdan Faber
- 975
66
votes
9 answers
What are some important but still unsolved problems in mathematical logic?
In the past, first-order logic and its completeness and whether arithmetic is complete was a major unsolved issues in logic . All of these problems were solved by Godel. Later on, independence of main controversial axioms were established by…
FNH
- 329
66
votes
1 answer
Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?
The infamous K3 surface has many constructions in many fields ranging from algebraic geometry to algebraic topology. Its many properties are well known. For this question I am really interested in the K3 surface from a algebraic topology perspective…
Chris Schommer-Pries
- 27,144
66
votes
1 answer
Why can't a nonabelian group be 75% abelian?
This question asks for intuition, not a proof.
An earlier question,
Measures of non-abelian-ness
was thoroughly answered by Arturo Magidin.
A paper by Gustafson1
proves that, for a nonabelian group,
the probability that two randomly…
Joseph O'Rourke
- 149,182
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66
votes
3 answers
Does linearization of categories reflect isomorphism?
Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between two objects $x$ and $y$ is the free $R$-module…
Tilman
- 6,042
65
votes
4 answers
How Does My Radio Work?
Bear with me for a moment while I invoke the real world; the main question at the end is purely mathematical.
I live in an area with $n$ AM radio stations and $m$ FM radio stations.
AM station number $j$ wants to send me the signal $\phi_j(t)$. …
Steven Landsburg
- 22,477
65
votes
13 answers
Video lectures for algebraic geometry
Are there any good video lectures for studying algebraic geometry?
john
- 1,257
65
votes
9 answers
List of Classifying Spaces and Covers
I am looking for a list of classifying spaces $BG$ of groups $G$ (discrete and/or topological) along with associated covers $EG$; there does not seem to be such cataloging on the web. Or if not a list, just some further fundamental examples. For…
Chris Gerig
- 17,130