Title: The ubiquity of smooth Hilbert schemes

Abstract:
What are the geometric properties of a typical Hilbert scheme? 
In this talk, we consider Hilbert schemes parametrizing closed subschemes 
in some projective space with a fixed Hilbert polynomial.  
We first explain how to make the collection of all such Hilbert schemes 
into a discrete probability space.  Exploiting the underlying combinatorial 
structure, we show why a random Hilbert scheme is smooth and irreducible 
with probability greater than 0.5.