I am slightly confused as to whether or not it's worth my time to apply to top tier math PhD schools. I don't know whether I have a viable chance at what are considered "top tier" schools, or if I should "stay in my place" where I am now. The reason why I ask is because professors have told me that I have a glimmering shot at good schools, but my academic past has been a slow start. I haven't talked too much to advisers, and have recently considered getting a PhD. I am interested in discrete structures.
My academic past involves starting off at a smaller community college and taking remedial courses in mathematics after a bad high school start. My math SAT is on my CC transcript, and it's pretty poor. However, I worked my rear off and got a CC GPA of 4.0/4.0 and after summer classes, I worked my way up to the Cal I-III series before I transferred to a university. I also took CS courses at the CC, and tutored calculus mathematics as an employee there.
At my University, I earned a transfer scholarship as well as a scholarship from the DoD, in exchange for going to the DoD lab in the summer for research in constructing algorithms.
I have 2 REUs in discrete structures, one with applications in physics from a top 20 school. The dean of admissions there told me when I went there over the summer that I was the only CC student there in the history of the school's REU program. He said my statement of interest was well written.
I will graduate 3 semesters from now due to taking interdisciplinary coursework. I have finished most of my math requirements, including netting A's in linear algebra, the real analysis sequence, and a graduate discrete course. My GPA is ~ 3.85 as of now (slightly higher).
My question is, do I have a shot at a top tier school, or should I just stay in my place and stick around at my uni? Does my CC past and my remedial courses poison my application? My thinking is, I've already made it this far as an aspiring mathematician, and I believe I could have more in store.
How do schools view students who have turned over a new leaf? My current uni isn't prestigious, it's an average school, probably only the 3rd most notable state school in my state.