Are undergrads applying to top math Phd programs expected to have taken core graduate clases?

Question

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.

Answer 1

No, possibly except for Princeton.

Berkeley in particular has a long tradition of accepting domestic graduate students who have less background but strong potential. (I had not learned Galois theory or anything about manifolds when I started graduate school there, and I still have never learned the Lebesgue integral.)

Graduate admissions committees in the US trust what recommendation letters say (with evidence) about the potential of students, and take that into account a great deal, in many cases attaching more importance to this than any actual achievements of students. Their statistics based on how students have done in their programs and after finishing is that how much math you know coming in actually doesn't make much of a difference; much more important is your demonstrated capacity to learn math and solve problems.

Keep in mind that it is quite possible to do quite impressive undergraduate theses despite not knowing much mathematics if one has a good advisor (who can explain how the problem can be reduced to an elementary(*) one), works hard, and is clever. (It's quite common for PhD advisors to complain that their weaker students don't actually understand their own dissertations.) Keep in mind that undergraduate admissions at Harvard or Stanford is extremely selective - generally more so than graduate admissions in fact - and every student at Harvard or Stanford is very clever.

(*) elementary meaning not requiring knowing lots of mathematics, not meaning easy.

Answer 2

Not necessarily, but it could help. Examples would include getting exposure to a field of interest that you wouldn't be able to get in your undergrad, or if you are doing it as a combined undergraduate/graduate degree program where doing so would count for credit towards both degrees.

If you are applying to a PhD program, you'll take the graduate courses only during the first couple of years generally, then coursework will take the backseat to research, which is what the degree is all about (unless, you leave early with a masters, which you still will likely get research experience out of). So while it's not a bad thing to take graduate courses, don't fret too much on this, and try and get some research experience/exposure as well, since that will be primarily how you get the recommendation letters that you need to apply to PhD programs.

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

This is very dependent on the situation with regards to where you did you undergraduate. If you did it at Harvard or another very top research institution, you'll have such opportunities to do research or take graduate classes. At smaller liberal arts colleges, from my understanding, people don't have as much of these opportunities readily available at their own institution. Graduate committees are generally aware of this and factor this in when making admission decisions.

So no, you're not expected to take graduate classes as an undergraduate (although if you have the time to do so, you could definitely get your feet wet in this), but you should have some research experience in order to demonstrate the ability to do research at the graduate/PhD level.