Is doing well in the standard core undergraduate math courses (complex analysis up to the Riemann mapping theorem, abstract algebra including Galois theory, point-set topology, and algebraic topology, as well as integration on real manifolds and Stokes' theorem) good enough to get into a top-20 math PhD program?
I started with linear algebra after placing out of second-semester calculus. I ended up taking almost all the undergraduate courses offered at the liberal arts college I went to but I still think it's not enough, because most of the Harvard PhD students apparently take lots of graduate courses while undergrads.
To be competitive, are applicants expected to have undergraduate research and do additional math programs in other countries (Budapest Semester, for example), and/or take graduate courses while still an undergrad?
In American universities there are more distribution requirements, whereas in universities in Europe, my understanding is that you declare a major right away and only take classes in that subject. Because of that, they learn a lot more math than American students. I've also noticed that at places like Stanford or Harvard, the undergrad senior theses that students write are super advanced and get into current research. Here are some examples: http://abel.harvard.edu/theses/index.html
Are math undergrads expected to know things like Galois cohomology, the Local Langlands Correspondence for tori or Lubin-Tate theory or the Jacquet-Langlands correspondence?
What could I have done as an undergrad to be able to write a senior thesis on stable homotopy theory or the moduli stack of G-bundles? I am thinking that I did not get a good math education in college.