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I believe this is the right place to ask this, so I was wondering if anyone could give me advice on research at the undergraduate level.

I was recently accepted into the McNair Scholars program. It is a preparatory program for students who want to go on to graduate school. I am expected to submit a research topic proposal in the middle of the spring semester and study it during the summer with a mentor.

Since I am currently in the B.S. Mathematics program and I want to get my Masters later. I figured that while my topic can be in any area, it should be in math since it is my main interest as well.

I am a junior at the moment and taking: One-Dimensional Real Analysis, Intro to Numerical Methods, and Abstract Algebra. I frequently search MathWorld and Wikipedia for topics that interest me, although I don't consider myself a brilliant student or particularly strong. I have begun speaking with professors about their research also.

I have not met any other students doing undergraduate math research and my current feeling is that many or all the problems in math are far beyond my ability to research them. This may seem a little defeatist but it seems mathematics is progressively becoming more specialized. I know that there are many areas emerging in Applied mathematics but they seem to be using much higher mathematics as well.

My current interest is Abstract Algebra and Game Theory and I have been considering if there are possibilities to apply the former to the latter.

So my questions are: 1) Are my beliefs about the possibilities of undergraduate research unfounded? 2) Where can I find online math journals? 3) How can I go about finding what has been explored in areas of interest. Should I search through Wikipedia and MathWorld bibliographies and or look in the library for research?

Thanks I hope someone can help to clarify and guide me.

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This question should be community wiki in my opinion. –  Grétar Amazeen Nov 12 '10 at 10:38
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All of the papers published in the journal Involve have a significant student contribution. You can read the abstracts at their website (involvemath.org). I am not saying this is the place to go to find a research topic (or that you should attempt to find a topic on your own via the internet). But I know from experience that it can be fun and inspirational to see that your peers are researching and publishing in a variety of areas. –  user4977 Nov 13 '10 at 2:25
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Note to the advice-givers: there is also a user Metahominid at the math.SE site who has asked a few questions about basic abstract algebra and real analysis. Reading over them gives some information (far from definitive, of course) about the student and his current level, which might result in more personalized advice. –  Pete L. Clark Nov 13 '10 at 19:26
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That is me. They are basic because I am in the class. I am not trying to give the impression I can do any significant research. There have been almost no math McNair students, and I suspect this because of the difficulty. The point of the program from what I have been told so far is to look into a topic of interest, and pursue it. There is only the expectation to submit, and not that it be good or profound. Mainly it is to prepare students for the research experience. The only other math McNair students have done math education, I would prefer not to. Thank you to everyone. –  Metahominid Nov 14 '10 at 0:22
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Congratulations for making it into the program. There are already tons of answers, so I won't add much to them, except to say first that undergraduate research can be extremely valuable for your development (to help you appreciate the difference between solving exercises and tackling real questions, even if someone already knows the answer btw). Ans second, a point that has already been made but cannot be stressed enough, that a successful UG research experience requires a good mentor, so find her or him first, and then worry about the topic. –  Thierry Zell Nov 16 '10 at 18:14

16 Answers 16

up vote 27 down vote accepted

Since you are a student who's already interested in going on to graduate school and is specifically asking about finding a topic to study at your undergraduate level program at McNair, please disregard the negative nattering nabobs whose answers and comments suggest that undergraduates have no place or business in trying to perform research, whether it's research as defined for all scientists or the "research experience" that is put together for undergraduates and for advanced high-school students. Undergraduates can definitely perform research, or even benefit from going through a structured and well-administered "research experience".

I agree with Peter Shor about finding a mentor, or multiple mentors, as soon as possible. There's no reason you have to be limited to getting advice from just one professor or teacher.

I agree with Ben Webster, specifically about speaking with professors in order to get a reasonable idea about the level of work that would be needed for you to perform useful research at an undergraduate level. A few other suggestions come to mind:

  • if you are at an institution that offers Masters and Ph.D. level degrees in mathematics, then your institution's library should have multiple research journals in hard-copy. I have found that it is much easier to go to the stacks in the library and browse through one or two year's worth of Tables of Contents and Abstracts in one journal in an afternoon or evening. This will familiarize you with the types of research papers being published currently, and make you aware of what "quanta" of research is enough to be a single research article.

  • make sure to attend Seminars, Colloquia, and (if your school's graduate students have one) any graduate research seminar courses that you can find time for. This will allow you to become more familiar with various subtopics within the topics of your interests, and to see what the current areas of interest are for local and visiting faculty members.

  • Colloquia are great as they often start by including a brief history of the topic by an expert in that field.

  • Seminars are great because they allow students to see the social aspect of math, including the give-and-take and the critical comments and requests for more detail and explanation, even by tenured faculty who don't follow a speaker's thought processes.

