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 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)

Acknowl­edge­ments

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

Con­tacts

Talk to any upper­class­man, but, in par­tic­u­lar:
Max Rabi­novich ’13 (mrabinov@)