Quantitative ell-adic sheaf theory

Sheaf cohomology is a powerful tool both in algebraic 
geometry and its applications to other fields. Often, one wants to 
prove bounds for the dimension of sheaf cohomology groups. Katz gave 
bounds for the dimension of the étale cohomology groups of a variety 
in terms of its defining equations (degree, number of equations, 
number of variables). But the utility of sheaf cohomology arises less 
from the ability to compute the cohomology of varieties and more from 
the toolbox of functors that let us construct new sheaves from old, 
which we often apply in quite complicated sequences. In joint work 
with Arthur Forey, Javier Fresán, and Emmanuel Kowalski, we prove 
bounds for the dimensions of étale cohomology groups which are 
compatible with the six functors formalism (and other functors 
besides) in the sense that we define the “complexity” of a sheaf and 
control how much the complexity can grow when we apply one of these 
operations.