How should I start Undergraduate Research in Mathematics?

Question

I am a college sophomore in US with double majors in mathematics and microbiology. My algorithmic biology research got me passionate about number theory and analysis. I started pursuing a mathematics major this Spring semester. I have been independently self-studying the number theory textbooks written by Niven/Zuckerman/Montgomery, Apostol, and Ireland/Rosen during this semester. As this semester progressed, I discovered that I am more interested in pure mathematics than applied aspects (computational biology, cryptography, etc.). I want to pursue a career as analytic number theorist and prove the Collatz conjecture and Erdos-Straus conjecture.

I have been thinking about doing number-theory research in my university (research university; huge mathematics department). I have been self-studying NT by myself and also regularly attending professional and graduate seminars on number theory. But, I did not do any pure mathematics research as an undergraduate. Should I visit NT professors in my university and ask them if I can do undergraduate research under them? If research is not possible (perhaps due to me lacking maturity), should I request to do independent reading under them and later proceed with research? How should I ask them? What should I address? If even independent reading is not desirable to them, what should I ask them or do by myself?

As for my mathematical background, I have been taking Calculus II (computational) and discrete mathematics. I will be taking Calculus III (vector calc.) over Summer, followed by Analysis I, Probability, Theoretical Linear Algebra in Fall 2015. As for my self-studying on this semester, I have been studying NT textbooks (mentioned above), proof methodologies, and basic linear algebra.

Answer 1

While it is possible for a highly competent mathematician to dole out a doable problem for an undergraduate to solve over the summer, empirical data has proven otherwise - i.e., it's rather hard for an undergrad to prove anything original in number theory if only given a few weeks during the summer.

For an REU in NT, it might be more realistic to expect to read some interesting topics in number theory or perform some numerical analysis. Take it as a bonus if you obtain any original result.

Number theory is known to be a very difficult topic to get into. Before you decide to commit, take courses in complex analysis and abstract algebra. They are crucial if you want to read more advanced texts in number theory.

Good luck!

Answer 2

This depends a bit on where you are. You mention that you are at a large research university, presumably with a large graduate program - in these places I expect that it is pretty rare for an undergraduate to do research directly with a faculty member unless they are exceptionally talented. Faculty members have their own research programs and their own graduate students to direct, and coming up with interesting yet tractable problems is hard! However, it can't hurt to ask. Since you've been regularly attending seminars, you should know who the regular attendees are - send an email to someone to set up a meeting to chat (or drop by an open office hour), tell them what you've told us, and ask if they have any suggestions for what you should do next. Sending the email first gives them the chance to think about your meeting ahead of time, and if they're not interested they could just send an email back saying so. As for what exactly to ask, I would recommend just asking for suggestions and see what they come up with. If they don't seem inclined to do so, asking for suggestions on what to read/which courses to take is a good plan.

In a smaller liberal arts place, perhaps surprisingly, there are more opportunities for undergraduate research, and just generally more opportunities for interacting with faculty members.

However, there are other opportunities to get into research other than working directly with faculty. In a large research university, there are probably lots of graduate students and postdocs (who probably also attend the same seminars as you). They, particularly if they are interested in a more teaching-focused direction, might be willing to talk to you on some regular basis about your readings. (I recently heard of the University of Chicago Directed Reading program, which sounds pretty cool.)

Lastly, I know that you mentioned you wanted to do research at your own institution, but I'd like to ask you to reconsider. There are several summer research opportunities in mathematics, and they are a fantastic opportunity not only to learn math and get the experience of tackling a problem on your own (or in a small group), it's also good to just meet other people at your stage in life with similar interests, as well as mathematicians from other universities. If you're curious, here is a list of summer REUs (REU stands for Research Experiences for Undergraduates).

Answer 3

Should I visit NT professors in my university and ask them about if I can do undergraduate research under them? If research is not possible (perhaps due to my lacking maturity), should I request of doing independent reading under them and later proceed with the research? How should I ask them? What should I address? If even independent reading is not desirable to them, what should I ask to them or do in my own?

Don't be shy. Go talk to some professors! You can start the conversation by telling them what books you've been working with, and ask for some academic advising. This means the professor will go over your plan (which courses to take, when, and in what order) with you. The professor will likely confirm the wisdom of the tentative plan you came up with -- and then the conversation will blossom from there. A professor may make some specific suggestions for coursework and/or independent study. But maybe you should wait until you've got more coursework under your belt before proposing a research project. (Still, someone may surprise you!)

You can start with a short email saying that you have fallen in love with number theory, and would like to make an appointment for some academic advising.

It is always a pleasure when two people who love the same thing get together for a chat. You have nothing to fear.