TL;DR: I'm applying for a PhD program in mathematics, but my academic performance was bad for a long while and I already have a PhD attempt that I ended early. (However, I now have good enough results to apply again.) Given this background, should I spend any time in my SoP addressing the reasons for these poor outcomes? If so, is it a good idea to only mention them as indicators of how serious I am about getting a PhD? Or, should I leave it out altogether?
More detail: I'm filling out applications for math PhDs (to start fall of 2024), but I'm a fairly non-traditional student in a lot of ways and could use some pointers on how to address that in my personal statement, if at all.
For a little background, I grew up extremely poor and was functionally a high school dropout by the time I got out of public school. I took an extra year to graduate, was placed in "at risk" courses, had a 1.2 GPA; just all around bad. I started community college a couple years after high school, starting out in basically 7th grade math and it took eight years to finish my undergrad. Maybe unsurprisingly, much of my undergrad was pretty rocky; most of my math classes were C's, with a couple of B's and a couple of D's and a few withdrawals. I had a handful of life circumstances that got in the way, like having to work full time to support my mom, a family death and a fiancé that ended up having an affair (I know for sure not to mention that last one in my statement), but more fundamentally it took a long time to master the essential soft skills like how to study. In my last three semesters of undergrad my math performance improved substantially and I started getting mostly A's in my upper division courses, with two B's. I did well enough to get into a masters degree in statistics, which was fairly theory focused and in which I continued to do well (final GPA was a 3.8 in the masters). I also had some extra room for elective course work, so I used it to take some courses like topology and abstract algebra, which I also got A's in.
After the master's, I applied to Math PhD's and managed to get into a couple of programs. But, covid hit six months later and I did not deal well with the isolation at all. My mental health completely tanked, and I couldn't work on anything for more than 5 minutes at a time. I left that program at the end of my second year and started working in industry, but I wanted to go back to a PhD eventually. This fall, I enrolled in a graduate analysis class at my alma mater (as a non-degree seeker) so I could get back into the swing of things, and I'm continuing to do very well. The professor has only posted a handful of the distributions on a couple of assignments and one midterm, but based on that and how discussions in class go I estimate that I might be one of the top three students in the class; my professor volunteered to be a letter writer for me when he asked about my long term plans and my background.
I have a few ideas about possible areas of research I want to go into; something that broadly fits somewhere in the umbrella of analysis, like probability or dynamical systems. Or possibly PDEs, but I don't have a very strong familiarity with that area yet, so I can't say that too confidently. Post-PhD the dream would be to work in academia, but given the need to post-doc and how tight the academic job market is that might be unrealistic for me, so working in a national lab might be up my alley. I don't think I can say anything more concrete than that until I'm a couple years into a program and I have a clearer view of what kinds of opportunities would be available to me given wherever I happen to be.
I know it's kind of a mess, but I am extremely determined to get a PhD in math. If I don't get in now, I'm just going to keep trying to improve my application until I do get in. Math has been life changing for me. I want to continue working with it in substantial and meaningful ways as an essential part of my career and continue to stay engaged with the math community in general. Any thoughts, suggestions or advice on what to say about this would be greatly appreciated.