  • Graduate student seminar presentations are great because a student observes how graduate students can falter during presentations, how they are quizzed/coached/criticized/mentored/assisted by faculty during their presentations.

  • I'll admit that I'm not sure attending dissertation defenses would be of any serious benefit to the undergraduate student, other than observing the interaction level (animosity level?) between faculty and graduate students.

  • absolutely make sure to schedule some time to meet with mathematics professors who specialize in the fields of your interest, and communicate your desire to do research while you are an undergraduate, and communicate your desire to go on to graduate studies in mathematics.

  • look on the internet and search for undergraduate opportunities for research in mathematics. I guarantee you will find quite a number of web sites that can give you more information. MIT has an undergraduate research opportunity program that many of their students take advantage of. Your institution may have professors who can speak with you and give you advice.

Also, make sure to speak with more than one professor, and do not take any single person's advice as being the final word. Mathematicians are human beings too, and subject to the foibles and inclinations and disinclinations that all human beings have. If you run into disgruntled and critical individuals, do not let that dissuade you from going on into mathematics or decrease your desires. If you run into overly optimistic individuals who praise you too much and are too eager to take you on to do "scut work" computer programming, thank them for their time and let them know you'll come back to speak with them after you've spoken with other professors and weighed your options. Don't turn anyone down immediately. Always be polite in speaking with professors and teachers. Ask them how they chose their topics for their degrees, and you'll learn a lot.

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I'm sorry, but this seems totally unrealistic. Anyone who, as an undergraduate taking real analysis and abstract algebra, can follow research-level seminars, either A) should expect the Fields Medal within a few years, or B) doesn't exist. Most of the WORDS in most seminars would be literally meaningless! For a student who already seems discouraged about their ability, this is a recipe for crushing disappointment. My advice: don't worry so much about research as an undergraduate. Find a problem you are interested in, open or not, and a professor at your school you feel you can talk to. –  Tom Church Nov 12 '10 at 13:34
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@Tom-Church, much like pre-med students who can volunteer and observe at hospitals without being able to participate (or understand all of the details) at the level of medical doctors, it is possible for students to attend research-level seminars to get an idea of the type of topics which are being discussed and are at the fore-front of research currently. How is the student going to find a problem to be interested in without at a minimum glancing at the field, reading abstracts, and seeing if the topics tickle his/her fancy? This isn't unrealistic. I, and many others, published as undergrads. –  sleepless in beantown Nov 12 '10 at 14:06
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I would encourage enthusiastic undergraduates to attend colloquia (meant for general audiences of mathematicians), not necessarily just to follow research trends, but also to watch interactions between mathematicians. Eavesdropping on mathematicians at tea can be interesting too. The point is that mathematics is intensely social, and it's good to see what mathematics and mathematicians are like away from books, articles, and lectures. If OP maintains a healthy balance between "I've got a lot to learn" and "this looks like fun, and something I want to do", where's the harm in it? –  Todd Trimble Nov 12 '10 at 16:36
    
Colloquia are great as they often start by including a brief history of the topic by an expert in that field. Seminars are great because they allow students to see the social aspect of math, including the give-and-take and the critical comments and requests for more detail and explanation, even by tenured faculty who don't follow a speaker's thought processes. Graduate student seminar presentations are great because a student observes how graduate students can falter during presentations, how they are quizzed/coached/criticized/mentored by faculty during their presentations. –  sleepless in beantown Nov 13 '10 at 0:20
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sleepless: your advice is the essence of good common sense, and I am sure that the OP will benefit from it if they take it at heart. When I was an undergrad, we attended talks made for us, but the faculty would attend too and the questions at the end could get serious (this is where I heard the term "gauge theory" for the first time). We were also encouraged to attend regular seminars with no expectations that we would understand anything, but, as one person said, for the music. You get the music first, then you add in the lyrics. –  Thierry Zell Nov 16 '10 at 18:29

As an undergraduate in the US with some research experience, let me offer my take on the situation.

1) I think it's important not to have a finished product (that is, a piece of original research) as the end goal. This past summer I did some research through MIT's SPUR program, and this is their definition of success:

Significant progress, relative to one's own background and experience, in developing interests, satisfaction, skill, and ideas, rather than getting the complete solution to a problem.

I think this is a really nice sentiment. The goal is not for you to start making serious contributions to mathematics but to prepare you in several ways for a graduate experience. (If you're curious, I blogged a little about my research here and for several posts afterwards. I did not prove a new result, but I learned a lot and thought it was a valuable experience.)

2) It depends on what your institution has subscriptions to. Click around on scholar.google.com to see what you can access without paying for. Many institutions, for example, have access to JSTOR or SpringerLink.

3) Find someone who knows the subject and ask them to mentor you. Or, find a mentor and ask them for a subject. This is hard to do without guidance.

