I am a rising junior, looking to apply for Ph.D. programs in Math or CS when my time comes. I heard that my friend got into the two prestigious universities for a Math Ph.D. program without much research. All he had was an undergraduate senior capstone paper in Mathematics (written in the fall semester of one's senior year to complete your math major) -- that is all for research --, a decent GPA (3.55), and I'm assuming decent recommendations. He graduated from a top 30 liberal arts college. I've seen people emphasizing the importance of research to get into Ph.D. programs but why are some undergraduate applicants accepted with very little experience?