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Results tagged with soft-question
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user 766
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
19
votes
Trichotomies in mathematics
After passing to a subsequence if necessary, a sequence of real numbers either (a) converges to a real number; (b) diverges to $+\infty$; or (c) diverges to $-\infty$. In a similar vein, a sequence …
- 4,713
53
votes
Colloquial catchy statements encoding serious mathematics
Complete disorder is impossible.
This is the standard way of summing up Ramsey theory in a succinct sentence (according to that Wikipedia article, the above quote is due to Motzkin).
- 4,713
91
votes
Colloquial catchy statements encoding serious mathematics
"Can you hear the shape of a drum?"
This was Kac's famous way of asking whether the shape of a two-dimensional domain could be reconstructed from the spectrum of the Laplacian on that domain. (The …
- 4,713
44
votes
Mathematicians who were late learners?-list
According to this Notices article, Raoul Bott was undistinguished in high school, but displayed impressive talent once he reached graduate school (though his thesis was actually in electrical engineer …
- 4,713
57
votes
What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?
This is a bit speculative, and perhaps too challenging for an undergraduate project, but I wonder if an AlphaGeometry type approach might be possible for the task of automatically upper bounding sums …
- 114k
26
votes
Oddities of evenness
The hairy ball theorem is only valid for even-dimensional spheres (or odd-dimensional ambient Euclidean spaces).
Similarly, the strong Huygens principle is only valid in odd-dimensional physical space …
- 114k
26
votes
Examples of conjectures that were widely believed to be true but later proved false
I believe that Fefferman's disproof in 1971 of the $L^p$ boundedness of disc multiplier for any $p \neq 2$ was considered a great surprise at the time; it showed that the classical result of Bochner a …
- 4,713
347
votes
Accepted
What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed t...
Mathematical notation in a given mathematical field $X$ is basically a correspondence
$$ \mathrm{Notation}: \{ \hbox{well-formed expressions}\} \to \{ \hbox{abstract objects in } X \}$$
between mathem …
- 114k
22
votes
Examples of "unsuccessful" theories with afterlives
(Converted from a comment to an answer as requested.)
Non-Euclidean geometry was initially developed in hopes of deriving the parallel postulate from the other axioms of Euclidean geometry, as can be …
- 114k
10
votes
Examples of problems where considering "discrete analogues" has provided insight or led to a...
In my paper
Tao, Terence, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dyn. Syst. 28, No. 2, 657-688 (2008). ZBL1181.37004.
I was able to settle a ques …
- 114k
9
votes
What is the most useful non-existing object of your field?
Non-trivial approximate subrings of ${\bf R}$ or of ${\bf F}_p$.
The existence of such objects is ruled out by a number of "sum-product theorems", a typical one of which asserts that given a subset $ …
- 114k
7
votes
Accepted
"Universal" differential identities
Let's work in two dimensions for notational simplicity. We claim that there is no non-trivial polynomial identity of the form
$$ P( f, f_x, f_y, f_{xx}, f_{xy}, \dots ) = 0$$
relating some finite num …
- 114k
24
votes
Accepted
Why is free probability a generalization of probability theory?
Quite a lot of questions here!
It is perhaps worth making a distinction between scalar classical probability theory - the study of scalar classical random variables - and more general classical proba …
- 114k
141
votes
Intuitive crutches for higher dimensional thinking
I can't help you much with high-dimensional topology - it's not my field, and I've not picked up the various tricks topologists use to get a grip on the subject - but when dealing with the geometry of …
194
votes
Thinking and Explaining
I find there is a world of difference between explaining things to a colleague, and explaining things to a close collaborator. With the latter, one really can communicate at the intuitive level, beca …
- 114k