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Math Graduate school for math

I was wondering what kind of graduate programs a math graduate can get into?

Is it required that for example, programs in applied math or statistics need a large amount of applied math and statistics courses in the undergrad?

Does having a pure math degree limit what you can do for graduate school?

I am asking this as I remembered my girlfriend's parents being engineers in china (msc level) and doing a masters in stats in canada.
Can anyone elaborate on this? How related does your graduate work have to be to your undergrad?
I am currently a PhD student in mathematics so I know at least something about all of this.

For any kind of math, applied math, statistics included, a strong pure math background is required. For me, in pure path, I honestly had no more than one applied math course in my undergrad, and it was fine. For someone going into applied math though, they will expect you to have more than just one pure math course, as well as some applied math.

What I'm trying to say is that basically, you just need a lot of math, but it doesn't have to be a large amount of applied math/stats for an applied math PhD (or Masters), but that's only if you have lots of pure math instead. Grad schools like well rounded candidates, so you should have taken courses in a broad range of topics in mathematics, several algebras, some analysis, and, especially in your case, some statistics/probability theory/measure theory.

A pure math degree is probably the best undergrad degree you can have, honestly. You can get into any applied math program, most likely, and, according to my physicist friends, grad schools prefer taking students in physics/astronomy from pure math backgrounds too. Pure math is always the way to go.

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