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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?

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    Look into summer undergraduate research opportunities with faculty at other universities, who are active in the areas of research you are interested in. They will be in a much better position to advise you in conducting high-quality research, and their letters of recommendation for graduate programs will carry more weight than that of someone who has no experience supervising PhD students, or sending students on to top departments.
    – ff524
    Jan 9, 2017 at 7:20

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The reputation of your previous program will be relevant when applying to PhD somewhere else. However, this by no chance eliminates your chances of getting into a good PhD program if your CV shows your competence compared to other applicants from other schools. In other words, in cases where the schools (and probably professors) are not well known, having publications in internationally reputable venues (not easy to do in undergrad) along with good standardized test scores might carry more weight than school-related attributes such as GPA and maybe even recommendations. (These will still matter, but less so when compared with GPAs and recommendations from highly known institutions/professors)

I will add this however. Being in a small and newly-established program can have other potential drawbacks. Assuming that a new and small program equates to less experienced professors and less budget, you will be missing out on the higher quality mentoring and availability of the required labs and equipment. Also, the student body will also be smaller, so you will have less chances to find people to collaborate with and share ideas. These drawbacks can be just as bad as the perceived reputation of the school for PhD admissions.

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