Moduli spaces of higher dimensional varieties                                                                                 
-- Zsolt Patakfalvi, October 7, 2011

<p> 

The moduli spaces of smooth curves of genus at least two and
its compactification, the space of stable curves, are             
one of the most investigated objects of algebraic geometry. In
the past two decades, natural higher dimensional                
generalizations, the moduli space of canonically polarized
manifolds and of stable schemes have been constructed.              
After giving a short introduction to the above spaces, I will
talk about their global geometry, in particular about            
the hyperbolicity properties of the moduli space of canonically
polarized manifolds.