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My question is about "soft-skills", i.e. knowing how to handle Latex, giving a speech or giving a lecture, or other knowledge that could be useful for a career in mathematics.

Skills that (all) mathematicians need, but which are not necessarily something that one learns studying the mathematics of text books or lectures.

This knowledge should therefore benefit mathematicians of varying disciplines and not only the algebraic topologists or the number theorists.

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    The term "soft skills" usually refers to interpersonal skills. Things like LaTeX and general writing skills would usually be excluded. All of these are of course useful. Furthermore your premise is not entirely correct: all these skills can be (at least partly) learned from books --- there are excellent books on LaTeX, general writing, presentation skills, and even on how to make friends. – Boris Bukh Jun 29 at 19:45
  • That's the reason why I put it in quotes. I could not think of a more suitable term, so I explained the thing that I am looking for more clearly in the question body. – Danijar Dreger Jun 29 at 19:50
  • " Studying the mathematics of text books or lectures". I didn't exclude learning skills from books, but I excluded specific mathematical knowledge. @BorisBukh – Danijar Dreger Jun 29 at 19:51
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A few that come to mind, and are probably valid in other fields also --

  • Learn how to give good talks.

  • Learn how to describe your research informally. Say you meet someone at a conference, and they ask you "What do you work on?" Can you convince them that what you work on is interesting?

  • Learn how to come up with interesting questions that you don't know how to answer.

  • Learn how to read a paper. Can you figure out "the big idea" without getting bogged down in the technicalities?

  • Learn how to meaningfully participate in a math discussion when you're less than 100% sure what's going on.

  • Develop a gut feeling for when a subject will come in useful. For example, "This lemma feels like I could use algebraic geometry to prove it", without knowing initially what that form would take.

  • Get a sense for what other researchers in your field consider interesting. What is worth writing a paper about, and what is worth giving a talk about?

There are many, many more -- and none of the above are easy.

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    "how to meaningfully participate in a math discussion when you're less than 100% sure what's going on" I would put it more bluntly: learn to ask questions when you don't understand something instead of nodding with a clever face while not having the slightest clue on what the other person is talking about. The best time to acquire this skill is when you are still a graduate student, but it is never too late :-) – fedja Jun 30 at 11:21
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    @fedja: I certainly agree that's important when you understand 0%. I was actually referring to when you understand some, but not all of what the other person is saying. I've often found it productive to keep the discussion going -- I get some idea for the big picture, and can fill in details later on my own. – academic Jun 30 at 13:10
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Understanding the research process and how to overcome hurdles and frustrations. Some of these points have already been addressed in other answers, but I think an explicit list of things around this is good. This means:

  • Knowing how to find interesting research questions.

  • Knowing how to get started on the research process.

  • Understanding what it means to put time and effort into solving a research problem.

  • Knowing how to find collaborators that would be interested in working with you, and then working with them in a productive manner.

... and many other things

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