How to go About Undergraduate Research I apologize in advance if this question is out of the scope or focus here.
I was just wondering about the whole prospect of researching as an undergraduate. How to do it? Who to talk to in my department (UCLA) and what to do in general? To give you an idea of where my knowledge of math is and my interests, I have gone through a few graduate texts on algebra and group theory. I have been trying to read the papers on arXiv and am able to follow them (albeit taking a few hours and multiple Wikipedia searches).
Most of my learning has been outside of a classroom setting though, and I am only taking an introduction to algebra class at this point. Adding to my problems is both the fact that I have just transferred into my school  and that I have generally never known how to interact with my teachers. I am a rising junior by the way.
I really have no idea what to do at this point. I would like to start researching, but don't really know how to do it. 
Any help would be useful. Thank you all very much.
 A: In the United States, one can participate in mathematics REUs.  They are generally done the summer between junior and senior year, sometimes the summer between sophomore and junior year, and only on rare occasions the summer between the freshman and summer year.  These can be hard to gain admission to, especially if you have a substandard GPA or go to an unexceptional college (but even in this case it can be done).  Best advice for this option is to apply for a ton of programs, especially those which match your research interests.
If you don't do an REU, you can still do similar summer program type stuff at your own university by working with a professor in the math department or getting a job working with another department such as CS or physics.  Just talk to people about this to learn about options specific to your university.  Don't stick with one advisor if they don't know of any options - keep asking around, maybe a professor will have connections to industry or another department in which you could get a scientific internship doing mathematics.
If all else fails, start speed reading advanced material in a topic you are interested in but don't know a lot about yet.  One of my biggest current research problems began by flipping through a book on character theory (which I knew jack about at the time), seeing a peripheral conjecture mentioned that I found intruiging, and tracing back through the definitions to figure out what it meant.  Then I read the surrounding chapters trying to figure out the "place" of the conjecture in the literature - that is, what other theorems it was related to, what exactly made it so complicated, which parts were still open, which approaches had been attempted to solve it, which parts were trivial to prove, etc.  Through all this targeted reading, I learned a ton about a subject I hadn't ever studied before, found a few interesting published papers to read on the internet, and through these papers eventually found a related, more entry-level problem which I could make progress on just by following my nose.  This is what you do in REU type programs anyway, so if you have the motivation, you can still do this to break into research if you are denied all other options.  (It also helps to find a professor - either at your university or elsewhere, by email - whom you can ask technical questions when you get stuck. Don't be discouraged if you can't find one right way. The more you read, the more professors you'll be able to talk to.)
A: I am currently doing mathematics research as an undergraduate, and all it took was asking my topology professor if anyone in the department would take me under his/her wing over the summer. One of the easiest ways to get into research is to talk to a professor you like and ask what's out there for you.
That said, since you say you've just transferred in the problem is most likely that you don't know any professors at UCLA well enough. My suggestion: give it a year. Most undergraduate research occurs over the summer, and in one year you'll have plenty of time to get to know a few professors. To this end, you probably want to take more than one mathematics course per semester so you can meet more professors.
Use the school year to lay the foundations for a summer of research. Over the school year, the amount of "research" (as in working on some novel result) an undergraduate can do is likely to be limited, and truth be told focusing on your schoolwork is probably best. The general consensus among professors I've gone to for advice is to be well-grounded in the basics - learn to crawl before you try to walk.
However, if you can make time to do some work outside of class, I recommend you look for some independent reading opportunity. Find or have someone suggest a topic that normally wouldn't be covered in class, and either read independently or (my preference) arrange some guided reading with someone. My institution has a program that pairs undergraduate mathematics students with graduate students to read in some interesting topic - you can find the details here http://www.math.rutgers.edu/undergrad/Activities/drp/. This can be a great way to explore your field of interest and find directions for your coursework and future research. (If the topic and pairing work out, this is also a highly entertaining way to do some pretty strange mathematics.)
All throughout, keep in mind the eventual goal: getting a summer research opportunity. If you're thinking about a REU, look for established REU programs early, and think about which ones you'd like to apply for.
I also want to throw in my two cents about your current coursework here (as opposed to making a new comment). If you know the material in the intro to algebra course (and you say you've gone far beyond it), there is no need to waste a semester doing review, unless there is a degree/graduation reason behind it (e.g. required for the major). I find that with mathematics courses, prerequisites are a flexible thing, and what really matters is getting permission from the professor and department. Brian mentioned that UCLA has some stingy policies regarding getting credit by examination, but if the only reason you're taking intro to algebra is to fill a prerequisite then it's probably better to simply request an override and get your credits via some other course.
A: I know a few students who did undergraduate research at UCLA with Professor Robert Brown and Professor Liam Watson. They do mostly topology. Just talk to your professors during office hours and ask them about undergraduate research. Even if they are not actively involved in anything, they will know who is, and they may be able to give you a recommendation to get you into a program.
There are also some summer programs, although you are passed the deadline now.
UCLA also has a Logic summer program which supposedly brings you to research level in logic, http://www.math.ucla.edu/~ineeman/Summer-school/
RIPS is a summer undergraduate research program. http://www.ipam.ucla.edu/rips/
Also towards middle of spring quarter, you will get emails about money available for summer undergraduate research. You will need to get a professor as an adviser, but it is my understanding that you mostly get to choose what you want to work on.
Most summer programs are applied to at the beginning of Winter quarter, so go to your Professor's office hours in the Fall and make friends for letters of recommendation. Ask in the math undergraduate office, and they will also tell you what programs you can apply for and what the deadlines are.
Edit: It might also be worth emailing ugrad@math.ucla.edu now, and asking about what types of programs are available.
