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Summer REU/Internships & Algebraic Topology/Category Theory Questions

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Hello,

I have a few questions that I feel would help me better understand a few things. The first set of questions relate to the type of research opportunities some of you guys will be involved with this summer. The second set of questions are a bit different and are related to math courses in preparation for Algebraic Topology/Category Theory.

First how common is it for undergraduate students to get involved in research? I was under the impression some undergraduate research topics are topics that professors feel can be completed within a few semesters. Nothing as serious as research at the graduate level, is this generally true, despite the field (ie math, physics)? I know some research conducted at the undergraduate level is very serious. Any research period must look good on an applicant's transcript! But would it hinder the transcript if the applicant conducted research for a few semesters but not publish anything? How hard is it to get your results published if you feel like you made substantial progress?

This leads to my second question, what summer REU/Internships will some of you be doing this summer? What kind of math/physics background do you guys have course wise? What do you feel made you stand out over the other applicants, and what kind of various research projects will you guys be working on?

I'd like to eventually study Algebraic Topology and Category Theory. I am thinking about taking a group theory, a real analysis and a topology course. Would this be the natural progression one would need to have under their belt before they study AT? What books would you guys recommend to study AT with? I figure once I learn group theory I can go back and learn the basics of ring theory by myself. Basically I am wondering what mathematical foundations do I need to have in order to study AT and Category Theory? Are these courses generally taught at the undergraduate level?

Thanks!
 
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I took a research class in Algebraic Topology last semester. Most of the background material I spent a lot of time reading was ring and module theory. I'm continuing with this in the fall and this summer I'm going some stuff in basic Algebraic Geometry. Basically to reinforce ring theory and since the AT stuff I was doing has a lot of overlap with AG. Category theory was mentioned a few times in the AT class, but I don't plan on studying it in any real depth until this Fall when Lang's Algebra forces me to. The summer program is not an REU, but a VIGRE thing being held at my school so there wasn't any admissions competition. I didn't apply to any REU's this year, but I plan to next year. I did an REU last summer as a freshman, but I'm still working on that paper, and I doubt it's going to get published in something better than an undergrad "journal".
 
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Basically I am wondering what mathematical foundations do I need to have in order to study AT and Category Theory? Are these courses generally taught at the undergraduate level?
I wonder if it would depend on the school to offer undergrad courses in AT and CT...but me personally have only seen Topology being offered as an undergrad course and it was only like once every two years or something like that.
 
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What do REUs typically look for in a math student?

I was also wondering if anyone attended the Math in Moscow or Budapest Semesters in Mathematics program?
I am interested in these type of programs and would be very interested in reading the experience of those that have participated in an REU or a study semester program.
 
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