What do grad schools want to hear about why I'm changing PhD programs?

Question

TL;DR: I'm reapplying to schools in the US for Fall 2021 because (1) the research project I'm getting paid to work on isn't in my area(s) of interest, (2) in general, mathematics at this university is heavily applied and computational, so there's little scope for me to do something else I'd like here, (3) I'm making much less than minimum wage (assuming a 40-hour work week for 52 weeks a year), and (4) I don't like living here. These are listed in order of importance to me. The question is, which of these points do I emphasise to an admissions committee?

More context: I'm an international student (for US and Canadian schools). I graduated with a Master's in math from a well-known university in Europe. I was "on-track" to start a PhD in Fall 2020, but due to personal reasons and global events I couldn't go to any schools in the US or Canada. I didn't want to take a year off from math, and I needed to make at least some money, so the best option seemed to be to enroll in a PhD program in Europe.

I plan to reapply to US and Canadian schools (for the reasons listed above). Only, I'm not sure how to explain that despite already being enrolled in a PhD program, I want to "transfer" to a different one. I have a solid academic record otherwise---how will this affect my application? How do I explain this in my statement of purpose?

Answer 1

I'd suggest that, as a guess, 95-99% of your application should be just as it was the first time, focused on your skills and likelihood of success in any new program. I think it would be a mistake to spend valuable words giving a bedsheet of reasons that you are unhappy. Even ignoring the issue altogether might be fine, if a bit risky, especially if you are crossing borders as part of your switch. You are "looking for opportunities not available locally". If it comes up in an interview you might want to raise it.

The one thing you say, however, that might resonate, is that COVID limited your choices last time around and that you picked a less than ideal situation because of it. Also, if you are interested in pure, rather than applied, math, then a quick/short statement that your current institution supports that poorly might be fine.

But keep the focus on the skills and success part, just as you would if this is your first application. People won't choose you because they feel sorry for you but because you seem to be a good fit for their programs and a good bet for success.