First of all, I'm a second year pure mathematics major/computer science minor with a 4.0 math g.p.a. and a 3.878 overall g.p.a. I'm looking to go to graduate school right after I get my B.S. I am very studious and I am constantly studying. I've been asking all my professors about research opportunities for math undergraduates, and most have led me to believe there isn't very much opportunity outside the REU program. So here are a few questions. I've heard some of the REU programs are "lazy", if you have any experience with the REU program, could you recommend a good one. I've began to study analysis and abstract algebra already but I’m not going to take a course in either until fall 2008, what year do REU programs and any math research programs usually require, ( most applications do not specify ), but are there any standards? I've heard that the Math in Budapest program is quite excellent for undergrads, has any one here done this program, or heard anything about it from someone who has. Finally, how common is it for undergrads to do pure math research and how much are colleges looking for research? I want to attend one of the top 5 math schools, the purpose of this is to help me assess the likelihood of attending these schools based on my current progress and future opportunities. Thank you for you help.