Let me also give you some advice you didn't ask for. If you are seriously planning on graduate studies, I think you should expand your mathematical worldview as much as possible beforehand. The easiest way to do that, in my opinion, is to read math blogs. I recommend starting with Terence Tao's and Tim Gowers' and working from there, and I also recommend John Baez's This Week's Finds (actually, read the rest of his stuff too). Math blogs are a valuable source of insight into mathematics and how mathematicians work, and these three are particularly interesting and well-written. Terence Tao's blog also contains his career advice, which is worth a read.

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@Qiaochu-Yuan, excellent answer,+1. Also, the mathematics department may have its own internal library room/area, with hardcopy issues of journals that may not be in the main library or in the Barker engineering library. I still find that quickly skimming through the table of contents and abstracts is a good way to get an overview of what's going on in a particular field, and then being able to drill down into a topic by reading an article if the abstract catches my interest. The blogs you pointed out are excellent, particularly for the career advice. –  sleepless in beantown Nov 13 '10 at 0:36
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Yes, very good. The "problem-solving" model is invidious, as is the "no-previous-experience-required" idea. Awareness (not "originality", not fussiness over small, long-done details) is perhaps the most important goal for a beginning grad student. All the sadder that some REU's and other potentially encouraging, uplifting experiences give people the idea that they're "already done" in terms of knowledge, awareness, etc. –  paul garrett Aug 27 '13 at 1:18

Your beliefs are somewhat unfounded; lots of people (for example, me) do research in various fora as undergrads, and in fact, the NSF is pushing undergrad research quite hard nowadays.

On the other hand, it's not something that's easy to do on your own. While you may get some other reasonable suggestions from people, there's an obvious first step here, which is talking to a professor (possibly several). Decide on the mentor, and then have them help you prepare the research proposal. As an undergraduate, trying to go out and find articles on your own without any direction is like searching for a needle in a haystack. You might find something cool, but I wouldn't recommend it as a first approach.

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You say "lots of people (for example, me) do research in various fora as undergrads..." Interesting, but extremely unusual in my experience. To my knowledge, almost no undergrads at my university (or any subsequent places I went to as a graduate and teacher) did (or would have been capable of) any proper research at all (apart from small little "projects" and "investigations"). Still, that was back in the nineties in England, so maybe it doesn't apply here. The NSF sounds totally crazy to me, but I admit I'm not qualified to judge. –  Zen Harper Nov 12 '10 at 10:07
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In my (limited but recent) experience, undergrad research is encouraged much more in the US than the UK, and thought of as more achievable. In the UK (my own undergrad was Cambridge, early 2000’s) we weren’t for the most part encouraged to think of research as something accessible to us at all as undergraduates, like Zen Harper says. In the US (my own grad school, and what I've seen at other schools) it's widely encouraged (though not ubiquitous or essential); and it turns out that undergrads, with good mentoring, can often reach some non-trivial original work. (ct’d) –  Peter LeFanu Lumsdaine Nov 12 '10 at 17:30
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…so I think there may be a bit of “if you don’t believe you can do it, then you can’t”: in the UK, we don’t have expectations that it’s achievable, and we don't have much of a model or experience of how to do it, without which our expectations of impossiblity are pretty much correct! (I don’t mean to disparage the teachers I had in the UK, by the way: they were excellent, and encouraged us in many useful directions; undergraduate research just wasn’t one of them.) –  Peter LeFanu Lumsdaine Nov 12 '10 at 17:36
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Felipe- I (obviously) don't think this advice is over-optimistic at all. Maybe the McNair Scholars program is too optimistic, but that's an issue you should take up with them. I'm not sure "over-optimistic" is really the right characterization of the problem in MO advice; I still think its more of an issue of getting advice from people who don't really understand your situation, as Zen as demonstrated. There are so many details necessary for getting good advice on these professional issues that "Go talk to someone who actually knows you" is essentially always the right advice. –  Ben Webster Nov 12 '10 at 19:48
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Qiaochu: while I hope you are enjoying Cambridge, I suggest you try the sentence "the structure of the UK system means that students are all at fairly similar levels of mathematical maturity and knowledge as they progress" out on some people and see what their reaction is ;-) Let me just say that Cambridge is not a representative sample of UK tertiary maths education... –  Yemon Choi Nov 13 '10 at 0:23

There are quite a few undergrads who have done significant research in mathematics at your level. Even if you don't end up with a published paper (you shouldn't expect this, although it probably happens more often than you might expect), you will gain significant experience into what doing original research in mathematics means. However, I think expecting to find a problem to work on yourself, rather than have a mentor suggest one (or several) to you, is absolutely unrealistic. Maybe it's realistic for other fields, but in my opinion not mathematics.

