I'm a non-EU resident and I want to pursue a PhD in pure math in Germany. I am currently in the process of writing to as many professors in my field of interest (functional analysis) as I can requesting that they take me as a doctoral student under their supervision.
I don't want to do research in the same area that I specialized in during my masters education (in fact, I want to switch from algebraic geometry to functional analysis). My question is how can I convince someone to take me on as their student if I'm making such a big leap and hardly have a clue as to what research in functional analysis entails? Professors naturally look for students that are already very familiar with their research area and don't need much training before they start working.
Usually when I email a professor requesting them to take me as a student, I attach a short CV (not that impressive either) and transcripts of my masters education. I don't have a research proposal for them and I feel that's hindering my likelyhood of getting an acceptance letter. I'm sure I'm not the first student with this problem and people in my situation do get admitted into a PhD program doing work unlike that which they did during their masters so I was wondering how do I become one of those people?
Thank you for your help.
@PaulGarrett: Since I can't comment below, I'll clarify here: I have some research experience in the area of fixed point theory (which is close enough) working as a research assistant for a professor in our department. I can get a reference letter if there is a need. I don't have a paper published, however, but I did learn a lot and so this is how I believe that that's what I want to work on. I of course realize that saying I've read a few articles and enjoy the work is one thing but convincing someone to pay me for 3/4 years to do some work in this area is something else, but that's all I can say and I can sense it doesn't really sound very convincing. I hope my situation isn't so hopeless that I either have to force myself to learn to enjoy algebraic geometry or start again with first getting another masters degree.