This project aims to stimulate and actively develop research on algebraic stacks. Algebraic stacks are widely used, especially for thinking about moduli spaces, and are one of the most important tools in algebraic geometry today. The research will focus on foundational questions and open questions related to global geometric problems. Two questions are: How far can you get developing the theory with a minimal set of hypotheses? In what generality can we develop intersection theory for algebraic stacks? Two geometric open problems are: Which algebraic stacks are quotient stacks? Are the Brauer group and the cohomological Brauer group of a separated scheme of finite type over a field the same?
The principal investigator will visit mathematics departments and give talks to advertise these problems. The principal investigator will attack some of the research problems, and collaborate on others with graduate students, post-docs, and algebraic geometers in the international mathematical community. As an integral part of the proposal the principal investigator will maintain a web-based collaborative algebraic geometry project, called the "Stacks Project." The goal is to develop and document theory, with the aim of closing the gap between graduate education and current research. The stacks project intends to be partly a high level text book, partly a foundational reference on algebraic stacks.
