The 2005 AMS Summer Institute on Algebraic Geometry in Seattle was an enormous event. With over 500 participants, including many of the world's leading experts, it was perhaps the largest conference on algebraic geometry ever held. These two proceedings volumes present research and expository papers by some of the most outstanding speakers at the meeting, vividly conveying the grandeur and vigor of the subject. The most exciting topics in current algebraic geometry research receive very ample treatment. For instance, there is enlightening information on many of the latest technical tools, from jet schemes and derived categories to algebraic stacks. Numerous papers delve into the geometry of various moduli spaces, including those of stable curves, stable maps, coherent sheaves, and abelian varieties. Other papers discuss the recent dramatic advances in higher-dimensional bi rational geometry, while still others trace the influence of quantum field theory on algebraic geometry via mirror symmetry, Gromov - Witten invariants, and symplectic geometry. The proceedings of earlier algebraic geometry AMS Institutes, held at Woods Hole, Arcata, Bowdoin, and Santa Cruz, have become classics. The present volumes promise to be equally influential. They present the state of the art in algebraic geometry in papers that will have broad interest and enduring value.
