I am hoping to apply to graduate schools in mathematics this fall, and I am in the process of deciding where I'm going to apply. I have a 3.95 GPA (overall and in math). I go to a top 30 math school. I've taken upper division in pretty much every topic (about 15 undergraduate) and have taken 5 graduate course (and plan to take about 10 by the time I graduate). I consider my coursework one of the strongest points of my application. My GRE scores are General: 800 Quantitative, 590 Verbal. Subject: 810 (87%). As a side question, should I retake the subject test? I was scoring near perfect on all the practice tests, so I believe I could do better, but is my score good enough? I did one REU last summer and have submitted results for publication. I've been to several major conferences where I gave a talk on these results. I did independent study with a famous mathematician last semester and am continuing this work this summer, which will lead into a senior thesis with him this coming year. I've also been a TA for the past two semesters (not sure if this is worth anything) My letters of recommendations should be good to great, being from professors I did research with and professors I have had multiple grad classes with. My interest is in topology, specifically algebraic topology. Right now my list of schools I want to apply to are as follows: Berkeley Chicago Columbia Michigan UPenn Washington Harvard MIT Northwestern I would like to have my list critiqued. Do I have a chance of getting into these school? Is my list too top heavy, i.e. should I add some lower ranked school? Any opinions on these schools with regard to my research interests? Thanks for any type of input!