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I want to apply for a PhD in mathematics in the United States, I did my BSc. in Physics at one of the best universities in my country, where I enjoyed studying the mathematics behind physics theories, but I only had to take three pure math courses. That's why after that I started a Master's in mathematics. Since I had to work to pay for my studies I did not fully dedicate myself to learning math, in the master's degree I took some one-semester courses on analysis, differential geometry, and algebra, but I don't know if this is enough for a PhD. I know that in the US you have to take hard QR exams (in algebra, analysis, or topology) in the first two years, and I'm wondering if I could pass these QR exams without the basic knowledge.

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  • What do you mean by "approve " a QR exam?
    – Allure
    Commented Jun 3 at 2:23
  • I mean obtaining the passing score on the respective qualifying exams (analysis, algebra....). Commented Jun 3 at 3:08
  • You got two years to learn the material no? So presumably it's possible (if that's what you're asking).
    – Allure
    Commented Jun 3 at 3:09

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I can't speak for all programs, but my experience is that qualifying exams in math are very difficult and are based on advanced, not elementary, material. The courses you have taken are unlikely to be sufficient to pass any of them. Qualifiers are difficult even for those with a much deeper math background than you suggest you have.

On the other hand, most programs provide advanced courses that lead to the qualifiers. But it will take a year or more to take those courses and they are also quite difficult even for those who have majored in math as undergraduates.

It isn't impossible, assuming you get admitted, but it will be difficult. But if you are admitted then someone has predicted that you can get through it. And note also that it is fairly common for US students to change fields between undergraduate and graduate studies. This is partly explained by the broad nature of undergraduate studies here.

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