Undergraduate Research

Research in mathematics and its allies takes a variety of forms: from the most abstract algebraic geometry to the most concrete problems in finance and everything in between. Accordingly, there are many ways to get involved in mathematical research. In general, as an underclassman, the best way to do so is to participate in an REU or other research program during the summer. After that, in your junior and senior years, Princeton provides a natural avenue to research: the junior papers (JP), one per semester during your junior year and, of course, the senior thesis, a year-long project undertaken in your final year. Depending on your background and level of interest in research, however, you might want to consider looking for research opportunities during the year, even in your first two years.

Getting Started [Show]Getting Started [Hide]
How do you get started doing research? In general, the answer is to build background through relevant coursework first. Particularly in pure mathematics, it’s very difficult to jump in without having good preparation. Choosing courses that will prepare you for what you want to do, and help you figure out what you want to do for that matter, is important. Necessarily, the details will vary from person to person, but some advice applies across the board. First: choose carefully and plan ahead. Choose your courses thoughtfully. That means thinking about what you want to get out of each course, how your courses in any given semester fit together (in terms of workload, etc.), and what trajectory you’re trying to follow. A lot of this will be uncertain, especially at the start, but thinking about these issues will help remove that uncertainty. Second: consult your peers, advisers, and professors (not necessarily in that order). They have been where you are now, and they can help you get where you are trying to go. You will of course have to choose whom you ask, and have to combine multiple—often conflicting—sources of advice, but the people around you are an invaluable resource in finding your path.
That being said, you should take advantage of opportunities to get involved in research early on: it’s often possible to find a good project to work on (at an REU, other research program, or at Princeton) even as a freshman. Early in your undergraduate years, you should be open to the possibilities; research experience, even if it isn’t in the area you ultimately want to pursue, is very useful, notably in helping you find your ultimate interests. As always, consult your academic advisers, professors, and friends (especially older friends) for advice.

Seminars, Lectures, and Colloquia [Show]Seminars, Lectures, and Colloquia [Hide]
Attending talks is an important way to find areas of research that interest you. These come in at least three varieties: the Undergraduate Colloquium, which includes faculty, graduate, and undergraduate speakers; the Graduate Student Seminar (GSS), given by graduate students on their research for the benefit of their peers and undergrads; and the various department seminars. If you’re interested in the former two, join the Math Club Listserv and talk to LeeAnn Coleman in the department office to find out how to be added to the GSS mailing list. To get an idea of what goes on in the latter, just look at the seminar listings on the math website at the beginning of every week and see if anything looks interesting. Most seminars provide abstracts, and these will give you an idea about whether you will understand the talk. Look out especially for the department colloquia, because these are usually pitched at a non-specialist level and reasonably accessible—not to mention generally given by very good speakers. The colloquium speaker also sometimes gives a lunchtime talk the day of the colloquium, a practice unique to Princeton; these are also worth attending.
As you attend talks, keep in mind that you will sometimes misgauge the difficulty of a talk and at times the speaker will not be very engaging. This will inevitably happen some of the time, but don’t let it discourage you. Try to get at least a little bit out of each talk you go to; an excellent mathematician once remarked to me that he was satisfied if he could come away from a talk with a single sentence of new knowledge: this is a bit extreme, but the general idea is important. In the end, you should attend seminars because you find their subject matter interesting (or intriguing). While they may help you get ideas for research, when you are just starting out, they will more likely point you to areas that are worth looking at. Moreover, attending seminars has long-term benefits for the aspiring researcher.

