Raymond Cheng, Department of Pure Mathematics, University of Waterloo
"Topologies in Algebraic Geometry"
Schemes, being geometric objects, have an underlying topological space. However, the usual topology on schemes, the Zariski topology, is extremely coarse, making it difficult to apply geometric intuition that one may have from manifold theory. When the base field is the complex numbers, say, this situation can be repaired by lapsing to the analytic topology in complex space; there are precise statements as to how one might import the analytic topology to the algebraic setting. In consider arbitrary schemes, however, this construction is not available. Nonetheless, other constructions exist. This talk, the first in this seminar series, will attempt to sketch how other topologies on schemes come about and what they might be useful for. This talk is aimed at those with only cursory knowledge of sheaf and scheme theory, so expect more pictures than rigor.