Can I pursue a PhD in Operator Algebras after a Master's in PDEs? [closed]

Question

I am about to start my second year of a Master's program in Partial Differential Equations (PDEs), which I originally entered due to limited options. However, my true interests lie in operator algebras and related areas, which I have mainly self-studied because they are not popular in my location. I have a solid foundation in the fundamental mathematics subjects expected of a student in these areas, but I lack formal coursework or an "official" background in more specialized topics within the field. (For example I have taken functional analysis and spectral theory, but not a course in operator algebras which i self studied.)

Due to financial constraints, I couldn't apply for a master's abroad and chose the PDE program at my university. While I enjoy aspects of PDEs (such as elliptic PDEs and Sobolev spaces), I am concerned that this decision may limit my chances of pursuing a PhD in operator algebras or other pure areas of analysis. Most of my letter writers and connections are from the PDE field, so I am unsure about my chances of transitioning.

Is it still realistic to apply for PhDs in operator algebras or similar fields, given my situation? How can I build connections with experts in these areas if my network is mostly PDE-focused? Cold emailing feels like my only option, but I’m unsure if that approach is reasonable. I also can't afford to simply travel around to conferences.

I understand the importance of being flexible and open-minded as a PhD student since I have likely only scratched the surface of these massive fields. However, I still want to steer myself toward an area of mathematics that I find interesting, which I can only describe as abstract functional analysis.

Answer 1

In most places, that shouldn't be a huge problem. You have at least gotten some generalized mathematical insights. Specific insight within a narrow subfield requires more work, but that is necessary in any doctoral program.

The only real difficulty is getting accepted to a program, and that, might require finding a supervisor who is willing to take you on. But you don't learn that until you try it. Note that requirements differ widely around the world, both for application and for earning the degree. You need to learn those requirements and make a good application.

Note that competition for many programs is very high. And, you don't enter a doctoral program to prove that you don't need it. You are expected to learn what is necessary.

To see specific requirements typical of various countries see: How does the admissions process work for Ph.D. programs in Country X?