Junior Seminars [Show]Junior Seminars [Hide]
Juniors in the department are required either to write a junior paper or participate in a junior seminar during both semesters. Junior seminars are a learning environment with which you are probably completely unfamiliar. First, it’s largely up to you how much to engage with the lectures and the course overall. For this reason, it’s especially important to pick a seminar whose subject matter interests you. This may be a challenge, as very few seminars are offered, but do your best. Actively following the speaker’s exposition and asking questions where appropriate—never be afraid to ask a question if something is unclear!—are great ways to stay focused on lectures while attending them. To keep up with the course overall, you will want to do the assigned readings, even those that are for talks other than your own.
At some point during the semester, you will spend a couple of weeks learning a certain bit of the main topic of the seminar, often contained in a chapter of a textbook or a journal article, and then make a presentation at the seminar. It’s important to give a good talk: it’s your turn to teach your classmates the material. Whether this is your first talk, or you are a veteran lecturer, the best way to ensure a good talk is practice, practice, practice. Practice it at least twice before you give it in the seminar, ideally with another student or even the instructor (feel free to ask).
Junior Seminars culminate in a final paper on some topic related to the theme of the seminar. The details vary from seminar to seminar, but, unlike the junior paper and senior thesis, which sometimes include original components, the paper will be purely expository and will generally represent a much more significant effort than a problem set. For advice on finding a topic, consult the section below. Your adviser in this context will be the seminar instructor. Also keep in mind that your paper will not be as elaborate as a senior thesis or junior paper.

What Type of Project is a Senior Thesis (or Junior Paper)? [Show]What Type of Project is a Senior Thesis (or Junior Paper)? [Hide]
It’s probably a good idea to start off by saying a bit about what type of “project” a senior thesis and junior paper is. Even if you have participated in an REU or another summer math program for undergraduates, your biggest question might just be, “What type of project does one do with a faculty adviser?”
Most theses and JP’s center on a particular important result, research area or program, or major conjecture. The thesis could give an exposition of a proof of a major result (perhaps extending it to slightly more cases), or of major partial results, or of recent results in a research area. It is also possible to undertake original research—though if this is what you want to do, you should ideally prepare yourself thoroughly for it during your first three years (see above). Junior papers are similar, though necessarily less involved, and the project can vary from a specific unsolved problem to an introductory exploration of an area unfamiliar to the student. Ultimately, a thesis or JP is an early step in the career of an aspiring researcher and, accordingly, it is foundational—you will not prove the Riemann hypothesis (probably), you might not end up proving anything, but you will gain valuable experience and learn mathematics that will continue to be useful to you later on. For specific examples, look up old theses in the library and, as always, talk to your older friends!
Finally, remember that both a senior thesis and JP require significant writing. Mathematical writing is rather different from the writing you have probably done in other contexts. Various resources on the subject can be found on Terry Tao’s blog, here.
In this guide, we’ll just point out that writing clear proofs and definitions makes the relevant concepts much clearer to you and can help you notice subtleties of—and subtle mistakes in—your arguments. One way to help yourself do this, and to find a style to start yourself off with, is to recall a textbook you read that was particularly written and model your writing on its style (not the content, though).

Finding an Adviser [Show]Finding an Adviser [Hide]
Starting a thesis or JP requires two major steps: choosing an adviser and choosing a topic. In the overwhelming majority of cases, the former comes first. Choosing your adviser carefully is important. Your adviser’s style and the compatibility between the two of you will deeply influence the quality of your experience. Since you have virtually no time as an upperclassman to experiment with possible advisers—though this will likely happen accidentally anyway—you should consult older students and the faculty academic advisers to figure out which professors might be a good fit for you in terms of research interests and advising style. As always, you will have to be proactive to ensure your experience is all that it can be; ask your peers many questions: about the frequency and content of meetings, the expectations for an undergraduate project (too low? too high?), the level of preparation expected, and so on.
Once you have an adviser, you will still need to find an effective way to work together. Sometimes, this will come naturally; that’s especially likely if your adviser often takes undergraduate students. Be that as it may, figure out how often meeting with your adviser is productive; once a week is standard, but some professors prefer biweekly meetings. Even if you have nothing to report, meeting with your adviser helps both of you stay in touch with the project and is an integral part of the research experience. You will also want to prepare for your meetings so as to get the most out of them. While meetings will be your primary interaction with your adviser, e-mails and other day-to-day interactions can be nearly as important. These generally take the form of questions and, here again, you will need to figure out how to make them work best for you. Experience is, for better or for worse, the only real way to learn how to do this. Finally, whoever your adviser is, you will benefit from making friends with their graduate students and postdocs (short for “postdoctoral fellows,” researchers who have recently obtained Ph.D.’s); they can serve as secondary advisers who can help you on a day-to-day basis—and share their experiences with early career research.

