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I am a postdoc in a field of mathematics A. Recently, I began talking with a professor that works in field B that is completely unrelated to A. He had a conjecture, and after a month or so, I was able to prove it. It is possible that this will be an important paper (although I am not sure if personal conjectures are as important as "known" or "famous" conjectures).

Will this affect my faculty job applications at all, assuming that I will be applying for positions in my original mathematical field? Will the committee just look upon it as a curiosity and then move on? Or does it provide some evidence of "well roundedness" that might help overall?

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    It will look good, especially if your paper is published by the time you apply for jobs. Jan 26 at 19:20
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    If my limited experience is any guide, it'll get counted as "Good paper in a good journal" and nothing more unless someone in the target dept. is specifically interested in the topic. But that's every paper. A paper in a month of work can hardly be a bad thing.
    – user176372
    Jan 26 at 19:23
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    @matilda Does it matter? It's not like you can trade it. You'll apply with the papers you've published in your CV, not with hypothetical other papers.
    – Bryan Krause
    Jan 26 at 19:37
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    "field B that is completely unrelated to A" Just as a side note: Are there indeed mathematical fields that are completely unrelated? Jan 27 at 8:26
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    Cherish it, Matilda! Such things happen in science all the time. One interdisciplinary area I have kept an eye on is the interplay of telecommunications and math. What happens frequently is that the engineers run into a practical problem, and think of some solution. When the problem is described to math people, they take over, and use their tools to improve upon the solutions (and/or the range of parameters where solutions exist). If you are lucky to be the right contact, just cherish it. It won't compromise your pure mathematician's soul to apply math like this occasionally. Jan 28 at 4:44

2 Answers 2

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Congratulations, this sounds very cool and I love stories like yours. The simple, literal-minded answer is that any good paper proving someone's conjecture will surely help your career.

However, I think what you are asking about is less about whether this paper will help you, and more about whether this paper will help you more or less than some other paper you could be writing if you invested the same amount of time working on your "original" field. In other words, this is a question about opportunity costs.

It's hard to give a definite answer, since you can find successful mathematicians who spend their whole careers studying a relatively well-defined topic, and you can find successful mathematicians who move around a lot and work on lots of different topics. And you can also find not-very-successful mathematicians of both of those categories.

Personally I think it's healthy to always be actively aspiring to learn more about other areas. This does take effort, which could potentially be more efficiently invested in obtaining new results in your "current" area, but in the longer term it yields dividends and helps you grow and mature as a mathematician and ultimately produce better research. However, one must be careful not to take this mindset to an unhealthy extreme - certainly some focus is required so you don't spread yourself too thin and are able to consistently publish good work, particularly in pure math where many mainstream areas of research require an intense level of focus and concentration to make progress on.

Another thing to consider is that how focused to be with your research may be a personal inclination tied to your personality, and that trying to fight your personal inclination is not necessarily a good idea, and may even be futile. In other words, even if perhaps there is a theoretically optimal level of focus one should aim for in order to maximize career success, you may discover that this is not a level of focus you are interested in, or are even capable of pursuing. (I speak from personal experience here.)

I'll end with a philosophical — almost mystical — thought: I believe that each mathematician has talents that are well-suited for some kinds of mathematics research, and less suited to other kinds. if you made an important discovery in field B even though your "official" research is in field A, one way of thinking about what's happening is that the universe is sending you a signal that you are the kind of person who can do really good work in field B, or even that you are more generally the kind of person who can do really good work in new fields that are far away from those you already mastered. I believe you should pay attention to such signals - they tell you something important about yourself, and by listening to them you can improve your intuition about what fields to pursue research in, what problems to work on etc. In the long run, this will lead you to become the best mathematician that you can be, and to maximal personal fulfillment. And hopefully to career success as well!

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I think it is a definite positive in more ways than one. Any good published paper is good on the CV, but in math, being known for some breadth of insight is an excellent qualification.

And, beyond the image itself, is the essence that insight in more than one subfield can bring insight to either and, beyond.

Assuming that you are published in a reputable journal, I can see no downside. Quite the contrary, being too narrow can be a detriment and leaves some doors closed that might be opened.

Paul Erdős, for example, was open to ideas in many fields. Nobody much thought of him as "unfocused".

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    Ha, no, actually everyone thinks that Erdos is sort of the prototype of the unfocused mathematician. He never stuck with a project, usually didn't do the writing and sorting out the details, and moved on to the next exciting thing. What set him apart is that he was good enough to get away with it :-) Jan 27 at 1:36

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