Generally speaking, studying for a doctorate in the United States depends on scientific research results. Although the school has no clear requirements on the quality of papers or mathematics, in fact, there are no more than two papers on hand in the mainstream of scientific research, and tutors generally don't let people go. Therefore, no matter what discipline, doctoral students should have scientific research innovation.

For applied mathematics, although the word application is included, what is actually done is not as applied as expected. Discipline is definitely the application of a certain discipline or field, including biology, chemistry, engineering, computer, finance and so on, but everything just stays on the theoretical basis. For example, before I was a doctor, my subject direction was chemical physics, and my papers were all about chemical physics, which was a very practical subject, but I had never really done experiments and other similar experimental subjects, so all scientific research was based on abstracting specific physical and chemical experimental models into mathematical models, and then solving these mathematical problems by various methods of calculation or computational simulation. Therefore, in terms of difficulty, it is much less unpredictable and complicated than the experimental discipline, but it requires a strong understanding of mathematical thinking and deep problem-solving ability.

Generally speaking, if the project develops normally, a doctor needs 4-6 years, and 5 -6 years is normal. Especially after that, if you want to be a postdoctoral fellow or find a teaching post, you need some success in scientific research, so it will take at least five years.