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Results tagged with soft-question
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user 2926
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.
29
votes
Is a come back to mathematical research possible?
I hesitate to posit myself as an example, but I was out of academia from 2001 to 2019, when I decided to become a stay-at-home dad while my wife became the breadwinner. (I won't go into the details of …
3
votes
What's a great christmas present for someone with a PhD in Mathematics?
A book of interviews of famous mathematicians could be good. I have in mind particularly More Mathematical People, which I've gotten a lot of mileage from here at MathOverflow.
6
votes
Would it be simpler, pedagogically speaking, if textbook writers introduced root systems as ...
You might find useful certain aspects of Alissa Crans's thesis,
Alissa Crans, Lie 2-Algebras, Dissertation, U. Cal. Riverside, 2004 (link).
For instances, Lie groups and their associated Lie algebra …
- 53.3k
16
votes
A map of non-pathological topology?
I'll go ahead and say that Polish spaces are an interesting and almost sui generis class. There is a rich literature of applications to and from descriptive set theory, with layers of "pathology" hier …
- 53.3k
9
votes
Examples of notably long or difficult proofs that only improve upon existing results by a sm...
The following example is described in The Man Who Loved Only Numbers by Paul Hoffman.
There is a reasonably short proof, found by Esther Klein (later Szekeres) in 1932, that given 5 points in the pl …
4
votes
Accepted
Technical term for representing object of a presheaf determined by a left-adjoint?
As requested, I'll turn my comments into an answer. There were two questions, the first being what we call the representing object if a presheaf $c \mapsto \mathcal{D}(F c, d)$ is representable, and t …
- 53.3k
14
votes
Problems where we can't make a canonical choice, solved by looking at all choices at once
General topology as found in textbooks seems to be chock-full of examples where the axiom of choice seems to be (unconsciously?) invoked, and unnecessarily if one follows Eilenberg's advice to avoid s …
12
votes
What is your favorite proof of Tychonoff's Theorem?
I won't swear it's my absolute favorite, but today I learned of a nice proof due to Clementino and Tholen who take as their starting point the closed-projection characterization of compactness, viz. t …
4
votes
Locales and Topology.
As for question 2.: it's hardly recent, but since I didn't see it mentioned in this thread, let me mention that the Tychonoff theorem for locales does not require the axiom of choice and is a piece of …
19
votes
Why is Set, and not Rel, so ubiquitous in mathematics?
Taking up remarks near the end of the OP, and somewhat in line with Mike Shulman's answer, I'd like to underline the structural interplay between $\mathbf{Set}$ and $\mathbf{Rel}$ to indicate one poin …
19
votes
Applications of the Cayley-Hamilton theorem
Cayley-Hamilton can be useful in commutative algebra. Related to its close connection with Nakayama's lemma as mentioned in a comment by Qiaochu (see also Wikipedia), see for example the development g …
6
votes
Categories of finite objects
I would say you could make good headway on this by looking over some of the research projects of Tom Leinster, Mark Meckes, and Simon Willerton (and others I may be forgetting), centering on various n …
12
votes
Parodies of abstruse mathematical writing
Of course, there's this old classic: http://bjornsmaths.blogspot.com/2005/11/how-to-catch-lion-in-sahara-desert.html
17
votes
Examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differe...
One way to understand $l^1(X)$ for a set $X$ with counting measure is that $l^1(-): Set \to Ban$ provides a left adjoint to the functor $\hom(k, -): Ban \to Set$. Here $k$ is the ground field and $Ban …
0
votes
Obscure Names in Mathematics
The Hauptsatz (due to Gentzen).