Realistically speaking, what league of schools does a hardworking, smart but not brilliant math major have a shot at for a PhD in pure math? Let's say I might apply with 8 graduate courses at a "lower Ivy", coauthored 1-2 papers, solid recommendations. I am not getting my hopes up for Stanford or MIT. But just how selective are Michigan and Wisconsin? Cornell? What about Berkeley? I will probably have a conversation very much like this with my professors in the not-too-distant future, but I would appreciate your thoughts!