<p>Dear Confused,</p>
<p>You are still confused.</p>
<p>At both Princeton and MIT you will have the opportunity to take a sufficient wide range of courses in math to prepare you for grad school. I think that you will do best in math where you ENJOY going to school. If math becomes a grind you will not do your best work. </p>
<p>In my humble opinion universities such as Cambridge, Princeton, MIT, and Harvard have such good math programs that if you do well as an undergraduate you will be able to get into a very good grad program in math. At Cambridge and MIT students are perhaps more dedicated to their math majors. At Princeton and Harvard there are so many other outstanding courses that math majors find interests in new subjects. </p>
<p>For example, Eric Lander '78 in high school he won the Westinghouse Prize (now the Intel Talent Science Search for his mathematics paper on quasiperfect numbers and was second in the country on a mathematics test. He graduated valedictorian at Princeton and received his doctorate in mathematics at Oxford on a Rhodes Scholarship. In 1987 he received a McArthur fellowship to to support his innovative application of statistics to the study of genetics. In 2001 his team at WICGR published a public draft of the human genome in the journal Nature. Lander is now one of the world leaders in bioinformatics and is currently the director of Center for Genome Research at MIT and Harvard. Mr. Lander has transitioned from pure math, then statistics,and now biology. Might you find new interests beyond pure math?</p>
<p>Princeton math majors are required to conduct independent research and write a thesis. Because you develop a close working relationship with your independent research advisors they are able to write personal and detailed letters of recommendation to math grad school committees. You will be able to take a sufficient number of math courses at both universities to demonstrate to grad schools your proficiency to do mathematics at the grad school level.</p>
<p>I do not agree that grad schools in the U.S. require specialization. For example, you can major in any subject at Princeton and be admitted to medical school. Christopher L. Eisgruber, the university provost majored in physics at Princeton, received a Rhodes Scholarship to study at Oxford University, where he obtained an M. Litt. in Politics and received a J.D. from the University of Chicago Law School. He did not specialized in law or politics. </p>
<p>There are more math majors at MIT than Princeton. Is that good or bad. Probably neither.</p>
<p>I encourage you to tell us what you enjoy doing outside of math so that we could tell you more about those interests at MIT and Princeton.</p>