So, I want to apply for PhD in pure math. I have an excellent GPA (4.0), a research experience (bachelor thesis and master thesis, both in operator theory, the second one in more advanced, both are unpublished) and I expect to receive pretty high scores on GRE math subject, GRE general scores are V153/Q168. However, I have one gap which I cannot fulfil - I cannot say in details what I want to study.
The situation is simple: my university specialize mostly in analytical areas - mathematical analysis, complex analysis, operator theory, PDE, mathematical physics, you name it. However, algebraic disciplines are presented as supplements for information security theory (signal coding theory, cryptography, etc). It turned out that I enjoy group theory, ring theory and things like that way more than operator theory. I took every course in algebra which I could but due to the family issues I cannot change a school for master degree and, as a consequence, cannot have a research in algebraic fields. Now I want to apply for a PhD program, but I cannot state directly "I want to study object X to prove the properties Y and Z, and contribute to theory A" because I have never done a research in algebra and I don't know open and perspective problems in it as well as modern techniques or results. I can only say general words that I want to study group theory\algebraic geometry\group representation theory\ect. I feel it is just not enough to be accepted to any school, and I don't know what to do.
Should I just write general phrases like "I am interested in X, because I took courses in it in which I performed excellent, and I read books Y and Z in it and now I am reading book A and book B in it, and I love all of it" in my personal statement? Are there anyone with the same problem to consult with?
Thank you for all responses in advance.