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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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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 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 at 2:51

2 Answers 2

You might explore Euclidean Ramsey problems, as championed by Ron Graham.

I wrote about these problems a decade ago in Comp Geom Column 46. Here is an example that gives you the flavor of the problems:


  Graham problem
So here, a 2-coloring suffices to avoid a monochromatic copy of the unit equilateral triangle.

See, e.g., "Monochromatic triangles in two-colored plane," Vit Jelinek, Jan Kyncl, Rudolf Stolar, Tomas Valla, arXiv link. And especially consult this recent survey,

Graham, Ron, and Eric Tressler. "Open problems in Euclidean Ramsey theory." In Ramsey Theory, pp. 115-120. Birkhäuser Boston, 2011. (Springer link)

This survey confirms that the conjecture I highlighted above remained open in 2011.

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This is a nice problem, and good general advice too. For the OP, combinatorics can be quite accessible for undergrads, especially when combined with geometry or number theory. Some of these problems need fresh new ideas; this is a perfect fit for you. So, in any case, a first combinatorics course may be enough to get going. You sound like you're off to an excellent start with your education, so I'd recommend being somewhat selective with your early research. (I wish I had been.) –  Peter Dukes Mar 11 at 2:43
    
Where can one find other open problems like this one? Also, is the $50 beside the problem supposed to indicate some sort of bounty? –  user89 Jun 10 at 17:33
    
@user89: The link I provided includes a few more problems, plus references. And, Yes, the \$50 is a US-dollar bounty offered by Ron Graham. –  Joseph O'Rourke Jun 10 at 19:28
    
Oh wow. P.S.: I absolutely love the book you co-authored with Dr. Devadoss! Big fan of yours. –  user89 Jun 11 at 19:46

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 at 2:40

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