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1500 questions
47
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0 answers
Concerning proofs from the axiom of choice that ℝ³ admits surprising geometrical decompositions: Can we prove there is no Borel decomposition?
This question follows up on a comment I made on Joseph O'Rourke's
recent question, one of several questions here on mathoverflow
concerning surprising geometric partitions of space using the axiom
of choice.
Joseph O'Rourke: Filling $\mathbb{R}^3$…
Joel David Hamkins
- 224,022
47
votes
6 answers
Non-enumerative proof that there are many derangements?
Recall that a derangement is a permutation $\pi: \{1,\ldots,n\} \to \{1,\ldots,n\}$ with no fixed points: $\pi(j) \neq j$ for all $j$. A classical application of the inclusion-exclusion principle tells us that out of all the $n!$ permutations, a…
Terry Tao
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47
votes
5 answers
Proof assistants for mathematics
This question is related to (maybe even the same in intent as) Intro to automatic theorem proving / logical foundations?, but none of the answers seem to address what I'm looking for.
There are a lot of resources available for people who want to use…
LSpice
- 11,423
47
votes
1 answer
Did Grothendieck have a plan for proving Riemann Existence algebraically?
A recent question reminded me of a question I've had in the back of my mind for a long time. It is said that Grothendieck wanted the center-piece of SGA1 to be a completely algebraic proof (without topology) of the following…
James D. Taylor
- 6,178
47
votes
1 answer
Which small finite simple groups are not yet known to be Galois groups over Q?
The subject line pretty much says it all. To expand just a little bit:
1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known to occur regularly over $\mathbb{Q}$; not known to…
Pete L. Clark
- 64,763
47
votes
1 answer
Do all exact $1 \to A \to A \times B \to B \to 1$ split for finite groups?
Let $A$, $B$ be finite groups. Is it true that all short exact sequences $1 \rightarrow A \rightarrow A \times B \rightarrow B \rightarrow 1$ split on the right?
In other words, do there exist finite groups $A$, $B$ and homomorphisms $f: A…
Dan Glasscock
- 859
47
votes
2 answers
Non isomorphic finite rings with isomorphic additive and multiplicative structure
About a year ago, a colleague asked me the following question:
Suppose $(R,+,\cdot)$ and $(S,\oplus,\odot)$ are two rings such that $(R,+)$ is isomorphic, as an abelian group, to $(S,\oplus)$, and $(R,\cdot)$ is isomorphic (as a semigroup/monoid)…
Arturo Magidin
- 7,012
47
votes
33 answers
What are the most overloaded words in mathematics?
This is community wiki. In each answer, please list one word at the top and below that list as many different meanings of that word in mathematics as you can think of, preferably with links or definitions. ("Adjective" and "adjective noun" count…
Qiaochu Yuan
- 114,941
47
votes
6 answers
True by accident (and therefore not amenable to proof)
The graph reconstruction conjecture claims that (barring trivial examples) a graph on n vertices is determined (up to isomorphism) by its collection of (n-1)-vertex induced subgraphs (again up to isomorphism).
The way it is phrased…
Gordon Royle
- 12,288
47
votes
4 answers
Why are planar graphs so exceptional?
As compared to classes of graphs embeddable in other surfaces.
Some ways in which they're exceptional:
Mac Lane's and Whitney's criteria are algebraic characterizations of planar graphs. (Well, mostly algebraic in the former case.) Before writing…
Harrison Brown
- 12,543
47
votes
6 answers
How do we study Iwasawa theory?
What papers should we read to start? What basic knowledge do we need to understand the question? What is this area really about? And what are people researching on it?
basic
- 473
47
votes
8 answers
How should one think about pushforward in cohomology?
Suppose f:X→Y. If I decorate that first sentence with appropriate adjectives, then I get a pushforward map in cohomology H*(X)→H*(Y).
For example, suppose that X and Y are oriented manifolds, and f is a submersion. Then such a pushforward map…
Peter McNamara
- 8,726
47
votes
7 answers
Classification of (compact) Lie groups
I would like to study/understand the (complete) classification of compact lie groups. I know there are a lot of books on this subject, but I'd like to hear what's the best route I can follow (in your opinion, obviously), since there are a lot of…
Gian Maria Dall'Ara
- 2,449
47
votes
10 answers
Possibility of an Elementary Differential Geometry Course
I have to admit I'm not sure if this is an appropriate question. It's related to research in math education, but not directly to math.
I've found that in talking to professional physicists and engineers, most of them find some use for differential…
Logan M
- 1,063
47
votes
3 answers
Testing whether an integer is the sum of two squares
Is there a fast (probabilistic or deterministic) algorithm for determining whether an integer $n$ is a sum of two squares?
By "fast" here I mean polynomial time (i.e. time $O((\log n)^{O(1)})$). Note that I am interested only in whether the integer…
H A Helfgott
- 19,290