Studying the decomposition theorem over the integers

The decomposition theorem is a fundamental result about the topology of algebraic maps. It has several spectacular applications in number theory, combinatorics and representation theory. (For example, it is the starting point for Ngo's proof of the fundamental lemma). A few years ago De Cataldo and Migliorini gave a Hodge theoretic proof of the decomposition theorem, and reading their proof carefully allows one to "decide" when it is true over the integers. The question as to when the decomposition theorem holds over the integers is a question about torsion in cohomology, and is the key to several difficult questions in modular representation theory. I will describe a recent theorem which uses this approach to show that the torsion in the intersection cohomology of Schubert varieties grows exponentially in the rank of the group.