A perspective on algebraic geometry

Abstract

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 Matematica Pura e Aplicada (plenary talk at the 23°- Coloquio Brasileiro de Matematica), Pontificia Universidade Catalica do Rio de Janeiro, Universidade de Brasilia, Universidade de Sao Paulo, Universidade Estadual de Campinas, Universidade Federal de Pernambuco, Universidade Federal do Amazonas, Universidade Federal do Ceara, 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. In Section A we collect together six survey talks on algebraic geometry starting with a talk on its historical footsteps. Section B consists of four general talks. In Section C we collect together six talks discussing the Galois theory coming out of algebraic fundamental groups. Section D consists of four talks on ramification and resolution. Finally Section E collects ten talks which I gave at various special sessions in the meetings of the American Mathematical Society. Each section starts with a brief description of its content. In a concluding section I shall list some of the open problems mentioned in the various talks.