Hi, (This is a long post, so hopefully someone has time to read it) I am trying to make my SOPs as good as possible for the last applications I have left. I already sent in them for 5 universities, but there's still a few to go... My background is that I am studying in Europe, and I really don't know personally anyone who knows anything about US admissions, because no one really applies to the US from my country (almost everyone continue in our own Ph.D. programs, but my interests aren't really represented here, so I need to go elsewhere). There are basically three things that I need advise with. First I've had a pretty rough year. I got very sick in August 2007. Thanks to incompetent doctors at my university health services, I got a thorough examination only in November after having lost 15kg of weight in just 4 weeks and was consequently hospitalized for a month where I was discovered to have ulcerative colitis. The illness got so bad that during last year I was pretty much a prisoner of my apartment (bathroom to be exact) and had constant agonizing stomach pain. Last year, I had planned to take the GRE subject test in October and the TOEFL and general GRE in November. Unfortunately, during the last week of September I was diagnosed with a precancerous condition in my colon and got scheduled for complete colon removal surgery in December. This messed me up pretty badly, because I had hoped some new medications would work, and I wasn't able to study at all for the subject test. I managed to get a 700 (66%) percent though. I got my self together for November though and after a week of studying got an 830 (91%) after retaking the math subject test as standby. This left 2 weeks total to study for the general GRE and TOEFL, which I got a 530V/790Q/4.5AW and 114/120 in TOEFL. Another problem was that writing these standardized exams was a real pain, because the only way to be able to sit down for the required time, without having to run for the toilet at least a few times, was to not eat anything the day before the exam and the morning before the exam, so I was pretty exhausted already before the exams. The question is, should I mention any of these problems on my application and in what extent? Even though I could have done much better, these results are still ok. I wrote a very brief statement on my previous SOPs just telling that this condition had an impact on my preparations, but nothing specific. Should I be more specific or do you think it would be interpreted as whining? The good thing is that the surgery really cured me and I should have taken it a lot sooner. Now the second problem is the interpretation of "research". After having googled about it, I noticed that e.g. the REU program at Chicago is pretty much nothing but writing an exposition about some "advanced" topic (like a proof of the Mordell-Weil theorem). In my country research is pretty much thought as something that has to produce a publishable result. Thus, I have done a M.Sc. thesis, but really nothing more special except having worked as a research assistant in a lab programming heuristic search algorithms (i.e. trial and error). I have taken a few courses by special arrangements which meant reading the course material and then writing a brief exposition about a related subject. The second question is thus, what size of a course project would you list as "research"? The third question is how specific should I be about my interests in my SOP? Should I write that I am interested in arithmetic geometry and number theory or should I be more explicit like I would like to learn about the links between questions in arithmetic geometry and questions about L-functions. Otherwise, my GPA was 4.92/5, which is probably the highest of about 150 students who began as math majors during the same year (though no rankings are calculated by the university). I should have two very strong recommendations and one that I really don't know about, because I haven't had much contact with him. The problem is that there's almost zero interaction between students and faculty and I have really worked with only two professors. Attending lectures is not mandatory and the local way of lecturing is having lots of taught hours going through every little detail which is slow, so I've read everything from books on my own... the end result being that few professors know me personally. The schools that are on the top of my list are UPenn, Columbia and Brown. I have also applied to Cornell, Chicago, Caltech, Johns Hopkins and Maryland. I have pretty much reasoned that if I go abroad, I want it to be a strong program or else I should just stay where I am.