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This summer I finished my BSc maths degree at Imperial College London and ranked around top 5% of the class.

I was considering the idea of applying to some US universities like Stanford, Harvard, etc., but they don't do master's applications, just straight up PhD ones for mathematics. I quite like the idea of doing a PhD, so it is not a big problem for me, but I was wondering about my chances here, coming off a BSc and applying straight to a PhD sounds like a big reach.

Has anybody had similar experiences and know if it's even worth trying?

Thanks for all the help.

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    Go for it. But bear in mind that competition for math PhD at places like Harvard and Stanford is so extremely fierce it is difficult to really grasp... Talk to your professors at Imperial and see whether anyone has any contacts and/or suggestions. Your letters need to be extremely strong.
    – user2705196
    Nov 16, 2020 at 17:14

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Most US students enter doctoral studies after "only" a bachelors degree. It isn't a reach at all. But expect there to be coursework and exams. The coursework is something like what you would get in a masters, but is intended primarily to prepare you for the qualifying examinations. Serious dissertation research often starts after you've done a couple of years of other things. But you aren't likely to finish in three years from where you are now. Expect something like twice that.

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