As someone switching rather late in the game to math, I'm wondering how to use my background in computer science, and my experience as a professional software developer, to offset a lack of math classes in an application.

My background:

I studied CS and graduated from a top SLAC in 2014. I worked professionally as a software developer and have some neat things to put on my resume along the lines of data science for the last few years.

Now I am taking classes at my local state university (which is a flagship state university, but not necessarily a famous one for math). By the time applications go in, I'll have gotten an A in analysis, algebra, ODEs, and a special topics course on Hilbert Spaces. And by the end of the fall, I'll have taken four more graduate courses (the graduate versions of analysis, algebra, topology, and some other class I haven't decided yet). I'll also have done an REU with a math professor here over the summer, but it was basically just programming and almost no math at all.

My goals:

I want to study pure math, not computational math, but I'm willing to make a detour toward that realm if it means increasing my chances of getting into the math world at all .

I don't have a complete idea of what I want to focus on yet (and I hope that's normal). So far my top interests are representation theory, functional analysis, and geometry, but that might just be what I've happened to study the most.

Questions:

- (*) Is a background in computer science any interest to a graduate committee in pure math at all? Would it depend on the particular subfield? If so, can you suggest subfields I look at?

(*) Should I be applying to applied math masters, instead of pure math PhDs? I realize they're very different, but the reason is that I don't know of any funded pure math PhDs, whereas there are a few applied math masters out there. And since I have a background in CS, they might look more favorably on that. Then once I have taken more math classes through the masters, I can apply to a PhD program from there.

(*) Should I be taking applied math classes in the meantime? For example, numerical analysis. The scheduling at the local university is a bit bizarre and everything overlaps with everything, so I'm afraid I'll have to choose between another semester of algebra and numerics, if I do decide to do numerics. I don't dislike numerics, I just want to learn pure math.

(*) Do I stand a chance at getting into a, say, top 50 pure math PhD program at all?

All that's a bit scattered, so any generic advice is appreciated as well.