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What are the Pros/cons of Harvard "non-degree/special student" application vs. PhD application upon graduating?

I am a second year mathematics/philosophy undergraduate student at a fourth tier university. I don't want to sound conceited or anything of that sort, but I suppose that I'm what you might call a "gifted" student; that is, I maintain a high GPA, participate in a good deal of academic and extra-curricular activities (e.g. community activism, Putnam competitor, debate, mathematics tutor, 6th grade math teaching assistant, presented papers at academic conferences, etc.). Unfortunately, I don't find myself well served (at all) at my current institution, Wayne State University. My departmental (mathematics) adviser is a nightmare, who essentially told me not to try and get into a top PhD program in maths. The facilities are somewhat sub-par, and the administration is a generally clunky, bureaucratic nightmare. Therefore (I am getting to the "question" part soon enough!), I am hoping to transfer, either to the University of Michigan, Ann Arbor (Go blue!), Michigan State U's honors college, UNC--Chapel Hill, or Vanderbilt. Due to this, there will be a lapse in my mathematics education for a while. Despite having a deep familiarity with high-level mathematics, up to the level of algebraic topology/homology theory (I am researching the general Grothendieck cohomology right now), I will not have as much de facto experience to show for it. My dream is a PhD program in pure maths or in logic (via applied maths) at Harvard, Columbia, MIT, Berkeley, or UCLA. However, I think that I would benefit from a year-or-so of advanced mathematics courses, beyond the basic undergraduate level. I wish to do something similar to the Tripos III program (Cambridge) PRIOR to taking the math subject GRE, in order to see if I have the wherewithal to commit to a graduate course in mathematics. I was wondering if I should simply apply, upon graduation, to non-degree/"special student status" programs at Harvard, Princeton, Yale, and other "reach" schools, as these are often easier to get into and provide greater flexibility, while bolstering the strength of one's curriculum vitae. Should I just take the subject GRE and apply directly to PhD programs? Should I take the GRE and apply to both? What would you suggest? Thank you!

Public Comments

1. This question is very difficult to answer without actually knowing what classes you will have taken and how much research you will have completed by your senior year. A year as a special student taking more advanced classes will likely help your application, but it is certainly possible that you could get accepted without it. There isn't enough information here to know, mostly because you aren't far enough along in your undergraduate degree yet. Most of the make-or-break work done by successful PhD applicants is done during their junior/senior years. My suggestion would be to get accepted to your transfer program first before worrying in detail about PhD admissions. Once you have started taking math classes at your new university, start doing research with a professor there. That professor will likely have insight as to what the best path would be. It sounds like you haven't been able to get help at your current university, but in my experience most professors in good college programs are very excited about encouraging students to apply for PhD programs. I bet that once you transfer you will be able to get good advice on how to prepare yourself best given the specifics of your situation. In general, there is no harm in re-applying to a PhD program. If your GRE scores significantly improve, note that in your essay for your re-application. GRE scores are rarely make or break though; your research is far more important. Don't spend an extra year trying to improve your GRE scores...spend it trying to do more research. It might be worthwhile to consider staying on at your undergraduate institution for a year and continuing to work with your thesis advisor. If you are unable to transfer, consider applying to REU (summer research) programs at other universities. That way you will get both research experience & access to more helpful professors.