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My background:
Applying for Ph.D. Math or Applied Math
International student, Ukraine, female.

Undergrad Institution: junior year - 20ish in Math globally (US institution), first two years - community college(US)
Double majoring in Math/CompSci
GPA 4.0/4.0, same for college
Some honors classes (hopefully will graduate with honors in math), but none graduate level.

Research experience - currently doing a summer research program, should result in a paper with 3 co-authors (all undergrads too)
Done a literature review in statistics with a pretty general topic.

I am a rising senior, planning to apply this Fall for Ph.D.
I am interested in graph theory, but I haven't done much outside of it, so I might as well become interested in something else.

My questions is:
What can I do to increase my chances of acceptance as of now if:
i. my schedule for the senior year is fixed, there is no way to change or add anything
ii. I'm an international student, so any paid/compensated job outside of my uni requires a working permit, which is unnecessarily hard to get?

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I honestly don't think you are going to have much trouble finding a good doctoral program. But, I have two suggestions to help maximize success, other than continuing to do well in your studies.

First, make sure that you, and your skills, are known to at least a few faculty members. If you haven't yet cultivated relationships with potential letter writers, do so. One way is just to ask questions in class. Another is to bring questions to office hours. But you want people to be able to honestly and sincerely predict your future success. A letter needs to say more than just "She got good grades in my courses."

The second is that when you apply, cast a wide net. Focusing on too narrow a range of universities could leave you without a bid if they all have very similar criteria.

You will have plenty of time in grad school to choose a research focus. Some might come from the advanced courses required to set you up to pass comprehensive exams.

Also be aware, as you go on, since you have a near perfect record up to now, that it might get much harder when (not if) you hit your natural boundaries and learning beyond that becomes extremely hard work. Some students don't make it over that hump and get burned out before they finish. I was "lucky" to learn my limits early and was able to push past the burnout.

Financing your doctorate will also probably be easily done through a (most likely) TA position. It takes some time and effort from your studies but provides good experience about academia in general and can even lead to some insight - especially in to learning. It provides a tuition waiver as well as sufficient funds to maintain a modest life style. There is high demand for TAs in fields that have a lot of undergraduate majors, so it should be a natural. They are also open to international students and don't raise additional visa issues.

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    "If you haven't yet cultivated relationships with potential letter writers, do so" - I think it's good to communicate with professors about grad school goals early, especially those leading smaller upper-level classes for specialists. I think professors are naturally biased towards students interested in further academic work (it's a shared interest!) and are often eager to help out, whether that means suggesting or offering research opportunities or student jobs, or just paying more attention that can help in writing a letter.
    – Bryan Krause
    Jun 21, 2021 at 22:19

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