Find a mentor who is willing to suggest a problem that you can tackle at your level, and (hopefully) give you ideas during the summer if you get stuck. Finding the right problems to work on is a major component of doing mathematical research, and sometimes the hardest one. If you find it on your own (rather than a mentor suggesting it), your mentor is likely not going to have any good ideas of how to attack it, it may end up a harder problem than is realistic for you to solve, and your mentor will be less motivated to help you. So my advice is to start looking for mentors now.

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Pity if the faculty in the environment are less able to think new thoughts than the student... though I suppose it happens all too often, given various dynamics. But, "srsly", one doesn't want a mentor to feel superior-to, in any case, indeed, for sure. –  paul garrett Aug 27 '13 at 1:15

So on the one hand I have a very strong cultural bias against undergraduate research programs. I don't think trying to emphasize originality is a good idea. I think it would be much better to give people problems to work on that have already been solved and so you know lead to good and interesting mathematics. By forcing people to work on "new" questions I think you are often forcing them to work on bad math.

On the other hand, just because I think it would be better for people to do other sorts of programs, REU-style programs are what exist and they seem to work reasonably well for a lot of people. Furthermore, they're certainly valuable as an alternative to classroom learning. Real math research is not like what happens at most REUs, but it's also not like what happens in a classroom, so doing an REU is still going to help you get closer to understanding the scope of what a graduate student does.

So yes it's certainly possible and somewhat valuable for undergraduates to try to do "research," but you shouldn't expect that research to be the same sort of research that mathematicians are really doing.

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What's "bad math"? –  Igor Belegradek Nov 13 '10 at 3:25
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Bad math may mean problems which are open because they seem uninteresting, or not clearly connected to anything. Working on odd, tedious but open problems may allow a student to do original work, but might be worse than a course for learning or for stimulating interest. –  Douglas Zare Nov 13 '10 at 4:00
    
Of course, I solved this issue during my REU experience by working on a problem that had been solved 40 years previously (and was actually easy, when viewed correctly), though neither my mentor or I knew this. –  Ben Webster Nov 13 '10 at 7:17
    
A mentor should have good taste and wisdom to involve students in rich problems, and there are plenty of those that are open. I have seen students solve long standing questions because nobody looked where they did. Working on a problem that you know have been solved is an entirely different activity; it isn't a trip into the unknown. Solving a problem and later finding out it was solved before leaves a bitter taste in one's mouth. –  Igor Belegradek Nov 13 '10 at 12:35
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Several points: "bad" math includes artificial problems, and problems posed as though they were mysteries, when they are not. Also, is it always about "problems"? This, too, is corruptive. I'd not tell people to work to understand something by_prescribed_means, but to understand it however they can. And that surely many important things are already understood (by hard-working, able people), but everyone has to get themselves caught up to the present. The "standard" that genuinely worthwhile new contributions be made by people who don't know anything is both a vicious fiction... –  paul garrett Aug 27 '13 at 1:12

You say:

"I am a junior at the moment and taking: One Dimensional Real Analysis, Intro to Numerical Methods, and Abstract Algebra".

Based on this information, I think it is a complete waste of your time even to consider research seriously at this stage; you need several more years of study as a minimum. Right now, you are still learning the basic language of mathematics. It's similar to, say, a student who wants to begin reading classical German literature, but only knows 100 words -- premature, to say the least. The maths you know right now is probably less than 1% of what you will need. Even after my Ph.D., I feel that my knowledge is very limited in comparison to most good researchers.

But do you really mean "research", i.e. new, original, nontrivial and interesting, and publishable in a good quality journal, i.e. one which your professors would publish in?

Or do you mean a kind of "investigation" or "project" instead? These are not required or expected to contain anything new or original. This would be highly worthwhile -- but only for your personal interest and satisfaction.

The question is, what do you expect to get out of it? If you're at a good university, their lecture courses should already provide you with all you need.

Please don't take offense, and apologies if I've formed the wrong impression, but it sounds to me (from your statement "I have begun speaking with professors about their research also") like you might be the kind of student that irritates professors, always bugging them and asking them questions about their own research, but lacking the knowledge to understand the answers. (But it's not your fault you lack knowledge - that's what you're at university to learn!) As an analogy, imagine a ten-year-old, knowing nothing more than how to add fractions, constantly harrassing you to teach them about calculus; my response (unless I were in a very good mood that day) would be: "go back to school and stop bothering me, for at least another 3 years!" Unless you're an exceptionally good, enthusiastic student, or your professors are far more patient than me, that might be what they're thinking also, but are too polite to tell you.

But just my opinion, don't take my word for it; why don't you ask them directly if that's what they're thinking?!

