In math, rec letters are the most important, and whether rightly or wrongly, math professors don't give a damn about the word of anyone other than other math professors.
Figure out how you will be spending your first two years, and the coursework in particular. At good programs it will be killer hard. Are you prepared for this? Your rec letters need to come from people who thoroughly understand how difficult graduate work is, and who can credibly assert that you will succeed in it. This needs to be based on your past success, presumably in your coursework.
In particular, top programs look for not only talent but also thorough preparation. Have you gone far beyond your university's requirements, and absolutely excelled?
If you want to get a Ph.D., and don't get into top schools, you can probably go to a mid-tier place, and if you are really thriving, file transfer applications your second year. Or just finish at the mid-tier place. I went to a non-top-ten university for my Ph.D. and am doing very well now.
Good luck to you!
Do you think open source work/self studying in the field I choose will have a large impact on my potential acceptance?
I regret not focusing in on CS and math earlier.
Self study probably won't help you get into good programs initially, but if you get accepted to a second-tier program, it will certainly help you get the ground running and get the most out of it, whether you try to transfer out or not.
Good luck!
Also, as stated above, you can always move into a top their school later in the game. For example, I went to a state school for my undergrad and PhD, but I've since moved onto Cornell, Columbia and Rockefeller. The school matters at some level, but you can make stuff happen when you apply yourself. As with anything else, it's not trivial, but it's kind of a fun game:) Best of luck, Keith
Best of luck with OmniSci!
A 'top graduate school' is actually somewhere that probably gets a large number of applications from students (Stanford etc.)
That means when you apply for a phd in that school, you are really trying to appease the admissions department, you may not be allocated to a 'phd supervisor' yet.
But if you find which area you'd like to go into and the various 'leaders' in that field, you will probably notice they are not aligned to a 'top graduate school' and possibly deal with admission enquiries directly. (To quantify a leader check someones h-index score on google scholar then take with big pinch of salt)
What's different in this case is that they will most likely appreciate any additional work you've done which shows how interested you are in that particular field.
An admissions department on the other hand will probably look for GRE, GPA, awards, scholarships as ways to differentiate people.
In math, anyway, graduate applications are evaluated by math faculty.
But in my experience, students approach the PI directly, and the PI makes the decision themselves.
There is no formal application process. This is in non 'top grad schools' but the PI would have 40+ h-index
I graduated with an MS in CS from Illinois in about 3 years while continuing to work full-time. When I attended, you could apply via the normal application process. Alternatively, I know of students who were able to take and complete 2-3 classes as non-degree students before submitting their formal application to the program. They could then apply and transfer these credits towards the program -- I'm guessing it's easier to get in too once you have a few decent grades for classes in the program.
For example, a school like Stanford is notorious for having a Masters program that is easier to get into than the undergrad program and is generally not terribly difficult to get into as long as you can pay.