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How do I take my mathematics to the next level?

  1. Apr 18, 2013 #1
    How do I take my mathematics to the "next level?"

    Let me explain what I mean:

    I am finishing my junior year in undergraduate. I consider myself "good" at math, in the sense that I make A's in my classes. But that's not enough.

    I want to be a world class math student because my dream is to go to a top 10 graduate school one day. I've looked at the preliminary exams, say Berkeley's for example, and I am not on that level yet. Despite taking around 18 math courses, I'm not skilled enough to tackle those problems.

    Also, I've taken the Putnam exam. I scored a 0. I'm not on that "next level" of mathematics yet.

    I know a great deal of natural talent is required to get into a school like Berkeley, but I believe that hard work goes a long way and can make up for not being a prodigy or genius.

    How do I take my math game to the next level? Obviously doing problems. But what does that entail? Drilling analysis and abstract algebra problems for a few hours everyday? What should my game plan be?
  2. jcsd
  3. Apr 18, 2013 #2
    The obvious way to prepare for the math required to enter grad school seems to be learning the math taught to undergrads at university.
  4. Apr 18, 2013 #3
    Yes, I'm aware of that. But does that mean I should whip open my analysis and algebra textbooks and just drill problems everyday for a few hours? What's the most efficient way?
  5. Apr 18, 2013 #4
    I would recommend attending courses and lectures. That is the best answer I can sensibly offer without drifting into the realm of "anything goes"-fantasy. But I am aware that this is not the answer you are looking for. So I better refrain from this thread and hope for others having ideas more to your taste.
  6. Apr 18, 2013 #5
    I did just that getting ready for my prelims in grad school. I knew my undergraduate stuff COLD. That way you have a good base when stuff starts getting weird in your graduate courses.
  7. Apr 18, 2013 #6
    I am in no position to give advice since I know nothing about prelim exams But I do have a suggestion. Maybe you can pick up some rigorous textbooks if you haven't already done so. I see spivak mentioned a lot and has quite the rep.
  8. Apr 18, 2013 #7
    The only Spivak book that might be useful would be his differential geometry / calculus on manifolds books, but those topics aren't covered at all on the preliminary exams, so I don't really need to worry about them.
  9. Apr 18, 2013 #8


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