I am an undergraduate engineering student in Azerbaijan, and I intend to pursue postgraduate prospects in mathematics. Unfortunately, the university I am studying at does not offer any advanced mathematics classes (e.g. Real Analysis, Complex Analysis, Topology, etc.); therefore, I had to self-study those branches of mathematics, and my independent study was approved by a distinguished professor. Can I convince graduate schools in the US that I studied those courses by a written statement by that professor? Thanks in advance.
It would probably depend on how strongly the professor supports you in a letter of recommendation, what they say, and their knowledge of the US educational system. But, I would think that many, if not most, graduate schools would accept a strong statement from a known and respected mathematician. It might depend on the interpretation of "distinguished", of course.
But, you use what you have and if you aren't successful initially, try again, using any feedback you get from early tries.
You might also use something like the Math GRE to validate that you have learned the necessary topics, though the exam is (or was) very difficult.
This depends on what exactly you mean by:
convince graduate schools in the US that I studied those courses
I would bet that a letter from a respected mathematician would suffice as "proof" that you have some advanced math experience in general (this is essentially a letter of recommendation). But I'm not sure that it would be accepted in lieu of a mandatory prerequisite course though.
Universities can be pretty bureaucratic and if a program specifically requires "Course X" you usually need to show that you have taken "Course X" or equivalent. The only way to know if self-study counts as equivalent is to ask the universities you are interested in. Whether or not they consider it equivalent might be influenced by a few things:
- Individual university policy i.e., how flexible they are with requirements
- How much they like you as an applicant/how many other qualified applicants they have
- How "approved" your self study was. There is a big difference between taking self-study credits (are they reflected on your transcripts?) and just reading a text book on your own (even if you have the support of a professor).