u/Critical-Deer-5342

Finished my first year in the master's program in mathematics at a top 50 R1 university. If everything goes well I'll graduate on time with a 3.4-3.5 gpa and a thesis project. I'd like to know what the realistic likelihood that I'd be accepted to a PhD program in mathematics at a R1 university (ideally related to my topic of interest). There is a specific area of research I'm really excited to get to the point where I can contribute. I know that acceptance is never guaranteed but I'd like to know more about the possibility.

I saw that every finite group PSL_2(F_p) for p prime is the Galois group for a Galois extension over Q. Is the realization of one of these groups of interest as a thesis project? Did some minimal research and saw that a portion of this is a computational problem. Recently, finished a course on Galois theory, I have a good amount of experience programming and some time to kill this summer. I'm trying to explore some ideas for a thesis project that aligns with my interests. Would anyone recommend pursuing this?