[dleeftink]

> seems like a highly particular / specialized area of mathematics. It's like computer science, can't know them all

As a non-mathy, I'm interested in whether the idea that being good/able to provide proofs in one area, automagically makes one proficient in another is customary in the field or rejected quite early on when choosing a math specialisation?

[svat]

It is not typical that being good in one area of mathematics makes one proficient in another — mathematics has a lot of depth, and to reach the frontier in any specialization requires years of study. However:

- There are skills that carry over; these are usually known by the name "mathematical maturity" https://en.wikipedia.org/w/index.php?title=Mathematical_matu...

- There is a story/legend told about Erdős, where he was so good at problem-solving/proofs that he once solved a problem in another area after asking for the definitions of the terms in the problem. (The fact that this story is told illustrates that it is not commonplace.)

[MichaelRo]

Well there is common stuff that one must be familiar with to be proficient and marginal stuff that you can get by without knowing because chances is you're not going to need it.

Like take for instance financial mathematics where I had some special interest, it's totally oblivious to areas such as geometry or number theory. I never had to figure out if a polynomial is divisible by 6 for instance :)

Like computer science, there's the common algorithms stuff but being an expert in web development doesn't help you much in writing high frequency trading server code, and the other way around.