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I'm very excited to start my undergraduate thesis and I've heard some recommendations for topic but I have a questions and I'd like to hear your opinions.

Should I look for the most advanced topic I can understand on an area I'm interested to work in graduate school or should I work on an elementary topic with the things I've learned so far, or something else?

Also, what should be the point on a undergraduate thesis, should I use it to work on the topics I'm interested to study in graduate school or to go deeper in topics I know or what?

Thanks in advance.

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I think that this is first and foremost a question for your advisor(s)/professors at your school. I don't mean to say that you should only ask them, but hopefully you're meeting people to discuss this, too. –  Jonas Meyer Dec 18 '12 at 1:34
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In addition to Jonas M's wonderful reply, it is okay to explore, but this can really only come from the school and advisors. Having said that, this is typically the sort of things that are considered, for example How do I select a topic?. In other words, there are guidelines. Here is another example at MIT Undergraduate Thesis. Does your school have a published policy or web site? Regards. –  Amzoti Dec 18 '12 at 1:58
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Some of Mexico's universitites are the only ones I know (this is important for debate's sake) that used to require thesis for undergraduates. Now there's another option in some but it still exists that condition. Anyway, from your best loved subject (say, Number Theory), choose a nice subject there (Say, Dirichlet's Theorem on primes in arithmetic progressions) and develop the whole material and the proof giving it a personal touch. Anyway, it is not usually expected to be original work. –  DonAntonio Dec 18 '12 at 3:09

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