Undergraduate math research 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.
 A: 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.
A: 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. 
A: 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.
A: 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:


*

*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!

*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.

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


*

*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.

*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?
A: 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.
A: 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.
A: 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! 
A: 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.
A: 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.
A: 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. 
A: 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.
A: 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.
A: 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?!
A: 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.
A: 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
