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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.
0
votes
Obscure Names in Mathematics
The Hauptsatz (due to Gentzen).
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.
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 …
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 …
23
votes
'Category-theory'-free areas of pure math, 'category-theory'-loaded areas of applied math
I would upvote Keerthi's comment multiple times if I could. Just find an area of mathematics that makes you smile and brings you happiness. If in the course of doing research you find that you need to …
15
votes
Categorification request
I tried to discuss this geometric series example of categorification in one of my answers to another MO question by Jan Weidner, here. I can't tell whether this reply was considered unsatisfactory, bu …
- 53.3k
14
votes
What are some examples of colorful language in serious mathematics papers?
From Jim Stasheff's Homotopy Associativity of H-spaces I, the magisterial-sounding
To study spaces which admit $A_n$-structures, we can work directly with the maps…. In the case of a topological gro …
3
votes
Major mathematical advances past age fifty
Charles Sanders Peirce (born 1839) explicitly declared his Existential Graphs (all three parts: Alpha, Beta, and Gamma) to be his chef d'oeuvre. This work on graphical logic began sometime in the earl …
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
Are there any "related rates" calculus problems that don't feel contrived?
Here's something you could try, based on a passage from Richard Feynman's "Surely You're Joking, Mr. Feynman!":
When I was in high school, I'd see water running out of a faucet growing narrower, …
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 …
8
votes
Individual mathematical objects whose study amounts to a (sub)discipline?
The braid group.
The Monster group.
The Steenrod algebra.
The representation ring of the symmetric group.
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
8
votes
Accepted
Concise definition of subobjects
Of course it's not necessary to make this identification, but it's fairly harmless since the groupoid of monomorphisms into an object $X$ is equivalent to the discrete category of subobjects, and it c …
- 53.3k
13
votes
Where is number theory used in the rest of mathematics?
Julia Robinson proved that the theory of fields is undecidable by showing that the natural numbers form a subset of the rationals definable by a first-order formula in the language of fields. The cons …