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Undergrad Overall: prestige is NOT everything, but if you are looking for prestige, the obvious choice is harvard, if ur looking for focus on undergraduates, brown and dartmouth are small, liberal arts, and undergrad focused as well as princeton bc none of them are dominated by graduate focus
Math: you really want to widen your perspectives on math bc caltech and mit are probably both superior to the ivies in fields of math bc of their intensive focus on it, however, one can usually study anything at any ivy league school and do well if prestige is the issue bc of name recognition
Medicine Fields: Harvard is at the apex of anything science, but once again all of them are very good at it so dont limit urself to harvard
Business: Harvard, Stanford, Upenn Wharton are all big name prestigious graduate business schools as well as supreme undergraduate forces, however, the University of Chicago may be the premier institution to study business in the world as the birthplace of economics
Engineering: Once again Caltech and MIT are superior to the ivies although an ivy engineering degree will ofcourse be excellent
Law: Yale and Harvard are equally at the top of law rankings
However, both of u need to quit looking at prestige as your primary focus's because there are many great institutions beyond the ivies that can present an abundance of oppurtunities
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