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I am currently an undergraduate junior. I have taken most of the standard undergraduate math courses and a few introductory graduate courses (measure theory, algebraic topology, complex analysis, analytic number theory). I am very interested in grad school but I have not completed any research and would like to get a start on some during my senior year. My interests are nearest to number theory and combinatorics, though my heart is far from set on either of those fields and I am completely open to looking into research in other areas. I have so far taken three introductory courses in number theory: elementary number theory, algebraic number theory, and analytic number theory. I have only taken a first course in combinatorics, so research in combinatorics may not be viable.

During spring break, I hope to get a glimpse of a variety of research level topics so I can see what interests me the most and get an idea of what I could work on next year, and potentially start on this summer. Are there any books or other resources that would be particularly well suited for someone in my situation?

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closed as off-topic by sandwich, probablyme, Claude Leibovici, Lee Mosher, Chris Godsil Jan 29 at 12:36

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Seeking personal advice. Questions about choosing a course, academic program, career path, etc. are off-topic. Such questions should be directed to those employed by the institution in question, or other qualified individuals who know your specific circumstances." – probablyme, Claude Leibovici, Lee Mosher
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Probably the best advice is try to do summer research (if it is available) and/or write a senior honors thesis. This means finding a good faculty member at your school to work with. If you have lots of flexibility on the subject you want to pursue, talk to any potential advisor at your school and pick the one seems like the best fit in your opinion. In terms of books, they will probably have books and papers for you to read over the spring/summer. Also, although many deadlines may have passed, you consider looking for an REU here: nsf.gov/crssprgm/reu/list_result.jsp?unitid=5044 – Neil Hoffman Mar 11 '14 at 0:20
    
A bigger list of undergraduate research programs is at ams.org/programs/students/undergrad/emp-reu. The difference is that the NSF only lists programs which get their funding in a certain way, the AMS lists programs regardless of their source of funding. – Michael Zieve Mar 15 '14 at 2:51

A very good introduction to elliptic curves, even though the book is over 20 years old now (in particular, at the time of publication, Fermat's Last Theorem was still open), is Silverman and Tate's introductory book to elliptic curves, Rational Points on Elliptic Curves (http://www.amazon.com/Rational-Points-Elliptic-Undergraduate-Mathematics/dp/0387978259). The study of elliptic curves opens one up to a large variety of contemporary research in number theory.

You may also find Apostol's book on analytic number theory useful (http://www.amazon.com/Introduction-Analytic-Number-Undergraduate-Mathematics/dp/0387901639). It is written at a level appropriate for an undergraduate audience, but it still gives a sufficiently good introduction to the subject. It's a bit dated now, but a lot of the techniques are still relevant.

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I second the recommendation of Silverman-Tate. That book, more than any other I've seen, makes advanced math accessible to undergraduates. It combines tools from several areas of math to achieve some very nice results, and thereby conveys a sense of how different areas interact with one another. I often recommend that book to undergraduates, and recently I used it in a reading course for a bright high school student. – Michael Zieve Mar 15 '14 at 2:40