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So I want to pursue research positions, I'm not sure what kind of research, but I know academic research is the goal for me right now, so I will eventually want to go onto a PhD program in something. As far as programming I've taken the the introductory programming sequence, data structures and algorithms, two classes in Computational physics, and will take Numerical Analysis (which I understand is foundational course more advanced scientific computing and computer science courses. I will be sure to take many coding/computer science oriented classes, I have been very interested in learning Machine Learning/AI but I'm not sure of any good books to try. Applied Math (as I understand it) is pretty broad which is something that appeals to me since there are so many potential problems to solve (Physics to finance). I'm not sure what kind of job or career would provide me that kind of breadth of research opportunities. Thank you for the response (I'm still very concerned not having Real Analysis, hopefully I can take a class at another university before I apply to some graduate programs)

In the spring I will be graduating with a degree in Physics and Math. I've spent most of my time in the physics department, but I have taken enough hours to be a Math major and have enjoyed a lot of the proof based math courses. I'm pretty sure I am a well qualified candidate for many mid-tier Physics PhD programs (REU, Home-University Research, tutoring, grades, undergrad researcher/student awards etc. in Physics) and competitive for some good schools. I have been doing a lot of experimental research in Physics, but I've always enjoyed the theoretical part of physics a lot more and I've only been doing the experimental stuff to fulfill a graduation requirement for honors. I've always enjoyed my math classes and I've really been enjoying the proof based classes more and I want to pursue more math oriented subjects, should I apply to an MS in math? Am I even qualified? I'll list my stats below, advice would be greatly appreciated. Math Classes: ~3.9 - Calc 1,2,3(Stewart) : A, A, B (Calc 2 and 3 were freshman year)( Methods of Math Proofs: B [this was same professor as the calc 3 professor, he never gave feedback or solutions to p-sets, tbh I don't think these grades reflect my potential] (Junior Year) Abstract Algebra (Judson): A Math Methods for Physics (cross-listed with a math class)(Boas): A (Sophmore) Linear Algebra: A Discrete Math: A Diff Eq: A PDE (Cain) : A Applied Math: A [spectral theory, applications of matrices, modelling techniques, MCMC, Runge Kutta, etc.] Intro to Stats/Prob: A Will Take/Currently Taking - Complex Analysis, Advanced Linear, Numerical Analysis Physics: GPA: ~3.9 Classical Mech (Taylor) : A ; QM ( McIntyre): A ; E&M (Griffths): A ; Advanced Lab: A; Computational Phys: A; Waves: A; Modern Phys: A; Thermo: A Advanced Computation (took with grad students): A [mcmc, fourier analysis, etc.] Currently Taking/ Will take - Graduate QM, Digital Logic I obviously probably missed a couple classes but I think that encompasses the 99% of the important stuff So the big concern: I'm missing Real Analysis and Topology and many 2 semester sequences - this is largely because the departments are not big enough and I have to wait 2 years for classes to be able to get enough demand to fill up with ~8 people. From what I understand not having real analysis is the real killer. Should I take Real analysis at another university after I graduate, then try to go to grad school ? Apply straight to MS? Try for theoretical Physics PhDs ?