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+1: Why was this answer down-voted? Are you suggesting that juniors are ready to do research? –  Douglas S. Stones Nov 12 '10 at 12:38
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I think it is possible to find a suitable project for a math-interested student at any level. For example, I would be happy to discuss calculus with any ten-year-old who was interested enough to learn about it; it would be an excuse to talk about graphs and rates of change and the concept of limits and the effect of minute changes. One can explain a part of these ideas even to someone with little background. Similarly, one can find an interesting suitable project for an undergraduate. The surreal numbers, for example, would be an attractive topic at the boundary of algebra and game theory. –  Joel David Hamkins Nov 12 '10 at 14:20
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The down vote might have been because the same thoughts could have been conveyed in a much more polite manner (just a guess). –  BCnrd Nov 12 '10 at 14:20
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Agree with BCnrd. "like you might be the kind of student that irritates professors, always bugging them and asking them questions about their own research, but lacking the knowledge to understand the answers" - isn't that reading a bit much into it? OP said "research at the undergraduate level", which I take to mean investigations into subjects he finds attractive, not publishing in the Annals as an undergraduate. Also, "their lecture courses should already provide you with all you need" - are books and papers all that pros need? One-on-one conversation is something we all benefit from. –  Todd Trimble Nov 12 '10 at 14:56
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-1: This answer latches on to one fact about the student while ignoring another big one: the OP is in a structured program, not just doing this on a lark. You can, of course, doubt whether that program will produce very high quality research; I think most of us do. But that's not really the point of such programs. They're mainly aimed at grad school preparation/promotion. Frankly, students need to do something over the summer, and they may as well have a experience that shows them mathematics from another angle than just the classroom. –  Ben Webster Nov 12 '10 at 17:46

Metahominid,

I am an undergraduate myself who was in a situation two years ago similar to the one you're in right now. As a sophomore (I'm a senior now), I wanted to do some sort of research but I hadn't taken that many courses. I was just taking real analysis and abstract algebra at the time. However, I asked around the math department for research opportunities algebra and game theory (yes, even my interests were similar!) and a professor recommended another professor to me who had just what I was looking for: an approach to combinatorial game theory using algebra (and a little bit of geometry). I have been working on this topic with him since the summer after my sophomore year. Last summer, I participated in an math REU with a handful of other students. Research in the REU was more independent, with the students doing pretty much all of the work while the mentors served more as helpful sounding boards than co-researchers. Here are the differences I have found between the two research experiences:

Research with professor

  • Since the problems I am working on are of direct interest to my professor as well, the topics I have to learn in order to even begin to approach the research tend to be more advanced, so I end up learning a lot of interesting theory (my research has led me into combinatorial commutative algebra and local cohomology)

  • Again, since the professor is working on this with me, I spend a great deal of time talking with him, bouncing ideas back and forth, and this creates a very strong mentor-student relationship that I feel is very beneficial. My mentor gives me advice not just on how to do math, but also on applying to graduate schools, writing good papers and abstracts, giving talks and presentations, etc.

REU

  • As the students were expected to work independently of the professor, I was thrown into the deep end basically. What entailed were weeks of intensive reading and thinking. Although the mentor was there to help me make sure I was sane by letting me bounce ideas off him, I was still responsible for all of the original thinking and problem solving. The benefits of this are absurdly great: my problem solving skills have improved greatly and I find it far easier to follow lectures and do homework problems now. In fact, math classes feel like nothing now that I have done some research on my own.

  • We were free to work with other students, and I did collaborate with a few students, which I think is an invaluable experience. By talking to these students every day, I learned different ways of thinking about things, and different approaches to solving problems. Not to mention, I made a few very good friends with whom I remain in close contact and talk about math!

  • I got to do some nontrivial original work (although I wouldn't say the problems I solved were important) and wrote some publishable material. This needn't happen, though. The point is to get the experience.

The bottom line is this: if you're eager to do research, ask around for professors who are looking for motivated undergraduate students. Make sure they know that you are motivated and willing to learn and work hard. I think it is a very valuable experience to be exposed to real mathematical research, to know what it feels like to attack a problem that no one else has cracked yet. It's completely different from coursework. I never considered myself particularly talented at mathematics, but over the years I have realized that being good at doing math is much more about practice and experience rather than some "natural" talent. I also highly recommend one of the NSF sponsored REUs, as the mentors are usually very skilled at picking problems that are at an appropriate level for undergraduate students. I hope this helped.

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The following might be more appropriate as a comment to sleepless in beantown's answer instead of an answer, but for some reason I am not able to comment.

The following website of the AMS contains numerous links to Research Experience for Undergraduates (REU) programs

http://www.ams.org/programs/students/undergrad/emp-reu

If I understand correctly, these REU programs are funded by the NSF to offer undergraduate students (not necessarily from the institution that is hosting the program) an opportunity to do research, under supervison, over the summer. (Since there was some debate what "research" should mean, I add that here "research" means, or at least can mean [and not only rarely], something that in the end is published in well-established mathematical research journals.)

