I am currently about two semesters away from receiving a Master's of Science in physics. Originally, I planned on pursuing a PhD in physics, but during my studies, I have learned that there is just something about mathematics that fascinates me. I find that more and more of my imagination and time is spent directed towards numerous fields/topics in mathematics. I have procured texts on model theory, topology, meta-mathematics, philosophy of mathematics, &c. &c., and find it hard to peel myself away from them even when my academic duties in physics demand it. I have been doing a lot of introspection and reflection regarding this over the past few years, and I have seen myself happiest when learning about and doing math. So, I am taking it upon myself to start researching the opportunities and challenges that I may face in considering going to school for a PhD in mathematics after receiving my master's.
Assuming that I were to follow this path, what sort of things should I begin doing to give myself every advantage possible moving forward? I know this is a very open-ended question, so I will provide some additional context.
I assume that, going into a PhD program in mathematics, I should expect things like diagnostic exams and qualifying exams. I would want to go into these doing as well as possible, naturally; in general, what sort of topics fall on these exams? Do they differ widely from program to program? I have yet to start looking into the math programs available at my current institution (and, honestly, I believe that I would prefer looking at programs elsewhere), but I plan to touch base with professors in the math department for their own particular insights regarding this.
Furthermore, as someone with formal training in most of the salient mathematical prerequisites for physics, I realize there are likely many topics I should get much more familiar with, likely on my own. Most of my training is in differential equations, linear algebra (of course, only so much of it has been prevalent in my physics career), and calculus (I, II, III). As someone with a bachelor's in philosophy and religion (and applied physics), I have some training in very rudimentary symbolic (and Aristotelian) logic. Highly salient topics like discrete mathematics and real analysis I have not received formal training in. Furthermore, I have no formal training in abstract algebra. Should I absolutely seek to attend some of these courses at my current institution before I receive my Master's? I have always been somewhat of a fruitful autodidact and would have no issue learning these on my own, but I assume that it would be much more attractive to math programs should I have taken these courses, or have at least audited them.
Apparently a lot of my concerns are with exposure to certain topics, but I do realize there are likely numerous other things that I should begin considering as soon as possible, as well. I welcome and would greatly appreciate any insight that you all may have regarding this. I'm certainly not averse to future research in fields with interfaces between physics and math; formal theory and the mathematical structure of physical theories such as quantum mechanics are both intriguing to me, too. So I haven't resigned myself to "wash my hands" of physics by any means!
Thanks in advance, and if you have any questions for me that will help me help you, let me know. I look forward to your replies!