Moving from mathematics to astronomy/astrophysics I'm currently studying for a PhD in pure mathematics (the topic is the cohomology of differential graded categories, just to give you an idea of how pure I mean), although the first year of my undergraduate degree was in mathematical physics before I transferred to mathematics, so I have a little (but unfortunately not that much) background in physics as well.
I'm seriously thinking about whether I want to pursue pure mathematics as a career, and since astronomy and astrophysics were, one could say, my first love when it comes to science-y type things, I've been wondering if it would be possible for me to switch fields, and what would be necessary to do so. Although I'm aware that astronomy/astrophysics is a wide area and could mean lots of different things in terms of a profession.
To give you a little more background, my current level of astrophysics knowledge is propped up mostly by my short time studying physics at university and a continuing interest and reading various non-technical books and low level textbooks of various topics. The topics I've found most interesting are things like supernovae, and the formations of stars and star systems, although I haven't studied the mathematics or technical details of these things in any serious depth. I should say that I split off from applied mathematics some time ago, so topics such as PDEs and statistical analysis are not areas I'm adept in (although I have no doubt that I would have relatively little trouble learning them at this stage). I've also spent a short amount of time (~ 1 year) working as a software developer, so I have some proficiency with programming. And actually I thoroughly enjoyed my time as a developer, and I know that software development/programming is an important aspect of work and research in this field for analysing images, data sets, creating simulations and that sort of thing. So something of that sort is of great interest to me, especially if it could directly involve mathematics. So perhaps working with simulations would be a good option for me?
Is a change of this kind possible for me? And if so what would you recommend I do to make it happen?
Apologies if this question is off-topic by the way. Please do let me know if there are more appropriate places to find this information.
There are fewer placements than qualified applicants so competition is strong.  Moreover, suppose this was the other way round:  A person doing a PhD in (say) planetary nebulae comes to the maths department saying that they are really interested in maths, they're good at equations and calculus and really like all that topology stuff (you know, rubber sheets and so on) so can they start doing cohomology of differential graded categories?
It's not impossible, and a strong foundation in maths is a prerequisite for progress in Astronomy but the question you should ask is why should a university hire you, when there are 5 others with PhDs in astrophysics?
Talk to your supervisor and careers advisor before burning any bridges.