Should I Change my Research Focus? I just finished my first semester in a PhD program in Pure and Applied mathematics. My initial focus was on probability, statistics, modeling, and machine learning. To this end I took a course in (Modern) Probability, Nonlinear Optimization, and the mandatory Functional Analysis. Based on my performance in these courses my advisor suggested that I should consider maybe switching to a different focus, as he thinks I will struggle to pass the qualifying exam (specifically in the two elective courses Probability and Nonlinear Optimization).
However, I do not believe my scores reflect my real capability or interest in the subjects, but rather reflect a gap in what was taught and what was tested. I am sure I could study really hard and succeed.
I am posting here to get a second opinion on whether or not I should change my focus, or if I should stick with it and double down and study really hard. I posted on math.stackexchange and was advised to post here for a second opinion. (https://math.stackexchange.com/questions/3984219/what-are-some-possible-phd-research-areas-based-on-these-course-pairs?noredirect=1#comment8221305_3984219)
What would you advise? Should I change my focus to another one that I might have a better synergy with? Thank you!
Your advisor's advice is probably worth taking. They know you better than we do.
But your current goal is to get a doctorate. The specific sub field is less important than that goal. You can always take up a different specialty later if it isn't too different.
Find a path to success. Examine the obstacles (exams, ...) and find a way through them. You may have made some judgmental errors, but don't make any more.
But to find that path, in mathematics, find a field in which you have true insight into how things work. Math insight isn't general. You can have deep insight into one field (analysis) and little in another (algebra).