Also, there is a somewhat recently founded journal Involve specifically dedicated to "showcasing and encouraging high quality mathematical research involving students (at all levels)"

http://pjm.math.berkeley.edu/involve/about/journal/about.html

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Hi, unknown (you can register and add a name or nickname also), you couldn't add a comment because you don't have 50 reputation points yet. Thanks for the comment/reply. I hadn't head about the AMS REU program web site. –  sleepless in beantown Nov 12 '10 at 14:19

I think the earlier one starts doing research the better, even if it is a research in plane geometry.

I directed a few REU's and it was great fun; the only problem is that it is hard to do anything significant in 8 weeks. I am confident that many US undergrads can produce a publishable work after focusing on a problem for 1-2 years.

In the place where I went in college (Novosibirk, Russia, mid 80s) students went through abstract algebra and real analysis in the first two years and the best of them them could do nontrivial work in the 3rd year. Quite a few people were going to research seminars in 3-4th year, which they had to be doing since a (master) thesis with original research was expected at the end of 5th year. As it happens for many students this thesis was largely expository, but those who later become professional mathematicians usually got something publishable in the 5th year. My own research in the 4th year went nowhere, but in the 5th year I proved something I am not ashamed of.

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Although the OP is in his 3rd year, it doesn't sound like he would fit into the third year of the college you describe by the courses he is currently taking. In many US colleges, mathematics majors spend a lot of their first two years taking classes outside mathematics. –  Douglas Zare Nov 13 '10 at 4:14
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Douglas, I am fully aware that US system is different (having taught here for many years). My points are (a) those who want to become mathmaticians should try research long before they pass comprehensive and oral exams in grad school (b) with proper mentoring math research is quite doable. –  Igor Belegradek Nov 13 '10 at 12:13
    
Thank you Igor. One of my good friends in my math classes is from Russia. From what he has told me and I have read both the secondary and post-secondary schooling emphasizes math more and introduces many things earlier. I wish that were the case here. I also wish the culture for chess were the same. –  Metahominid Nov 14 '10 at 3:06
    
@Metahominid, the system in Russia is different and I cannot say it is better, but it does prove that original research by undergraduates is quite possible. Some in this thread seem to argue that a student is better off by delaying research till grad school, and I disagree. –  Igor Belegradek Nov 14 '10 at 12:32
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There are two different issues being confounded throughout, for unfortunate semantic reasons. If "research" means "thinking" (as opposed to obeying, conforming, guessing what's on the final), well, yes, this is a fundamental scientific/intellectual trait. But if "research" means to do better than all existing professionals on an issue meaningful and interesting to them... well, let's think... maybe this is not something to be counted-on on a regular basis. Sometimes kids are in stark, low-energy situations, which is bad, but let's not deceive them about the larger world. Yes, thinking is good. –  paul garrett Aug 27 '13 at 1:22

The web site on the McNair Graduate Opportunity program gives no indication that mathematics students were in mind, and a quick survey of some past students' topics showed none in mathematics. I do not think the program was designed for mathematics students or the way mathematics research is done, even undergraduate mathematics research.

Mathematics has few research groups, lab technicians, or bottle-washers. We usually do mathematics with 1-2 people involved. In other areas, you can learn to run some tests, and collect data, and analyze it with the help of an advisor. You have a very high chance of accomplishing something, and meanwhile you can try to learn how your work fits into a larger picture. Most mathematical projects are more risky. It takes a lot of work as an advisor to create a project approachable by an above average mathematics major which has a good chance to produce new results the student can write up. Much of mathematics does not involve programming, but many projects designed for short-term results are programming exercises which may give a distorted picture of mathematics.

This is not to say that undergraduate research is not worth the attempt. It is one way to see that mathematics is alive and exciting, which may be hard to see from courses. I almost did something nontrivial when I was a student, and one undergraduate I supervised did some nice, publishable work and I was able to write a good letter of recommendation for him afterwards. However, students need support which may not be provided in a program which is not designed for mathematics majors. You need an advisor who will put a lot of effort in (even in choosing the topic) beyond what is needed in other areas. You should be aware that failing to solve the problem is often not a surprise, and that you may be severely handicapped with only one year of solid mathematics courses.

I think you should look at alternatives such as mathematics Research Experiences for Undergraduates (REUs) or setting up a reading course in which the goal is not to produce new research, but to understand some recent result or paper, perhaps to create a more accessible exposition of that topic.

