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EDIT: My question in the title was motivated by the following situation.

A mathematician from my field has solved a problem in this field which was considered to be important. In fact he even was awarded an international prize for this work. To solve this problem he has used methods from a completely different field of math which are not known to experts in his original field.

He managed to use these methods to solve few other open problems in the field, although less known than the first one. However he has failed to get other people to be interested in these tools since they require a really different education. For the same reason he cannot suggest such kind of problems to his students since they also would need to spend a lot of time to master the background from different fields, but even if they do, they would probably feel isolated.

Thus I am wondering to what extend this situation is typical, and whether there are other examples of mathematical works whose interdisciplinary character played a negative role in one's career.

  • I think it played an extremely positive role in the career you describe - at first. After all, it resulted in an international prize. However, things probably derailed a bit when he expected other people to want to learn his ultra-niche technique. The narrow applicability of the technique is demonstrated by the mathematician's own inability to apply it to other significant problems. It's great to do something interdisciplinary, but not great to try and shove it down other people's throats, who might not be interested in all of the involved disciplines. – DepressedDaniel Feb 5 '17 at 18:16
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    Your question doesn't add up to me. First: "In fact he was even awarded an international prize for this work." And then: "However he has failed to get other people to be interested in these tools[.]" I don't get it: receiving an international prize is close to the highest possible level of interest. Are you saying that the international mathematical community is very excited about his solution to the problem but no one cares about the tools he developed to solve it? Doesn't sound like the international mathematical community that I know. – Pete L. Clark Feb 5 '17 at 18:39
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    That helps, but I still don't quite understand. For instance, you suggest that he cannot suggest problems to his students, but I don't understand why. Most math students have to spend years learning background necessary to work on the problems posed to them by their advisor. So your colleague seems to be in a wonderful position: almost singlehandedly, he has expertise in multiple fields, which has already led to the solution of multiple problems, one of them very important, for which he won an international prize. I would expect him to be beating away potential students with a stick. – Pete L. Clark Feb 5 '17 at 20:02
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    When a mathematical subfield experiences an influx of ideas and techniques from a different subfield, your colleague's reaction is the most standard short-term reaction of researchers in the field. Students have the advantage here precisely because they can afford to spend several years learning the areas without publications. – Pete L. Clark Feb 5 '17 at 20:24
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    Also, in my experience, most areas of mathematics require students to "learn several areas": e.g. in order to do number theory (my field) a student must learn some amounts of: real analysis, complex analysis, finite group theory, Lie theory, commutative algebra, noncommutative algebra, homological algebra, algebraic geometry, differential geometry, algebraic topology....Most students spend five or six years in a PhD program taking several courses a semester. That's a lot of time to learn a lot of things. – Pete L. Clark Feb 5 '17 at 20:26
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It is clear that knowledge in other fields can be helpful, especially if one has an unusual combination of expertise such that one can see things others would miss even on well-studied problems. In fact, the variety of thoughts between colleagues of different background makes collaboration in math-related fields so much exciting.

It is hard to find examples where interdisciplinarity played a negative role in one's career, just for the same reason we don't learn about most of the useless approaches in mathematics (definitions that turned out to be useless, in particular not well-defined, propositions that turned out not to help for the problem they were thought for, etc.).

But from common sense I think that interdisciplinarity can never be negative unless you "cross paradigms", for example by mixing different contradicting systems of axioms to solve the same problem. Fortunately, in mathematics it's (today) quite clear what is not allowed.

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