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.

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.