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After looking over the McNair website, I agree completely. I didn't see anything oriented towards math specifically, and this makes me skeptical -- the way math is done is quite different from that of other academic endeavors. I was especially concerned by something on the website that said that creation of a paper of "publishable quality" was a requirement of the program. In my opinion this is not a realistic goal for undergrad summer research. If I were the OP, I would consider either doing the summer research in something else, or, if math is truly of interest, applying for an REU. –  Pete L. Clark Nov 13 '10 at 7:08
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I ran out of space in my last comment, but let me give one reason an REU is a better bet for summer math study than this McNair Program: in REUs there is a guaranteed cohort of other students to interact with and derive support from. –  Pete L. Clark Nov 13 '10 at 7:11
    
Sorry I didn't mention that. It is pretty much an REU. It is a TRIO program and I do have a cohort, although they are not in my field. I will be living with them in the summer as well as students from other universities. –  Metahominid Nov 14 '10 at 3:08
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@Metahominid: if the other students are not doing math, then they will be able to support you in some ways but not others -- for instance, you cannot ask them casual questions that you might not want to bother your mentor with. I think an REU would be better for this. –  Pete L. Clark Nov 14 '10 at 7:41
    
@Pete L. Clark I recognize this however I have already been accepted and I cannot do both this summer and I am not sure if I will be able to my senior year's summer. –  Metahominid Nov 14 '10 at 9:34

Yesterday I proved a small fact and asked a follow-up question in this answer:

Algebras over the little disks operad

It's fairly elementary, I'd be interested to know more about it, and I have never seen anything like it in the literature (although I have not searched). So you could look at that if you wanted to. More generally, there are plenty of problems like this, but they are not always easy to find. If you just start reading books and looking for things to research there is a danger that you will just be led along the best-travelled paths where enormous amounts of work have already been done. So I would second the advice to ask several professors for suggestions before deciding on a topic.

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I have to disagree with the sentiment that undergraduate research (that is, research done by students who are actually at the undergraduate level in their studies, like the OP) is premature or somehow not worthwhile. A month into my first abstract algebra class, I approached my professor to talk about research and what I should do to get to that level. Luckily for me, this professor went a step further and actually gave me a choice of things to work on.

Now granted, I doubt this is the norm. My college did not have a graduate program, which made undergraduates more of the center of attention and moreover this particular professor has a keen interest in fostering undergraduate research. However, I think it is worthwhile to a student to pursue it (even if your professors aren't interested or don't have a problem at your level to give you, as has been mentioned there are always REUs) for the following reasons:

  1. You see a different facet of mathematics then you typically see in a course or textbook. Those are realms of proven things (generally speaking; I am sure there are exceptions, but the usual undergraduate topics tend to be fully developed in my experience). On the other hand, research is messy, with missteps and "mathematicians block" and the thrill of showing something new. I think the potential mathematician should see that as soon as possible!

  2. You gain a wealth of valuable insight and skills. At least for me, I grew very comfortable with TeX, got a good deal of experience with presenting mathematics, and learned a lot about effectively explaining mathematics on paper as well.

  3. If you are lucky like I was, this initial collaboration can lead to more research, hence more time honing your intuition, research habits, and paper-writing skills.

In other words, sure, you are probably not going to see an undergraduate solve a particularly interesting problem, but surely it is worthwhile to promote growth of these skills as well (not to mention that, frankly, in the competitive world we live in it wouldn't hurt to have your name on some papers and some professors seeing you talk at workshops and things). On a personal note, I have to say I found that there was feedback between positive research and my courses: the more I learned, the more tools I had to attack problems obviously but the research aspect really improved my ability to see the solutions to exercises and grasp the larger picture of the courses I took.

Bottom line, for an aspiring mathematician there is nothing to lose and everything to gain, so you should definitely see what is out there and try to get involved with some research.

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You might want to start looking for a mentor before you get too deeply involved in developing your project. It's great to have some broad ideas, but it isn't a good idea to box yourself in so far that your project isn't a good fit for those on the faculty who might be interested in mentoring you over the summer. Also, potential mentors might have some ideas for projects that would be a good fit for both you and the mentor.

Before you start approaching potential mentors, be sure to check with your McNair program to find out what the program expects of the mentor (for example, the mentor might be expected to write a brief biweekly status report commenting on your progress, as well as validating that you have met milestones for your project). You might want to develop your project proposal with your mentor and to start working together on a realistic set of milestones. The McNair program here tied a large chunk of the summer stipend to meeting milestones.

Have there been other McNair scholars in math at your school in previous years? You might check with your school's McNair program, your department head, and/or your coursework advisor about this. If there have been others, you might also get some tips about potential mentors. You might also check with your coursework advisor or your department head about which faculty members have mentored undergraduate research students. Did you submit letters of recommendation as part of your application? If so, you might want to share your good news with your letter writers and ask their advice concerning potential mentors.

Please don't worry about finding a big result. This is an opportunity to get a taste of research and to learn about some new topics. You will probably be expected to write up what you learned at the end of the summer and present at a conference for McNair scholars. Good luck!

