Why do so many pure math PhD students in the USA drop out or leave academia, compared to applied mathematics PhDs in the USA? I notice that many pure math PhD students, doing things like algebraic geometry or representation theory, drop out and enter industry and become software developers. The ones who finish their pure math PhDs also drop out and do things such as blogging, tutoring, etc. These students are pretty "top of the heap" too, attending the top pure math PhD programs like at UC Berkeley and UCLA.

Then I notice that many applied math PhD students typically finish, and also get post-doc offers. These PhD students are typically doing numerical methods for the solutions of partial differential equations.

Why does this happen? Do pure math PhD tracks often lead students to dead ends? Is it too hard, even for the brightest PhD candidates at the best programs?

In numerical PDEs, it seems that just a little progress leads to a PhD degree, e.g. solving numerically a PDE in one space dimension. (I know firsthand of someone getting their applied math PhD from an Ivy League program for solving a PDE in 1D.)

I have friends in both pure math and applied math PhD programs and have noticed this trend for many years now.
Only my personal experience, but I hope it will help you a little bit:

I am almost in the exact situation you describe in your question. I finished my PhD in Algebra (coding theory, lot's of linear algebra and representation theory) and am now working as a software developer. As a fun fact, this wasn't always planned. When I started my PhD, I was still dreaming about getting a PostDoc and ending up as a tenured prof at a prestigious university.

For me, personally, there were three reasons to take this step:


The PhD was quite hard and taxing on my mental health. Having a rather theoretic topic, you sit at your desk all day and try to come up with an idea. There is often no way to track progress for a long time, as you either have a clever idea or you don't, you can't set up a step by step plan to see if you are doing ok for such things. There is also always the fear that you simply won't have the right idea before your time/funding runs out. Due to that reason, I decided already some time before finishing it that I will run from academia without looking back as soon as I am done.
Closely related to the first reason, I wanted to do something more applied. I found a job where I can use my skills to really help people/the climate/... (our company develops AI solutions for agriculture, allowing farmers to better use resources and minimizing the need for pesticides).
I still plan to become a professor, one day. But after that experience during the PhD, I'm aiming for a position that is more focused on teaching and applied research (e.g. with industrial partners). For this, at least here in Germany, you need some years of industry experience.