<p>I'm currently a math major finishing up my junior year at a not particularly high ranked state school. Before this, I was previously at community college where I managed to wipe out my Gen. Eds and earn a rather useless AAS in Computer Science. Since transferring, I've maintained a 4.0 and taken several graduate courses(Algebra I-II, Complex Analysis, Representation Theory, Algebraic Topology, ect) and independent studies beyond the standard B.S curriculum. I'll also be participating in an REU (My first REU) this summer in addition to honors thesis project in the fall(On algebraic geometry, hopefully). Beyond that, I'm quite comfortable with much of the material in the math GRE and have done fairly well on recent practice exams(760~ ish).</p>
<p>Assuming the REU and the GRE go well, and I get good recs from the professors I've worked with, I'd like to get a feel for which schools I should I apply to in the fall. My dream school is UCB, but that might be aiming a little too high. Other schools I'm looking at include Columbia, Cornell, Wisconsin, Michigan, Stony Brook, UCLA, NYU, Brown.</p>
<p>Mostly, what I'm wondering is if I should instead plan on applying to a broader spread of schools. Is it too much to hope that I'll get accepted to at least one of the above schools? What non-Top 25 schools should I look at as someone interested in Algebraic Geometry/Topology/Category theory?</p>