Paths to mathematics/university/research while dealing with a few difficult situations I'm a (just turned) 17 year old student (Year 11) from Australia in a bit of a tricky situation. Unfortunately, I spent most of the last few years and am currently in a psychiatric unit. I've also liked math for a while now, specifically certain subfields of algebraic topology and differential topology/geometry, like (topological) k-theory and cobordism for the former and spin geometry/index theory for the latter.
However, I've fallen very far behind in schoolwork and because of the risk associated with my going back to school, my the school admins decided to prevent me from returning  (it was not a good school  anyway - an all boys school (club) full of Neo-Nazis (that isn't even an exaggeration), rapists and the like. This leaves me in a difficult position because I don't thing it's possible for me finish school (I've been described/diagnosed by almost half of the DSM, MDD, Adjustment disorder with depressive features, GAD, SAD, BPD, cPTSD, NPD, OCD, ASD - all at different times).
I also can't really go back to home, so sometime next week I'm being put into a youth hostel/communal living place/refuge (this is bad because I might never get to see my cute dog (golden retriever) again). I really want to go to university to study math and eventually do research(?) in some of the areas I mentioned above but I don't really see a feasible way there. I know asking for life advice online is questionable but I really don't have any idea what else to do. Any suggestions/hard truths/similar experiences/comments in general would be really appreciated. Thank you all.
Universities typically demand qualifications, which you don't have. You could get some: You seem familiar with advanced areas of mathematics, perhaps you can study some revision guides, teach yourself whatever you need to know to get some qualifications. (I'm unsure which institutes you can enrol with in Australia. I presume such institutes must exist. They do elsewhere—not everyone follows the traditional route.) Alternatively, you could reach out to universities, explain your situation, negotiate why they should admit you without typically mandated qualifications. Perhaps you have someone who can help you, e.g., some staff member who happens to know an academic who is willing to help you navigate the system.