So I'm wondering what the chances are of me getting into a Ph.D program directly after undergraduate attendance at a school that is totally unknown on the computer science scene. The school is well renowned for it's Business, English, and Pre-med degrees, but their computer science program is tiny and completely unknown. There are 2 professors, only one of which currently has a Ph.D. Their math department is also small, but the professors have Ph.Ds and a a long history of industry work and research.
But here's the catch, I know the professors there extremely well, and they all think I'm a bright student and have said that they would allow me to study whatever I want under their supervision. I could go as deep into the subject as I want. So I'm looking at studying subjects like complexity theory and theoretical cryptography my second semester at the school. This appeals to me greatly because I've always been an autodidact, and I honestly prefer teaching myself things over learning in a classroom.
Because of this freedom, I am fairly confident that I could graduate with some basic research under my belt in one of those fields. As I see it, four years of study seems likely to culminate in at least a single interesting finding of some sort.
If I apply to a Ph.D program directly after my time at this school, would I have any chance at all of getting in? I can foresee my application having challenging coursework on it, along with a little bit of research, but my undergraduate degree is from a completely unknown school.
TL;DR: Any chance of getting into a computer science Ph.D with a little research and a C.S./Pure Math degree from a totally unknown school?