mathematics - How should I choose which graduate programs to apply to for the PhD?

This fall I will be an senior mathematics major at a small public liberal arts university. I'm trying to finalize the list of schools to which I will apply (to PhD programs in pure math), but I am finding it somewhat difficult. I know that I am not a candidate for admission at top-tier universities. My school is relatively unknown outside of its state, and the professors who will write my recommendation letters aren't very well-known either (although they do publish somewhat frequently in their respective fields). I have taken two semesters of abstract algebra, one semester of real analysis, and two semesters of topology, and received an A in all. By the time I graduate I'll have taken another semester in real analysis and a course in complex analysis, plus several other applied and discrete math classes. I haven't taken any graduate courses since my school doesn't offer them, but I have completed a research project with one of my professors in the area of math which I hope to study in grad school. My GPA is around 3.85.

I think (and please correct me if I am wrong) that I should focus on applying to "mid-tier" programs, but I find it very difficult to determine which programs are at this level. I know that one student from my school was accepted at a program ranked in the 40's by the U.S. News math grad school rankings.

Essentially, my question is this: Do the U.S. News rankings accurately reflect the selectivity of programs, and if so, is there some point in the rankings at which schools become "mid-tier" or at which I would be competitive for admission?

Answer

Yes, the rankings (roughly) reflect the selectivity of the program, as far as I know. Your choice to focus on mid-tier programs makes sense. (I think defining mid-tier as starting in the 40s is plausible, though I suspect you'll get conflicting answers from different people.) However

1. 
   

   Don't confuse the quality of the program overall with the quality of their specialty that you hope to study. Their specialty program could be much better or worse than the program overall. In math departments (as opposed to, say, some computer science departments), students are typically admitted to the department, rather than to a professor's research group or a specialty area. Taking advantage of this could get you into a specialty program that's rated higher than you "deserve".

   
   



2. 
   

   Don't exclude a school just because it's too highly rated. It's good to apply to a wide range of schools, some "reach" schools that you think it unlikely you'll be admitted to, and some "safety" schools that you would be quite surprised not to be admitted to. All sorts of factors influence how likely it is you'll be admitted to a given school in a given year, many of them completely unknowable to you. For example, maybe a new faculty member will be coming and looking for students (or maybe one will be leaving). Maybe the program has all their TA positions tied up with current students who haven't finished yet, or maybe not. Maybe a dean wants to grow the program, etc. Many of these things you just can't know. Most likely, you won't get into MIT, Harvard, Princeton, or Stanford. But once you get into the 20s, 30s and below, the outlook is less clear.

   
   


3. 
   

   Focus on fit, rather than solely on ranking. Think about where the school is located. If you plan to live there for 5 or 6 years, you don't want to hate the place. This can include proximity to your family or friends, climate, scenery, nightlife, etc. If you know what specialty you want to study, the school should have at least 1, but hopefully 2 or 3 folks that you would potentially like as an adviser.

   
   


4. 
   

   Focus on aspects of your application that you can still change. With a transcript no stronger than yours, I got into a top-25 school. The cool part is that for the specialty I chose, they were top-10. At this point, most of your transcript (and much of your application in total) is fixed. But you didn't mention your GRE scores. (The surprising thing about the GRE is that you can do pretty well if you're just really good at Calculus (through multivariable), Dif. Eq., and Linear Algebra.) I think my subject test was something like 65th percentile. That's not terribly good, but I think it was enough to convince the admissions committee that even though I was coming from a small LAC that no one had heard of, I did know something. The other thing I did, which I highly recommend you try, is get a letter of recommendation from a faculty member at that school (where I was admitted). I wrote code for him for 3 or 4 weeks (about 30 hours/week, I think) during Christmas break of my senior year. In exchange, he wrote a recommendation for me. I'm sure it didn't say that I was a math wunderkind, but whatever it said added just enough to my application to get me in.