I am wondering if having much knowledge in other (seemingly not directly related) branches of mathematics is helpful to a mathematician in his own research. Say there are two algebraic topologists A and B. A also has some knowledge in PDE (say he knows the basics of graduate level PDE well and can even teach a graduate course on PDE) and B does not even know the very basics of ODE. Does this make any difference in their ability in doing research in algebraic topology?

I know that some fields are somehow related; for example, it will help a group theorist if he knows algebraic topology or number theory; but how about other fields that are not apparently related, such as the example that I have quoted?

  • 3
    IMO this is better for MathOverflow.
    – xuq01
    Dec 11 '18 at 18:49
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    Your question is literally about if something is "helpful to a mathematician in his own research", so I'd say this sounds awefully like being about research level mathematics.
    – mlk
    Dec 11 '18 at 19:27
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    @Zuriel There's a label called "soft-question" there for questions not directly about the details of mathematics but of concern to mathematicians. There are plenty of questions like yours under that label :-) I'm afraid this question is the perfect fit for MO.
    – xuq01
    Dec 11 '18 at 19:53
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    Applications of PDE in algebraic topology can even win you a Fields medal. Dec 11 '18 at 20:20
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    @NateEldredge or two Dec 11 '18 at 20:55

The answer is yes. It helps. Mathematics is very broad. In the various major branches, such as Analysis, Algebra, etc. different insights are needed to really understand what is going on and what is important. In addition to the all-important insights, different proof techniques are common.

When you are doing real research in mathematics you are engaged in a very narrow study. But having a wide variety of ways to attack a problem, even if they might be suggested by ideas from another subfield, will give you an advantage.

Both breadth and depth are needed. Depth is usually what is most valued, but the breadth of knowledge helps you get there. So, don't completely neglect other mathematical subfields than the one you are most interested in. You don't need to be equally skilled in all of them, but being able to grasp the essential insights is a plus.

  • Indeed, just to provide evidence: the prime number theorem (a number theory problem) has deep connections to complex analysis, but on the surface, you wouldn't naturally connect the two.
    – Clayton
    Dec 11 '18 at 20:17

Here are two anecdotes from my own career. I wrote a thesis in summability theory, despite a background in algebra. In that thesis I used the 5-lemma to deduce an isomorphism between two subalgebras of a Banach algebra given an isomorphism between two related subalgebras. Later on, working in algebra, I used familiarity with matrix methods of summability to prove that a countably infinite dimensional vector space V over a field k is not Hopfian, that is there is an onto linear transformation T from V to itself that is not one-to-one. The transformation T mimicked the usual shift map that demonstrates this fact, but I did not have to know it based on my summability experience. So I was lucky to get cross-fertilization in two directions. I might have obtained the results in each case eventually, but it saved a great deal of time knowing something from a different area of mathematics.

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