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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.
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 …
325
votes
Awfully sophisticated proof for simple facts
An example that came up in my measure theory class today:
The harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ diverges, because otherwise the functions $f_n := \frac{1}{n} 1_{[0,n]}$ would be dominat …
252
votes
Examples of unexpected mathematical images
The third image below was certainly unexpected for my soon-to-be-collaborators, Emmanuel Candes and Justin Romberg. They started with a standard image in signal processing, the Logan-Shepp phantom:
…
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 …
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 …
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 …
86
votes
Mathematical habits of thought and action which would be of use to non-mathematicians
Here are some that came to mind:
Equivalence. Basically, the idea that two things can be functionally equivalent (or close to equivalent) even if they look very different (and conversely, that two t …
73
votes
Still Difficult After All These Years
Difficulty is not additive, and measuring the difficulty of proving a single result is not a good measure of the difficulty of understanding the body of work in a given field as a whole.
Suppose for …
68
votes
Accepted
Why are proofs so valuable, although we do not know that our axiom system is consistent?
If you like, you can view proofs of a statement in some formal system (e.g. ZFC) as a certificate that a counterexample cannot be found without demonstrating the inconsistency of ZFC, which would be a …
62
votes
Analogues of P vs. NP in the history of mathematics
This isn't an exact analogue to P != NP, in which two large classes exist and it is undecided whether they are equal or not; instead, two large "universes" exist, of which only one is the truth, with …
58
votes
Why are characters so well-behaved?
The trace is about the strongest general way we have to linearly project a non-abelian situation (matrices) to an abelian situation (scalars): tr(AB)=tr(BA). By using the trace, the representation th …
- 114k
58
votes
Accepted
Why do people use "formal calculation" to describe informal calculations?
Formal, adj. Relating to or involving outward form or structure, often in contrast to content or meaning.
In mathematics, a formal argument is one that manipulates the form of an expression with …
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
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).
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 …