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Most major topics have been covered in discussion, so just two remarks/experiences:

  1. While director of graduate studies at Northwestern (2007-2010), I led a committee which valued undergraduate preparedness over research experience. So at least as far as Northwestern was concerned during that time frame, research (especially research for which an undergrad may not be fully prepared) did not help as much as you might have thought.

  2. In trying to use RTG funds toward undergraduates, rather than try to simulate a research environment, I and my co-PI's created an "undergraduate conference" to try to offer a supplement to standard undergraduate curricula, without yet getting on toward research. Here is the link: http://www.math.northwestern.edu/summerconference/ Maybe we'll do it again next year?

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I can see why research experience is not a good way to select grad students, and that good bases are fundamental. But its usefulness is upstream: a serious research experience is a very good way for a student to figure out if they want to go to graduate school in the first place. In that respect, I think it plays a very educational role (I wish all of our secondary-math majors did REU-like intensive research, if only to have teachers out there with an understanding of what professional math is like.) –  Thierry Zell Nov 16 '10 at 18:33
    
We need a different name for "undergraduate mathematics research"... I am very enthusiastic about getting people out of textbook/school-math/artificial/adversarial settings, but "exploration" is not "research", or else the latter term has become meaningless. I rant endlessly to my students about the evils of "school-math", and also about accidentally believing that one's voyage of discovery is "research" that should be published... as necessary as this voyage is. Confounding substantially different things is not helpful. –  paul garrett Aug 27 '13 at 1:03

I cannot speak from the point of view of a Math major in US since I never was one. I completed my undergraduate studies in engineering and currently pursuing a Ph.D. in pure mathematics. In my opinion, applied mathematics (though admittedly this quite a generic term) would be more accessible to an undergraduate considering research than pure Mathematics. I ended up publishing two single author papers in respected journals while in my senior year. I had started working on both these problems during my junior and both of them were picked by me. When I though I had a good insight into the problems, I approached the faculty within my university for suggestions. I think it is safe to say that a lot of problems in applied mathematics require less sophisticated machinery than is used by most pure mathematicians. Many of my engineering friends started working on their Ph.D. thesis problems fresh out of a Bachelors in areas which could be termed as applied mathematics. This contrasts with most pure math grad students I know who usually spend between 1 to 3 years of coursework before starting to work on a concrete research problem. So it seems that "undergraduate level coursework" would be sufficient in handling a good number of applied math problems. So if you are advanced undergraduate student with a good background in one such allied area, I think it might be worthwhile to explore this possibility. After all you can gain valuable experience doing research even if you do decide to pursue some other area of math in your graduate life.

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I feel like anyone planning on becoming a pure-mathematics-oriented researcher stands to gain nothing (apart from, maybe, networking with other students and/or a summer job) by working on an applied math research project. That time could be much more usefully spent (at least in terms of mathematical knowledge) by sitting oneself down and working through something like Lang's Algebra or Rudin's Real and Complex analysis. –  Harry Gindi Nov 12 '10 at 21:48
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I was not implying that anyone who wishes to become a pure math researcher should consider applied mathematics research as an undergrad. This was specifically directed to the OP since the OP mentioned taking a class in numerical methods, an interest in game theory and having considered applied mathematics as a research option. –  Timothy Wagner Nov 12 '10 at 22:52
    
It is also worth considering that part of the purpose of summer REU's is to discover what sort of research one might or might not want to do later. Besides, knowing some applied math is almost certainly bound to help even the most "pure" mathematician, since the origins of much of the best mathematics lie in very real problems. (I spent two summers trying to learn quantum field theory under the ultimately mistaken impression I wanted to do physics, and I would not say the time was wasted.) –  Dave Anderson Nov 13 '10 at 4:47
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Harry- That's a fairly ridiculous position; intellectual development and careers do not proceed in an entirely linear manner. I'm not sure I would strongly recommend applied math research to someone who ultimately was planning on going into pure math, but it can surely still be a valuable experience for them. But more to the point, no undergrad should proceed with their life as though it was absolutely certain that they would end up in a particular career. Given the job situation at the moment, I bet a lot of people trained in pure mathematics would be happy to go back in time and (cont'd) –  Ben Webster Nov 14 '10 at 8:45
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spend more time getting experience with applied mathematics as an undergrad. –  Ben Webster Nov 14 '10 at 8:46

Since you are interested in game theory, one area you could consider is "Algorithmic Game Theory" (basically Algorithm Design + Game Theory) It is a now a fairly hot area in theoretical computer science but still seems relatively approachable to an undergraduate with knowledge of game theory. If you can find someone willing/able to mentor you in this area I think there is good potential for a productive experience.

There is a free textbook online and the blog of Noam Nisan (a leader in the field) is a good place to follow the latest developments.

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