Mathematics Colloquium

The Hilbert scheme $S^{[n]}$ of $n$ points on a surface $S$
provides a smooth compactification of the configuration space
parameterizing unordered $n$-tuples of distinct points. The geometry of
$S^{[n]}$ plays an important role in topology, combinatorics,
representation theory and algebraic geometry. After discussing the basic
properties of $S^{[n]}$, I will describe recent joint work with Arcara,
Bertram and Huizenga on the birational geometry of the Hilbert scheme of
points on the plane.