Finding a Project [Show]Finding a Project [Hide]
Choosing a project is just as important as choosing an adviser. Working on a problem that fascinates and excites you will make your research more enjoyable and rewarding: it is also the best way to ensure you stay motivated throughout the project. Your adviser will likely suggest at least one project at your first meeting, or over e-mail beforehand. Starting the discussion early, whether by e-mail or in person, is helpful; doing so will allow you to go through a few possibilities before committing to one. To decide if an idea is one you want to pursue, you should read about it online, in books, and in the technical literature (for specific places to look, consult your adviser). After you’ve had a chance to look into it, don’t hesitate to voice your doubts about a project if it seems over your head or not interesting to you! All this being said, you should choose a problem early in the term—certainly within the first two or three meetings with your adviser. The easiest way to achieve this is to approach potential advisers well before term starts and to be pro-active about finding a project in advance.
While it’s hard to give general advice about choosing a problem, there are a few things to keep in mind. First, choose a project in an area of active research. This will have many benefits, most notably perhaps that you will have access to a wide selection of papers that can help you as you work on your problem. Second, as mentioned above, choosing a project is to a large extent a judgment call: you should choose a topic you find interesting and beautiful if at all possible. Finally, keep in mind that in research, especially early on, things rarely go as planned: balance perseverance with flexibility, and don’t be afraid to change course if necessary, particularly as you figure out what you really want to do with your research.

Advice on the Research Process [Show]Advice on the Research Process [Hide]
Read, read, read. And Google. Take time to read about things related to your research. Google—especially Google Scholar—is an excellent way to find material useful to you, often better than arXiv or mathscinet, two popular online technical libraries. Being a Princeton undergraduate means getting access to most journal articles you’ll need, and you should definitely take advantage of the convenience! Google can help answer queries both big and small, from “Find a readable introduction to algebraic K-theory” (huge) to “What does convex cocompact mean?” (tiny). In fact, Googling is almost always the fastest way to resolve any confusion you have as you struggle to understand an idea or term. Ultimately, you may need to read in a more structured way—by focusing on a book or a specific collection of journal articles, for instance—but unstructured reading can help prepare you for that, and help give you a general idea of how the area you’re working in, well, works.
Daydreaming (about research) is good for you. If you have a specific problem you need to solve, whether it’s finishing a mostly finished proof, solving a tricky problem, or understand a particularly hard part of a paper you’re reading, it’s frequently useful to daydream about it at random quiet moments during the way: while you walk to class, shower, wait for a friend to meet you for dinner, or whatever. You’ll likely find, as countless mathematicians, scientists, and humanists before you have, that you have more idea during this unstructured time than during scheduled blocks set aside for research (though they have their place—see below).
You need structure, too. Regularly setting aside time to work on research is essential. Progress in research is nearly always incremental and non-linear: it takes time and patience. A thesis or a junior paper is a fundamentally long-term project, and to do your best work you will need to treat it that way. To keep yourself productive during structured time, it helps to focus on concrete tasks (e.g., “read this paper”, “work on this part of this proof”, “understand this technique I need”) and to work on answering specific questions.

Other Useful Resources

Terence Tao on time management

Ravi Vakil’s advice for graduate students (some of which is applicable to undergraduates)


Many thanks to John Pardon ’11 and Max Rabinovich ’13 for contributing this article.

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