A perspective on algebraic geometry

This article collects together material presented in various places during 1996–2001, namely Austria, Brazil, Canada, England, France, Germany, India, Israel, Italy, Japan, Oman, South Africa, Spain, and the United States of America, and includes all 13 talks given while visiting Brazil from June 17 to August 11, 2001 at the following institutions: Instituto de Matemática Pura e Aplicada (plenary talk at the 23° Colóquio Brasileiro de Matemática), Pontifícia Universidade Católica do Rio de Janeiro, Universidade de Brasília, Universidade de São Paulo, Universidade Estadual de Campinas, Universidade Federal de Pernambuco, Universidade Federal do Amazonas, Universidade Federal do Ceará, Universidade Federal Fluminense, and the Fourth Meeting of the Brazilian group in Commutative Algebra and Algebraic Geometry. It is a pleasure to record that my visit to Brazil was organized by my very first Ph.D. student, Alberto Azevedo, and gave us a welcome opportunity to solidify our friendship.