What should I do to prepare to apply to a math PhD program?

Question

A little background information first. I am about a year out of my undergraduate program where I double majored in Computer Science and Mathematics. My main focus was computer science and the math major was mostly for fun. Since I wasn't considering grad school during undergrad I never partook in any REU programs as I was planning on working in the computer science field long term. However after about a year working as a software engineer I miss studying Mathematics and am not loving my work as much as I thought I would. So now I am looking into applying for a PhD program not just because I am not loving my job but because of my love of Math. My love of Math is what led me to study Computer Science in the first place. The earliest would be starting in Fall 2020 but I think Fall 2021 might be more reasonable given the timeline.

This is where the bulk of my question comes in. Since I double majored, I didn't get to take as many math courses and so I have holes in my education such as Differential Equations and Analysis. I do have textbooks on these subjects that I have begun studying with but how can I show programs I apply to that I have the knowledge in these areas? Will not taking these classes affect my chances to get accepted? I should mention that I did take courses in Stats, Abstract Algebra, Number Theory, Combinatorics and Linear Algebra. My overall GPA was a 3.5 and I did well in most of my classes except for Abstract Algebra and Number Theory where I struggled more in, but I do have a reason for that as I was taking three Upper Level Computer Science classes at the same time as Abstract Algebra and then that led to struggling in Number Theory due to lack of understanding of some of the basics. Are there any other tips that you could give me to help improve my attractiveness to a program?

Answer 1

(US-specific) The REUs don't matter much for admission. The problem with not having analysis at all on your transcript is that most programs use it as a litmus test for whether you are able to write proofs and handle graduate coursework. The fact that your weaker courses were number theory and abstract algebra is also going to raise red flags in that regard.

There might be programs that would be willing to take you as an MS so that you could remediate, but probably the best course of action is to just register for a year as a post-bac or nonmatric somewhere that offers upper-div math courses.

Answer 2

I'll assume this is the US. It might apply elsewhere, but I have no experience. In the US, it is possible to apply for a PhD with "only" a BS/BA degree. Those degrees vary widely and some smaller schools don't have enough faculty to cover the very wide mathematics territory very deeply. Graduate admissions people understand that.

On the other hand, most such doctoral programs include at least the possibility of additional coursework before prelim/qualifying exams and research. Some schools require it. My first graduate level course was in Measure Theory (new to me) and I did little but coursework for three years. Another early course was in Topology, and was much deeper than anything I saw as an undergrad.

In addition, a doctoral dissertation requires a very deep dive into a very narrow part of mathematics, and, while breadth is useful, it isn't normally necessary to write a dissertation.

The conclusion of all of that is that you are probably already qualified for admission, other things being equal, though you will find a lot of competition from well prepared applicants.

My advice would be to apply ASAP to one or more programs and write up your background and experiences in the most positive way. You need to give evidence of your likelihood of success and your letters of recommendation need to support that.

But for the direct question - what to do -, you can find a way to improve your math knowledge and skill, either working toward breadth (analysis) or depth. But the way to do that most effectively is with a guide of some kind. If you are near a university you may be able to take individual courses and so obtain the guidance of a faculty member. You might be able to do it for no credit or grade. But the advantage of doing it formally is that you get both guidance and feedback. The feedback is essential, not just for looking at your work, but for suggesting what you should look at next.

Reading existing texts will not have the same effect, though reading along with a lot of problem solving will be better. But, again, feedback on your solutions to those problems can be extremely valuable. If you can only solve things awkwardly you need to find ways to "up your game". Practice and feedback.