This Summer Mathematics Institute is a six-week residential research summer school for about 25 minority undergraduates from the United States and Puerto Rico. The participants will have completed at least two years of collegiate mathematics with distinction by Summer 1991. Each student will participate in two intensive seminars which have been designed to allow the student to explore a mathematical subject in depth and to experience the excitement of creating mathematics. A principal common to all seminars is helping students to develop increased fluency with mathematical language, both written and oral. The overall goal of the Institute is to help the students make the important transition from learning mathematics to doing mathematics. The Institute will be held on the Berkeley campus and continues and builds on successful Institutes held in 1989 and 1990. These Institutes are an integral part of a comprehensive national strategy aimed at eliminating the under-representation of minorities in mathematics and mathematics-dependent professions.