Smooth Hilbert schemes

How can we understand the closed subschemes in a projective space? Hilbert schemes provide the geometric answer to this question. After surveying some features of these natural parameter spaces, we will classify the smooth Hilbert schemes. Time permitting, we will also describe the geometry of the nonsingular Hilbert schemes by interpreting them as suitable generalizations of partial flag varieties. This talk is based on joint work with Roy Skjelnes (KTH).