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This r/math post spurred this question. To concretize the meaning of "ace", assume

  1. success means earning professorial tenure at a Top 100 world university.

  2. your liked stronger branch doesn't exactly relate to your weaker branch.

E.g. can you ace Stochastic Calculus and Financial Mathematics, while flubbing Abstract Algebra?

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    $\begingroup$ As currently phrased, it's a yes/no question. Presumably you're looking for justification and/or examples. Please clarify. $\endgroup$ – J W May 14 '20 at 7:56
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    $\begingroup$ Also, I suggest thinking about which tags the question should have. As currently phrased, it does not seem to be about "self-learning". $\endgroup$ – J W May 14 '20 at 7:57
  • $\begingroup$ This question has nothing to do with mathematics education. $\endgroup$ – Dan Fox May 21 '20 at 7:39
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    $\begingroup$ I’m voting to close this question because it is not about mathematics education. Instead, it seems to be a yes/no question about abilities of individuals. There may be an education question buried in there somewhere, but as written, it is not clear what that question is. $\endgroup$ – Xander Henderson May 21 '20 at 12:18
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I think it matters what you mean by flub.

There are a couple specific exceptions, but I think any of my colleagues in my department could teach any of our undergraduate courses in an emergency. Would they be the best person at it? No. Could they do a competent job for an emergency situation and be helpful to almost all the students in learning the material, even perhaps as they are just reading the textbook a week ahead of the students? Yes.

There are general mathematical skills of reading and understanding theorems, proofs, and definitions, and coping with abstract concepts. They are usually enough to get through any undergraduate material, and almost all mathematicians have mastered these skills in a general context well enough to learn and even teach all undergraduate material on their own.

There are of course exceptions, but they are unusual.

At a higher level, I wouldn't feel at all comfortable teaching a first-year graduate class in all but a few areas of mathematics, and I think I would not have a reasonable chance of learning some of these areas in a reasonable amount of time (but other areas I could), and I would not be able to do decent research in them even if I took as much time as I needed to retrain.

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It often takes many years to develop a strong foundation and an accurate intuition in an area of mathematics to become steadily productive in that area. By that point, one might be proficient in, say, elliptic curves and modular forms, but relatively unschooled in, say, combinatorics on words. So I agree with @G.Allen: The answer to the OP's question is Yes.

There is a broader issue not easily identified with expertise in particular subfields of mathematics, discussed in Gower's The Two Cultures of Mathematics. The two cultures are, loosely,

"the distinction between mathematicians who regard their central aim as being to solve problems, and those who are more concerned with building and understanding theories."

But this takes us far from the OP's question.

Gowers, William Timothy. "The two cultures of mathematics." Mathematics: Frontiers and Perspectives (V. Arnold et al., eds) (2000): 65-78. PDF download.

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Absolutely.

Math is such a hugely diverse field that it's been over a century since anyone could reasonably claim to understand the whole thing. That kind of diversity creates subbranches and specialties that have absolutely nothing to do with each other. To get an idea of how diverse the field is, take a look at this great video.

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