I am currently a first year graduate student in a pure math PhD program in a top-100 university studying topology. I declined offers from top 20 schools to attend this school because I liked the professors at this university and the location of the school was good.
After going to a couple of conferences and talking to some post-docs/grad students from other universities, I found out that the career prospects for someone like me look quite terrible (if I wanted to stay in academia). Even grad students from Stanford/Berkeley (i.e. top schools) were having a bit of trouble finding a spot in academia. Given that I'm really far down the "food chain" and I want to stay in academia, it looks very unrealistic for me. Also, the math is really hard :)
I realize that I should go to a top-20ish school to have access to enough resources to make myself a competitive applicant in the future (I realize that I should focus on my work, but it's hard to grow a lot at my current university given the lack of learning seminars/seminars in general).
That said, I've been going to conferences and giving talks and "doing the right things." However, I am finding that the current research I am doing is quite abstract and I don't have people I can regularly talk to about it. I've recently discovered the subfields of topological data analysis and DNA topology and I'm finding those subjects really appealing in that I can actually see how my work might impact people and I can actually understand the material that is covered in seminars (the topics are very approachable).
From what the grad chair told me, I'm one of the strongest students that has entered this school's PhD program so I'm suspecting that, if I want to transfer, my letters of recommendation would be fairly strong.
However, the main issue for me is that I still do want to stay in academia. I know that I don't need to go to a top 20 school to do "good" work but I don't see myself finishing my PhD at a school that isn't in the top 20.
So, that's the background.
I really want to transfer to a top (whatever that might mean) applied math program from my current institution. I have only taken pure math classes. I don't know how to nicely approach my professors and say "I don't want to stay here. Please write me a letter of recommendation so I can get out." My current plan is to say something like "I've developed some interest in applied math and we don't do that here. I'm seeing myself doing research in applied math. Will you write me a letter (etc)?"
How realistic is it for me to transfer to a (top) applied math program? My GPA is currently a 4.0 and my undergrad was completed at a top 5 university (with a 3.8).
If it is realistic, how do I maximize my chance of being admitted into a (top) applied math program?
If it is realistic, what are some better ways to approach professors at my current university about this?
If it isn't realistic, I might just finish a PhD here. I know that I can get more specialized attention at my university, but I thought I'd do a preliminary pass here to get a sanity check before I ask people in my department. I am not familiar with people transferring from pure math to applied math programs.