<p>To get into Harvard, I would take at least the following graduate courses: analysis, algebra, algebraic topology, differential geometry, algebraic geometry (modulo your interests). You would pretty much need to get As in all of these courses. If you do this, then try to get a high 90% on the GRE subject, a few REUs with good recommendations, hopefully a publication or one in preparation and some impressive awards from your undergrad institution.</p>
<p>Students at Harvard, Princeton etc. usually take nothing but graduate courses after their second year of undergrad. Even at my school ranked around 10th most people have taken only grad courses in their last year or two of undergrad, several have publications, have graduated summa cum laude, and almost all have GPAs in the 3.9+ range. I also know that at least 4 of the 1st and 2nd year students have 800Q/800V on the GRE.</p>
<p>The competition for grad school in math is probably one of the toughest. The reason being that pretty much all applicants are among the very best students in math at their undergrad institution. You really don’t have mediocre people applying for math Ph.D. programs. Another issue is that most schools accept between 10-20 students per year (or even less). Math also doesn’t require language skills in the same fashion as e.g. the humanities and social sciences, so it attracts a significant amount of foreign applicants.</p>