What do I need to do to move from a CS major to a math PhD

Question

I am a first year computer science student in India. I have always liked mathematics and have performed well in it. I want to pursue research in pure mathematics. I did some research on the internet and read some answers on this website too. I found out that it is indeed possible and people have done it in the past by having some 'relevant undergraduate/graduate level math classes' under their belt.

The problem with this is that my college does not offer any math classes except linear algebra, calculus(including multi-variable) and discrete mathematics.

How should I go about things now? I have self-taught myself proofs and calculus and I am also good at algorithm analysis(my college has not taught me that yet).Here are the other ways, I found on the internet, that I think would help my cause -

1. Does self-preparing for GRE math subject test and scoring highly in it help showcase my skills?
2. Would doing a math heavy research internship in CS help?
3. Do letters of recommendation from CS faculty carry the same weight as a math faculty?
4. Is doing a masters in mathematics a better path?
5. What are the other ways in which I could show the admission committee that I have the required knowledge one would expect from a math major(assuming I can attain that through self-study) and that I can pass the qualifying exam by taking courses at that university?
6. What other math topics should I study? Do I copy a standard undergraduate curriculum or should I read advanced graduate texts too? Please specify the topics.

I want to do research in the US as there are much better institutions there. What advice could you give me for undertaking this journey? Feel free to even claim that admission in a top PhD program would not be possible if that is what you think.

Thanks for reading!

Answer 1

In the US, a student will typically do an undergraduate math degree and then apply for a doctoral program. The undergraduate degree typically has about 16 semester-long math courses along with lots of other things; writing, science, philosophy, history, etc.

The math courses will usually include, in addition to the ones you have taken, things like a year of abstract algebra (say, 2 courses), advanced analysis, topology, probability and statistics, etc. Each program varies. I took a (terrible) course in projective geometry, for example.

So, you are missing quite a lot. But some things will compensate.

You don't actually need an undergraduate degree in mathematics to join a doctoral program in math. Most will have done so, but exceptions are made, even quite often for CS majors.

Moreover, such a doctoral program is heavy in advanced coursework at the beginning. My best probability course was as a grad student. The coursework is there to help you prepare for qualifying/comprehensive exams that usually occur before you get serious about a dissertation research problem, though there may be some research earlier. My first two graduate courses were in Measure Theory, which is now often taught to undergraduates, and a really great Topology course, far beyond what I'd seen as an undergrad.

So, if you have a way to gain any of the knowledge of the missing courses, say through online work, it would help give you the background to succeed in a US doctoral program. It will be hard to show any credential for such study, but it is the knowledge and background that you also need.

But the only way to know if you can get accepted is to apply. I'd suggest that you spend some time now studying on your own, but also making application to a few programs at US R1 universities. State universities are usually pretty good. But I strongly suggest that you don't limit your applications to a narrow range of places. Some students are convinced they will only be happy in a TOP 10 institution, but this is unrealistic given the numbers of slots vs the number of highly qualified applicants. But you will learn something from applying, even if you are rejected. Try to find out, from any rejection, what might have made the difference.

And note, that the above is probably only valid for the US.